What is the (2,3) entry of the kernel matrix?
\(K_{23} = \) - 1 - + 1 + Show Answer @@ -1028,13 +1028,13 @@ Part 2) Let \(\vec x = (1, 1, 0)^T\) be a new point. What is \(H(\vec x)\)? - 14 - + 14 + Show Answer @@ -1127,13 +1127,13 @@ Part 2) Suppose \(\pr(Y = 1) = 0.5\) and \(\pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -1160,13 +1160,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.29 - + 0.29 + Show Answer @@ -1222,13 +1222,13 @@ Part 1) What is \(f(1.5)\)? - 0.2 - + 0.2 + Show Answer @@ -1243,13 +1243,13 @@ Part 2) What is \(f(7)\)? - .05 - + .05 + Show Answer @@ -1300,13 +1300,13 @@ Problem #036 What is the largest value that \(f(\vec x)\) can possibly have? - 1/6 - + 1/6 + Show Answer @@ -1400,13 +1400,13 @@ Problem #040 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\mu\) and the covariance matrix that must be estimated as its own parameter. - 8 - + 8 + Show Answer @@ -1573,13 +1573,13 @@ Problem #045 If Gaussian Naive Bayes is trained on this data, how many univariate Gaussians will be fit? - 20 - + 20 + Show Answer @@ -1735,13 +1735,13 @@ Problem #049 The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin? - 80 - + 80 + Show Answer @@ -1835,13 +1835,13 @@ Problem #053 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count). - 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Let \(\vec x = (1, 1, 0)^T\) be a new point. What is \(H(\vec x)\)?
- 14 - + 14 + Show Answer @@ -1127,13 +1127,13 @@ Part 2) Suppose \(\pr(Y = 1) = 0.5\) and \(\pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -1160,13 +1160,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.29 - + 0.29 + Show Answer @@ -1222,13 +1222,13 @@ Part 1) What is \(f(1.5)\)? - 0.2 - + 0.2 + Show Answer @@ -1243,13 +1243,13 @@ Part 2) What is \(f(7)\)? - .05 - + .05 + Show Answer @@ -1300,13 +1300,13 @@ Problem #036 What is the largest value that \(f(\vec x)\) can possibly have? - 1/6 - + 1/6 + Show Answer @@ -1400,13 +1400,13 @@ Problem #040 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\mu\) and the covariance matrix that must be estimated as its own parameter. - 8 - + 8 + Show Answer @@ -1573,13 +1573,13 @@ Problem #045 If Gaussian Naive Bayes is trained on this data, how many univariate Gaussians will be fit? - 20 - + 20 + Show Answer @@ -1735,13 +1735,13 @@ Problem #049 The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin? - 80 - + 80 + Show Answer @@ -1835,13 +1835,13 @@ Problem #053 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count). - 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Suppose \(\pr(Y = 1) = 0.5\) and \(\pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution?
- 0.25 - + 0.25 + Show Answer @@ -1160,13 +1160,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.29 - + 0.29 + Show Answer @@ -1222,13 +1222,13 @@ Part 1) What is \(f(1.5)\)? - 0.2 - + 0.2 + Show Answer @@ -1243,13 +1243,13 @@ Part 2) What is \(f(7)\)? - .05 - + .05 + Show Answer @@ -1300,13 +1300,13 @@ Problem #036 What is the largest value that \(f(\vec x)\) can possibly have? - 1/6 - + 1/6 + Show Answer @@ -1400,13 +1400,13 @@ Problem #040 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\mu\) and the covariance matrix that must be estimated as its own parameter. - 8 - + 8 + Show Answer @@ -1573,13 +1573,13 @@ Problem #045 If Gaussian Naive Bayes is trained on this data, how many univariate Gaussians will be fit? - 20 - + 20 + Show Answer @@ -1735,13 +1735,13 @@ Problem #049 The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin? - 80 - + 80 + Show Answer @@ -1835,13 +1835,13 @@ Problem #053 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count). - 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution?
