From 9a74b3b4858120606ff17b3dc69f60461b756eab Mon Sep 17 00:00:00 2001 From: ne3x7 Date: Fri, 18 Oct 2024 10:55:34 +0200 Subject: [PATCH] Webinar 6 and 7 + homeworks --- .../Seminar 6 (Expectation and Variance).html | 7963 +++++++++++++++++ ...Seminar 6 (Expectation and Variance).ipynb | 639 ++ .../Seminar 6 (Expectation and Variance).pdf | Bin 0 -> 441275 bytes ...r 6 (Expectation and Variance).slides.html | 7928 ++++++++++++++++ ...minar 7 (Continuous Random Variables).html | 7802 ++++++++++++++++ ...inar 7 (Continuous Random Variables).ipynb | 456 + ...eminar 7 (Continuous Random Variables).pdf | Bin 0 -> 90353 bytes ... (Continuous Random Variables).slides.html | 7812 ++++++++++++++++ home_assignments/MSAI_Prob24_HW6_7.pdf | Bin 0 -> 139014 bytes 9 files changed, 32600 insertions(+) create mode 100644 Seminar_materials/seminar_06/Seminar 6 (Expectation and Variance).html create mode 100755 Seminar_materials/seminar_06/Seminar 6 (Expectation and Variance).ipynb create mode 100644 Seminar_materials/seminar_06/Seminar 6 (Expectation and Variance).pdf create mode 100644 Seminar_materials/seminar_06/Seminar 6 (Expectation and Variance).slides.html create mode 100644 Seminar_materials/seminar_07/Seminar 7 (Continuous Random Variables).html create mode 100755 Seminar_materials/seminar_07/Seminar 7 (Continuous Random Variables).ipynb create mode 100644 Seminar_materials/seminar_07/Seminar 7 (Continuous Random Variables).pdf create mode 100644 Seminar_materials/seminar_07/Seminar 7 (Continuous Random Variables).slides.html create mode 100644 home_assignments/MSAI_Prob24_HW6_7.pdf diff --git a/Seminar_materials/seminar_06/Seminar 6 (Expectation and Variance).html b/Seminar_materials/seminar_06/Seminar 6 (Expectation and Variance).html new file mode 100644 index 0000000..41db190 --- /dev/null +++ b/Seminar_materials/seminar_06/Seminar 6 (Expectation and Variance).html @@ -0,0 +1,7963 @@ + + + + + +Seminar 6 (Expectation and Variance) + + + + + + + + + + + + +
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+ + diff --git a/Seminar_materials/seminar_06/Seminar 6 (Expectation and Variance).ipynb b/Seminar_materials/seminar_06/Seminar 6 (Expectation and Variance).ipynb new file mode 100755 index 0000000..14900a2 --- /dev/null +++ b/Seminar_materials/seminar_06/Seminar 6 (Expectation and Variance).ipynb @@ -0,0 +1,639 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "a8f7b639", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "# Seminar 6" + ] + }, + { + "cell_type": "markdown", + "id": "present-example", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Recap of random variables\n", + "\n", + "A **random variable** is a function from sample space to the real numbers $X: S \\to \\mathbb{R}$." + ] + }, + { + "cell_type": "markdown", + "id": "congressional-malaysia", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Recap of distributions\n", + "\n", + "For a random variable $X: S \\to \\mathbb{R}$, its distribution acts on numbers in $\\mathbb{R}$ in the same way as probability function $P$ acts on outcomes." + ] + }, + { + "cell_type": "markdown", + "id": "behind-startup", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Functions describing distributions\n", + "\n", + "- For any distribution we have **cumulative distribution function** (CDF) $F_X(x) = \\mathbb{P}(X \\leqslant x)$\n", + "- For discrete distributions we have **probability mass function** (PMF) $\\mathbb{P}_X(x) = \\mathbb{P}(X = x)$\n", + "- For continuous distributions we have **probability density function** (PDF) $f_X(x) = F'_X(x)$" + ] + }, + { + "cell_type": "markdown", + "id": "auburn-speaking", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Location-scale transformation\n", + "\n", + "Random variables transform like functions, i.e. if $Y = \\varphi(X)$, then $Y(\\omega) = \\varphi(X(\\omega))$.