- 0.29 - + 0.29 + Show Answer @@ -1222,13 +1222,13 @@ Part 1) What is \(f(1.5)\)? - 0.2 - + 0.2 + Show Answer @@ -1243,13 +1243,13 @@ Part 2) What is \(f(7)\)? - .05 - + .05 + Show Answer @@ -1300,13 +1300,13 @@ Problem #036 What is the largest value that \(f(\vec x)\) can possibly have? - 1/6 - + 1/6 + Show Answer @@ -1400,13 +1400,13 @@ Problem #040 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\mu\) and the covariance matrix that must be estimated as its own parameter. - 8 - + 8 + Show Answer @@ -1573,13 +1573,13 @@ Problem #045 If Gaussian Naive Bayes is trained on this data, how many univariate Gaussians will be fit? - 20 - + 20 + Show Answer @@ -1735,13 +1735,13 @@ Problem #049 The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin? - 80 - + 80 + Show Answer @@ -1835,13 +1835,13 @@ Problem #053 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count). - 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is \(f(1.5)\)?
- 0.2 - + 0.2 + Show Answer @@ -1243,13 +1243,13 @@ Part 2) What is \(f(7)\)? - .05 - + .05 + Show Answer @@ -1300,13 +1300,13 @@ Problem #036 What is the largest value that \(f(\vec x)\) can possibly have? - 1/6 - + 1/6 + Show Answer @@ -1400,13 +1400,13 @@ Problem #040 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\mu\) and the covariance matrix that must be estimated as its own parameter. - 8 - + 8 + Show Answer @@ -1573,13 +1573,13 @@ Problem #045 If Gaussian Naive Bayes is trained on this data, how many univariate Gaussians will be fit? - 20 - + 20 + Show Answer @@ -1735,13 +1735,13 @@ Problem #049 The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin? - 80 - + 80 + Show Answer @@ -1835,13 +1835,13 @@ Problem #053 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count). - 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is \(f(7)\)?
- .05 - + .05 + Show Answer @@ -1300,13 +1300,13 @@ Problem #036 What is the largest value that \(f(\vec x)\) can possibly have? - 1/6 - + 1/6 + Show Answer @@ -1400,13 +1400,13 @@ Problem #040 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\mu\) and the covariance matrix that must be estimated as its own parameter. - 8 - + 8 + Show Answer @@ -1573,13 +1573,13 @@ Problem #045 If Gaussian Naive Bayes is trained on this data, how many univariate Gaussians will be fit? - 20 - + 20 + Show Answer @@ -1735,13 +1735,13 @@ Problem #049 The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin? - 80 - + 80 + Show Answer @@ -1835,13 +1835,13 @@ Problem #053 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count). - 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the largest value that \(f(\vec x)\) can possibly have?
- 1/6 - + 1/6 + Show Answer @@ -1400,13 +1400,13 @@ Problem #040 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\mu\) and the covariance matrix that must be estimated as its own parameter. - 8 - + 8 + Show Answer @@ -1573,13 +1573,13 @@ Problem #045 If Gaussian Naive Bayes is trained on this data, how many univariate Gaussians will be fit? - 20 - + 20 + Show Answer @@ -1735,13 +1735,13 @@ Problem #049 The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin? - 80 - + 80 + Show Answer @@ -1835,13 +1835,13 @@ Problem #053 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count). - 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\mu\) and the covariance matrix that must be estimated as its own parameter.
- 8 - + 8 + Show Answer @@ -1573,13 +1573,13 @@ Problem #045 If Gaussian Naive Bayes is trained on this data, how many univariate Gaussians will be fit? - 20 - + 20 + Show Answer @@ -1735,13 +1735,13 @@ Problem #049 The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin? - 80 - + 80 + Show Answer @@ -1835,13 +1835,13 @@ Problem #053 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count). - 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
If Gaussian Naive Bayes is trained on this data, how many univariate Gaussians will be fit?
- 20 - + 20 + Show Answer @@ -1735,13 +1735,13 @@ Problem #049 The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin? - 80 - + 80 + Show Answer @@ -1835,13 +1835,13 @@ Problem #053 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count). - 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
The density estimate within a particular bin of the histogram is 0.1. How many data points from \(\mathcal D\) fall within that histogram bin?
- 80 - + 80 + Show Answer @@ -1835,13 +1835,13 @@ Problem #053 Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count). - 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Suppose a Gaussian with a diagonal covariance matrix is fit to 200 points in \(\mathbb R^4\) using the maximum likelihood estimators. How many parameters are estimated? Count each entry of \(\vec\mu\) and the covariance matrix that must be estimated as its own parameter (the off-diagonal elements of the covariance are zero, and shouldn't be included in your count).
- 8 - + 8 + Show Answer @@ -2128,13 +2128,13 @@ Problem #061 Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)? - 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Let \(\vec x = (1, 2)^T\). What is \(H(\vec x)\)?
- 3 - + 3 + Show Answer @@ -2206,13 +2206,13 @@ Problem #063 What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)? - 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the value of \(H\) at the point \(\nvec{x}{4} = (0, 1)^T\)?
- 0 - + 0 + Show Answer @@ -2366,13 +2366,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the totalsquare loss of \(H\) on this data set?
- 11.25 - + 11.25 + Show Answer @@ -2387,13 +2387,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the totalperceptron loss of \(H\) on this data set?
- 1 - + 1 + Show Answer @@ -2408,13 +2408,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the totalhinge loss of \(H\) on this data set?
- 3.5 - + 3.5 + Show Answer @@ -2451,13 +2451,13 @@ Part 1) What is the gradient of this function at the point \((0, 0)\)? - \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the gradient of this function at the point \((0, 0)\)?
- \((2, 1)^T\) - + \((2, 1)^T\) + Show Answer @@ -2506,13 +2506,13 @@ Problem #068 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent?
- \((1, 2)^T\) - + \((1, 2)^T\) + Show Answer @@ -2716,13 +2716,13 @@ Problem #073 What weight vector is the solution of the Hard SVM problem for this data set? - \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What weight vector is the solution of the Hard SVM problem for this data set?
- \((3, 0, \frac{1}{4})^T\) - + \((3, 0, \frac{1}{4})^T\) + Show Answer @@ -2756,13 +2756,13 @@ Problem #074 What is \(\frac{d}{d \vec w} f(\vec w)\)? - \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is \(\frac{d}{d \vec w} f(\vec w)\)?
- \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) - + \(\sum_{i=1}^n \nvec{x}{i} + 2 \vec w\) + Show Answer @@ -2988,13 +2988,13 @@ Problem #079 Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)? - \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Let \(\vec x = (1, 2, 1)^T\). What is \(H(\vec x)\)?
- \(-3\) - + \(-3\) + Show Answer @@ -3020,13 +3020,13 @@ Problem #080 What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number. - 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the mean squared error of a least squares classifier trained on this data set (without regularization)? Your answer should be a number.
- 0 - + 0 + Show Answer @@ -3108,13 +3108,13 @@ Part 1) What is the totalsquare loss of \(H\) on this data set? - 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
- 13.5 - + 13.5 + Show Answer @@ -3129,13 +3129,13 @@ Part 2) What is the totalperceptron loss of \(H\) on this data set? - 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
- 1.5 - + 1.5 + Show Answer @@ -3150,13 +3150,13 @@ Part 3) What is the totalhinge loss of \(H\) on this data set? - 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
- 4 - + 4 + Show Answer @@ -3195,13 +3195,13 @@ Part 1) What is the gradient of this function at the point \((1, 1)\)? - \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the gradient of this function at the point \((1, 1)\)?
- \((-1,3)^T\) - + \((-1,3)^T\) + Show Answer @@ -3216,13 +3216,13 @@ Part 2) What is the gradient of this function at the point \((-1, -1)\)? - \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the gradient of this function at the point \((-1, -1)\)?
- \((-2,0)^T\) - + \((-2,0)^T\) + Show Answer @@ -3287,13 +3287,13 @@ Problem #085 Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent? - \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Suppose that the initial weight vector is \(\vec w = (0, 0)^T\) and that the learning rate is \(\eta = 1\). What will be the weight vector after one iteration of subgradient descent?
- \((1, 3)^T\) - + \((1, 3)^T\) + Show Answer @@ -3439,13 +3439,13 @@ Problem #090 What is \(\frac{d}{d \vec x} f(\vec x)\)? - \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is \(\frac{d}{d \vec x} f(\vec x)\)?
- \(2 A \vec x + \vec b\) - + \(2 A \vec x + \vec b\) + Show Answer @@ -3526,13 +3526,13 @@ Part 1) What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)? - 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is \(\pr(1 \leq X \leq 3 \given Y = 0)\)?
- 0.6 - + 0.6 + Show Answer @@ -3559,13 +3559,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution?
- 0.30 - + 0.30 + Show Answer @@ -3592,13 +3592,13 @@ Part 5) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution?
- 0.27 - + 0.27 + Show Answer @@ -3635,13 +3635,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)? - 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the estimate of the Class 0 density at \(x = 3.5\)? That is, what is the estimate for \(p(3.5 \given Y = 0)\)?