\n", + "\n", + "For a $\\varphi(x) = ax + b$ and $a > 0$, we have\n", + "$$\n", + "F_Y(y) = \\mathbb{P}(Y \\leqslant y) = \\mathbb{P}(a X + b \\leqslant y) = \\mathbb{P}\\left(X \\leqslant \\frac{y - b}{a}\\right) = F_X\\left(\\frac{y - b}{a}\\right)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "laden-divorce", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Mathematical expectation\n", + "\n", + "Mathematical expectation generalizes the concept of mean. Consider probability space $(S, \\mathbb{P})$ and discrete random variable $X: S \\to \\mathbb{R}$. Then expected value of $X$ is then\n", + "$$\n", + "\\mathbb{E}\\left[X\\right] = \\sum_k x_k \\mathbb{P}(X = x_k)\n", + "$$\n", + " \n", + "It may be the case that $\\mathbb{E}\\left[X\\right] = \\pm \\infty$ or even does not exist." + ] + }, + { + "cell_type": "markdown", + "id": "rational-operator", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 1\n", + "\n", + "We roll a die and r.v. $X$ is the score of a roll. What is $\\mathbb{E}\\left[X\\right]$?" + ] + }, + { + "cell_type": "markdown", + "id": "returning-webcam", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 1\n", + "\n", + "$$\n", + "\\mathbb{E}\\left[X\\right] = \\sum_{k=1}^6 k \\cdot \\mathbb{P}(X = k) = \\frac16 \\sum_{k=1}^6 k = \\frac72\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "basic-feature", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 2\n", + "\n", + "We flip a non-symmetric coin and $X$ is the r.v. for heads, $X \\sim Be(p)$. What is $\\mathbb{E}\\left[X\\right]$?" + ] + }, + { + "cell_type": "markdown", + "id": "running-lemon", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 2\n", + "\n", + "$$\n", + "\\mathbb{E}\\left[X\\right] = 0 \\cdot \\mathbb{P}(X = 0) + 1 \\cdot \\mathbb{P}(X = 1) = p\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "interesting-topic", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 3\n", + "\n", + "Consider discrete r.v. $X$ with distribution $\\mathbb{P}(X = 2^n) = 2^{-n}$. What is $\\mathbb{E}\\left[X\\right]$?" + ] + }, + { + "cell_type": "markdown", + "id": "working-syria", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 3\n", + "\n", + "$$\n", + "\\mathbb{E}\\left[X\\right] = \\sum_{n} 2^n 2^{-n} = \\infty\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "dressed-bookmark", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 4\n", + "\n", + "Consider discrete r.v. $X$ with distribution $\\mathbb{P}(X = 2^n) = \\mathbb{P}(X = - 2^n) = 2^{-n-1}$. What is $\\mathbb{E}\\left[X\\right]$?" + ] + }, + { + "cell_type": "markdown", + "id": "prospective-partnership", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 4\n", + "\n", + "Expectation of r.v. $X$ exists if and only if $\\mathbb{E}\\left[|X|\\right] < \\infty$" + ] + }, + { + "cell_type": "markdown", + "id": "incredible-revolution", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 5\n", + "\n", + "Consider $X$ with **Poisson distribution** $X \\sim Pois(\\lambda)$:\n", + "$$\n", + "\\mathbb{P}(X = k) = \\frac{\\lambda^k}{k!} e^{-\\lambda}\n", + "$$\n", + "\n", + "What is $\\mathbb{E}\\left[X\\right]$?" + ] + }, + { + "cell_type": "markdown", + "id": "criminal-lighter", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 5\n", + "\n", + "$$\n", + "\\begin{aligned}\n", + "\\mathbb{E}\\left[X\\right] & = \\sum_{k=0}^\\infty k \\frac{\\lambda^k}{k!} e^{-\\lambda} = e^{-\\lambda} \\sum_{k=0}^\\infty k \\frac{\\lambda^k}{k!} = e^{-\\lambda} \\sum_{k=0}^\\infty \\frac{\\lambda^k}{(k - 1)!} = \\\\\n", + "& = e^{-\\lambda} \\sum_{k=0}^\\infty \\frac{\\lambda^{k-1} \\lambda}{(k - 1)!} = \\lambda e^{-\\lambda} \\sum_{j=0}^\\infty \\frac{\\lambda^j}{j!} = \\lambda e^{-\\lambda} e^\\lambda = \\lambda\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "missing-jurisdiction", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Properties of expectation\n", + "\n", + "Consider r.v.s $X$ and $Y$ with finite expectations. Then,\n", + "1. For any constants $a$ and $b$ it holds $\\mathbb{E}\\left[aX + b\\right] = a \\mathbb{E}\\left[X\\right] + b$\n", + "2. $\\mathbb{E}\\left[X + Y\\right] = \\mathbb{E}\\left[X\\right] + \\mathbb{E}\\left[Y\\right]$\n", + "3. If $X \\leqslant Y$ a.s., then $\\mathbb{E}\\left[X\\right] \\leqslant \\mathbb{E}\\left[Y\\right]$ ($X \\leqslant Y$ a.s. $\\Leftrightarrow \\mathbb{P}((x, y) : x > y) = 0$)\n", + "4. If $X \\perp Y$, then $\\mathbb{E}\\left[XY\\right] = \\mathbb{E}\\left[X\\right] \\mathbb{E}\\left[Y\\right]$" + ] + }, + { + "cell_type": "markdown", + "id": "blond-gibraltar", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 6\n", + "\n", + "Consider $X$ with binomial distribution $X \\sim Bi(n, p)$. What is $\\mathbb{E}\\left[X\\right]$?" + ] + }, + { + "cell_type": "markdown", + "id": "worst-mention", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 6\n", + "\n", + "- We know that $X = \\sum_{k=1}^n X_k$, where $X_k \\sim Be(p)$\n", + "- We know that $\\mathbb{E}\\left[X_k\\right] = p$\n", + "- Then, $\\mathbb{E}\\left[X\\right] = \\sum_{k=1}^n \\mathbb{E}\\left[X_k\\right] = np$" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "65a318d2-4684-4e86-91fe-7b156e94fd4c", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "skip" + }, + "tags": [] + }, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/var/folders/33/j0cl7y453td68qb96j7bqcj4cf41kc/T/ipykernel_28394/2600749600.py:8: DeprecationWarning: `set_matplotlib_formats` is deprecated since IPython 7.23, directly use `matplotlib_inline.backend_inline.set_matplotlib_formats()`\n", + " dp.set_matplotlib_formats(\"retina\")\n" + ] + }, + { + "data": { + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import numpy as np\n", + "import scipy.stats as sts\n", + "\n", + "import IPython.display as dp\n", + "import matplotlib.pyplot as plt\n", + "import seaborn as sns\n", + "\n", + "dp.set_matplotlib_formats(\"retina\")\n", + "sns.set(style=\"whitegrid\", font_scale=1.5)\n", + "sns.despine()\n", + "\n", + "%matplotlib inline" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "89e40c31-8fdb-4c25-b29f-f0cf481980d2", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "
" + ] + }, + "metadata": { + "image/png": { + "height": 747, + "width": 1317 + } + }, + "output_type": "display_data" + } + ], + "source": [ + "n, p = 9, 0.7\n", + "x = np.arange(-1, 10)\n", + "y = sts.binom(n, p).pmf(x)\n", + "\n", + "fig, ax = plt.subplots(figsize=(16,9))\n", + "ax.stem(x, y, label=\"PMF\")\n", + "ax.axvline(n * p, ls=\"--\", linewidth=3, color=\"red\", label=\"E[X]\")\n", + "ax.axvline(x[np.argmax(y)], ls=\"--\", linewidth=3, color=\"green\", label=\"mode\")\n", + "ax.legend();" + ] + }, + { + "cell_type": "markdown", + "id": "occupational-campbell", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Expectation of a function of a random variable (LOTUS)\n", + "\n", + "Consider discrete r.v. $X$ and $Y = \\varphi(X)$, then expectation of $Y$ is\n", + "$$\n", + "\\mathbb{E}\\left[Y\\right] = \\sum_n \\varphi(n) \\mathbb{P}\\left(X = n\\right)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "caroline-scoop", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Variance\n", + "\n", + "We call **variance** the following quantity of a r.v. $X$ with finite expectation:\n", + "$$\n", + "\\mathbb{V}\\text{ar}(X) = \\mathbb{E}\\left[\\left(X - \\mathbb{E}[X]\\right)^2\\right]\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "governing-ethiopia", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 7\n", + "\n", + "We flip a non-symmetric coin and $X$ is the r.v. for heads, $X \\sim Be(p)$. What is $\\mathbb{V}\\text{ar}\\left(X\\right)$?" + ] + }, + { + "cell_type": "markdown", + "id": "oriented-nirvana", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 7\n", + "\n", + "1. We know the formula\n", + " $$\n", + " \\mathbb{V}\\text{ar}\\left(X\\right) = \\mathbb{E}\\left[\\left(X - \\mathbb{E}\\left[X\\right]\\right)^2\\right]\n", + " $$\n", + "2. We know $\\mathbb{E}\\left[X\\right]$\n", + " $$\n", + " \\mathbb{V}\\text{ar}\\left(X\\right) = \\mathbb{E}\\left[\\left(X - p\\right)^2\\right] = \\mathbb{E}\\left[X^2 - 2 p X + p^2\\right]\n", + " $$\n", + "3. We know that expectation is linear\n", + " $$\n", + " \\mathbb{V}\\text{ar}\\left(X\\right) = \\mathbb{E}\\left[X^2\\right] - 2 p \\mathbb{E}\\left[X\\right] + p^2 = \\mathbb{E}\\left[X^2\\right] - p^2\n", + " $$\n", + "4. For $Y = X^2$ we can compute\n", + " $$\n", + " \\mathbb{E}\\left[Y\\right] = 0 \\cdot \\mathbb{P}(Y = 0) + 1 \\cdot \\mathbb{P}(Y = 1) = \\mathbb{P}(Y = 1) = \\mathbb{P}(X^2 = 1) = \\mathbb{P}(X = 1) = p\n", + " $$\n", + "5. Finally,\n", + " $$\n", + " \\mathbb{V}\\text{ar}\\left(X\\right) = p - p^2 = p (1 - p)\n", + " $$" + ] + }, + { + "cell_type": "markdown", + "id": "israeli-instruction", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Properties of variance\n", + "\n", + "1. $\\mathbb{V}\\text{ar}\\left(X\\right) \\geqslant 0$ and $\\mathbb{V}\\text{ar}\\left(X\\right) = 0$ if and only if $X = const$ a.s.\n", + "2. If holds\n", + " $$\n", + " \\mathbb{V}\\text{ar}\\left(X\\right) = \\mathbb{E}\\left[X^2\\right] - \\left(\\mathbb{E}\\left[X\\right]\\right)^2\n", + " $$\n", + "3. It holds\n", + " $$\n", + " \\mathbb{V}\\text{ar}\\left(aX + b\\right) = a^2 \\mathbb{V}\\text{ar}\\left(X\\right)\n", + " $$\n", + "4. If $X \\perp Y$, it holds\n", + "$$\n", + "\\mathbb{V}\\text{ar}\\left(X + Y\\right) = \\mathbb{V}\\text{ar}\\left(X\\right) + \\mathbb{V}\\text{ar}\\left(Y\\right)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "casual-happiness", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 8\n", + "\n", + "Consider $X$ with binomial distribution $X \\sim Bi(n, p)$. What is $\\mathbb{V}\\text{ar}\\left(X\\right)$?" + ] + }, + { + "cell_type": "markdown", + "id": "caroline-execution", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 8\n", + "\n", + "- We know that $X = \\sum_{k=1}^n X_k$, where $X_k \\sim Be(p)$\n", + "- We know that $\\mathbb{V}\\text{ar}\\left(X_k\\right) = p(1-p)$\n", + "- Then, $\\mathbb{V}\\text{ar}\\left(X\\right) = \\mathbb{V}\\text{ar}\\left(\\sum_{k=1}^n X_k\\right) = \\sum_{k=1}^n \\mathbb{V}\\text{ar}\\left(X_k\\right) = np(1-p)$" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "64169f26-1dfe-4d5b-88f8-80dfee6e1294", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "
" + ] + }, + "metadata": { + "image/png": { + "height": 747, + "width": 1317 + } + }, + "output_type": "display_data" + } + ], + "source": [ + "n, p = 9, 0.7\n", + "x = np.arange(-1, 10)\n", + "y = sts.binom(n, p).pmf(x)\n", + "\n", + "fig, ax = plt.subplots(figsize=(16,9))\n", + "ax.stem(x, y, label=\"PMF\")\n", + "ax.axvline(n * p, ls=\"--\", color=\"k\", label=\"E[X]\")\n", + "ax.axvline(n * p ** 2, ls=\"--\", linewidth=3, color=\"red\", label=\"E[X] - Var[X]\")\n", + "ax.axvline(n * p * (2 - p), ls=\"--\", linewidth=3, color=\"green\", label=\"E[X] + Var[X]\")\n", + "ax.legend();" + ] + } + ], + "metadata": { + "celltoolbar": "Slideshow", + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + 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    + + + diff --git a/Seminar_materials/seminar_07/Seminar 7 (Continuous Random Variables).html b/Seminar_materials/seminar_07/Seminar 7 (Continuous Random Variables).html new file mode 100644 index 0000000..b7699b6 --- /dev/null +++ b/Seminar_materials/seminar_07/Seminar 7 (Continuous Random Variables).html @@ -0,0 +1,7802 @@ + + + + + +Seminar 7 (Continuous Random Variables) + + + + + + + + + + + + +