- 0.1 - + 0.1 + Show Answer @@ -3656,13 +3656,13 @@ Part 2) Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)? - 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Using the same histogram estimator, what is the estimate of \(\pr(Y = 1 \given x = 3.5)\)?
- 0.5 - + 0.5 + Show Answer @@ -3695,13 +3695,13 @@ Problem #095 Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)? - 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Consider a new point \(\vec x = (1, 1, 1)\). What is \(H(\vec x)\)?
- 50 - + 50 + Show Answer @@ -3963,13 +3963,13 @@ Part 1) L(6, 5)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood.
- \(10^{-5}\) - + \(10^{-5}\) + Show Answer @@ -3984,13 +3984,13 @@ Part 2) What is \(\mathcal L(3, 2)\)? - 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is \(\mathcal L(3, 2)\)?
- 0 - + 0 + Show Answer @@ -4005,13 +4005,13 @@ Part 3) What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)? \(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What are the maximum likelihood estimates of \(\alpha\) and \(\beta\)?
\(\alpha\): - 5 - + 5 + Show Answer @@ -4020,13 +4020,13 @@ Part 3) \(\beta\): - 4 - + 4 + Show Answer @@ -4210,13 +4210,13 @@ Part 1) What is the (1,2) entry of the sample covariance matrix? - 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the (1,2) entry of the sample covariance matrix?
- 1 - + 1 + Show Answer @@ -4231,13 +4231,13 @@ Part 2) What is the (2,2) entry of the sample covariance matrix? - 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the (2,2) entry of the sample covariance matrix?
- 8/6 - + 8/6 + Show Answer @@ -4351,13 +4351,13 @@ Part 1) Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)? - 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Suppose that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is \(\pr(1 \leq X \leq 3)\)?
- 0.3 - + 0.3 + Show Answer @@ -4384,13 +4384,13 @@ Part 3) Suppose again that \(\pr(Y = 1) = \pr(Y = 0) = 0.5\). What is the Bayes error with respect to this distribution? - 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
- 0.25 - + 0.25 + Show Answer @@ -4405,13 +4405,13 @@ Part 4) Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)? - 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Now suppose \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is \(\pr(1 \leq X \leq 3)\)?
- 0.34 - + 0.34 + Show Answer @@ -4438,13 +4438,13 @@ Part 6) Suppose again that \(\pr(Y = 1) = 0.7\) and \(\pr(Y = 0) = 0.3\). What is the Bayes error with respect to this distribution? - 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
- 0.28 - + 0.28 + Show Answer @@ -4476,13 +4476,13 @@ Part 1) What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)? - 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the estimate of the Class 0 density at \(x = 0.6\)? That is, what is the estimate \(\hat p(0.6 \given Y = 0)\)?
- 5/3 - + 5/3 + Show Answer @@ -4497,13 +4497,13 @@ Part 2) Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)? - 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Using the same histogram estimator, what is the estimate \(\hat\pr(Y = 1 \given x = 0.35)\)?
- 3/5 = 0.6 - + 3/5 = 0.6 + Show Answer @@ -4518,13 +4518,13 @@ Part 3) What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)? - 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the estimate of the marginal density of \(x\) at \(x = 0.1\)? That is, what is \(\hat p(0.1)\)?
- 7/(19*0.25) = 1.47 - + 7/(19*0.25) = 1.47 + Show Answer @@ -4541,13 +4541,13 @@ Part 4) \] - 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
- 1 - + 1 + Show Answer @@ -4575,13 +4575,13 @@ Problem #112 Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)? - 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Consider a new point \(\vec x = (2, 0, 1)^T\). What is \(H(\vec x)\)?
- 37 - + 37 + Show Answer @@ -4615,13 +4615,13 @@ Part 1) What is the (1,3) entry of the sample covariance matrix? - 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
What is the (1,3) entry of the sample covariance matrix?
- 9/4 - + 9/4 + Show Answer @@ -4636,13 +4636,13 @@ Part 2) What is the (1,2) entry of the sample covariance matrix? - -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood. - 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
- -3/4 - + -3/4 + Show Answer @@ -4770,13 +4770,13 @@ Part 1) L(0, 2)\)? Note that \(\mathcal L\) is the likelihood, not the log-likelihood.
- 0 - + 0 + Show Answer @@ -4791,13 +4791,13 @@ Part 2) Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)? -
Let \(\mathcal X = \{0, 1, 2\}\) be the same data set as in the previous part. What is \(\mathcal L(0, 4)\)?
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