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    + + diff --git a/Seminar_materials/seminar_07/Seminar 7 (Continuous Random Variables).ipynb b/Seminar_materials/seminar_07/Seminar 7 (Continuous Random Variables).ipynb new file mode 100755 index 0000000..9822cd0 --- /dev/null +++ b/Seminar_materials/seminar_07/Seminar 7 (Continuous Random Variables).ipynb @@ -0,0 +1,456 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "a8f7b639", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "# Seminar 7" + ] + }, + { + "cell_type": "markdown", + "id": "7bb7a2e9", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Recap of random variables\n", + "\n", + "A **random variable** is a function from sample space to the real numbers $X: S \\to \\mathbb{R}$." + ] + }, + { + "cell_type": "markdown", + "id": "concrete-petroleum", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Recap of distributions\n", + "\n", + "For a random variable $X: S \\to \\mathbb{R}$, its distribution acts on numbers in $\\mathbb{R}$ in the same way as probability function $P$ acts on outcomes." + ] + }, + { + "cell_type": "markdown", + "id": "increasing-delaware", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Recap of functions describing distributions\n", + "\n", + "- For any distribution we have **cumulative distribution function** (CDF) $F_X(x) = \\mathbb{P}(X \\leqslant x)$\n", + "- For discrete distributions we have **probability mass function** (PMF) $\\mathbb{P}_X(x) = \\mathbb{P}(X = x)$" + ] + }, + { + "cell_type": "markdown", + "id": "printable-haiti", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Probability density function\n", + "\n", + "- If $X$ has a discrete distribution, then $F_X$ has a countable number of jumps $p_i = \\mathbb{P}(X = x_i)$ and at $x = x_i$ it is continuous\n", + "- If $X$ has absolutely continuous distribution, then $F_X$ is differentiable a.e. and can be recovered from its derivative:\n", + " $$\n", + " F_X(x) = \\int\\limits_{-\\infty}^x f_X(t) dt\n", + " $$\n", + " \n", + " where $f_X(t)$ is the probability density function and $f_X(t) = F'_X(x)$ a.e." + ] + }, + { + "cell_type": "markdown", + "id": "vocational-chinese", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 1\n", + "\n", + "We say that random variable $X$ is distributed uniformly on $[a, b]$ and write $X \\sim U([a, b])$ if\n", + "$$\n", + "f_X(x) = \\begin{cases}\n", + "\\frac{1}{b-a}, a \\leqslant x \\leqslant b, \\\\\n", + "0, \\text{else}\n", + "\\end{cases}\n", + "$$\n", + "\n", + "What is $F_X(x)$?" + ] + }, + { + "cell_type": "markdown", + "id": "1214e30a-1cc4-41ea-974e-12e44e8b6716", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "fragment" + }, + "tags": [] + }, + "source": [ + "$$\n", + "F_X(x) = \\int\\limits_{-\\infty}^{x} \\frac{1}{b - a} {\\rm d} x = \\frac{1}{b - a} \\int\\limits_{a}^{x} {\\rm d} x = \\frac{x - a}{b - a}\n", + "$$\n", + "\n", + "e.g. if $X \\sim U([0, 1])$, then $F_X(x) = x$." + ] + }, + { + "cell_type": "markdown", + "id": "adjusted-acrylic", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Continuous convolution formula\n", + "\n", + "Consider $X$ and $Y$ independent random variables with PDFs $f_X$ and $f_Y$ respectively. Then, their sum $Z = X + Y$ has absolutely continuous distribution with density\n", + "$$\n", + "f_Z(z) = \\int f_X(x) f_Y(z-x) dx\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "local-minnesota", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 2\n", + "\n", + "Let $X, Y \\sim U([0, 1])$ and $Z = X + Y$. Find $f_Z(z)$." + ] + }, + { + "cell_type": "markdown", + "id": "wireless-commerce", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 2\n", + "\n", + "$$\n", + "f_Z(z) = \\int\\limits_{0}^1 f_X(x) f_Y(z-x) dx = \\int\\limits_{0}^1 f_Y(z-x) dx = \\begin{cases}\n", + "z, & 0 \\leqslant z \\leqslant 1, \\\\\n", + "2 - z, & 1 \\leqslant z \\leqslant 2, \\\\\n", + "0, & \\text{else}\n", + "\\end{cases}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "95675940", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Functions of continuous random variables\n", + "\n", + "Random variables transform like functions, i.e. if $Y = \\varphi(X)$, then $Y(\\omega) = \\varphi(X(\\omega))$.\n", + "\n", + "For a smooth $\\varphi$, the density will be:\n", + "$$\n", + "f_Y(y) = \\sum\\limits_{\\varphi(x) = y} \\frac{f_X(x)}{|\\varphi'(x)|}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "79501593-45f8-4864-82d6-d607ed7fd24d", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## LOTUS for continuous random variables\n", + "\n", + "If $X$ is a continuous r.v. with PDF $f_X(x)$ and $g(\\cdot)$ is a function, then,\n", + "$$\n", + "\\mathbb{E}[g(X)] = \\int g(x) f_X(x) {\\rm d} x\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "alpine-motor", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 3\n", + "\n", + "Let $X$ be a **normally distributed** random variable with parameters $m$ and $\\sigma^2$:\n", + "$$\n", + "f_X(x) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \\exp \\left( - \\frac{(x - m)^2}{2\\sigma^2} \\right)\n", + "$$\n", + "\n", + "Find PDF of $Y = X^2$." + ] + }, + { + "cell_type": "markdown", + "id": "narrow-seven", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 3\n", + "\n", + "According to the formula,\n", + "$$\n", + "f_Y(y) = \\sum\\limits_{x^2 = y} \\frac{f_X(x)}{|\\varphi'(x)|} = \\sum\\limits_{x = \\pm \\sqrt{y}} \\frac{f_X(x)}{|2 x|} = \\frac{f_X(-\\sqrt{y}) + f_X(\\sqrt{y})}{2 \\sqrt{y}}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "9eb4d737", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Recap of expectation\n", + "\n", + "- If $X$ is discrete, then\n", + " $$\n", + " \\mathbb{E}\\left[X\\right] = \\sum_k x_k \\mathbb{P}(X = x_k)\n", + " $$\n", + "- If $X$ is continuous, then\n", + " $$\n", + " \\mathbb{E}\\left[X\\right] = \\int_{-\\infty}^{+\\infty} x f_X(x) dx\n", + " $$" + ] + }, + { + "cell_type": "markdown", + "id": "3d0c668e", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Recap of variance\n", + "\n", + "We call **variance** the following quantity of a r.v. $X$ with finite expectation:\n", + "$$\n", + "\\mathbb{V}\\text{ar}(X) = \\mathbb{E}\\left[\\left(X - \\mathbb{E}[X]\\right)^2\\right]\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e3c846bd", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 4\n", + "\n", + "Find the expectation and variance of uniform distribution. Draw its CDF and PDF." + ] + }, + { + "cell_type": "markdown", + "id": "8d720d76", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 4\n", + "\n", + "Let $X \\sim U[a, b]$, then $f_X(x) = \\frac{1}{b - a}$\n", + "\n", + "$$\n", + "\\mathbb{E}\\left[X\\right] = \\int\\limits_a^b x \\frac{1}{b - a} dx = \\frac{1}{b - a} \\left( \\frac{b^2}{2} - \\frac{a^2}{2} \\right) = \\frac{a+b}{2}\n", + "$$\n", + "\n", + "$$\n", + "\\mathbb{E}\\left[X^2\\right] = \\int\\limits_a^b x^2 \\frac{1}{b - a} dx = \\frac{1}{b - a} \\left( \\frac{b^3}{3} - \\frac{a^3}{3} \\right) = \\frac13 \\frac{b^3-a^3}{b-a}\n", + "$$\n", + "\n", + "$$\n", + "\\mathbb{V}\\text{ar}\\left(X\\right) = \\frac13 \\frac{b^3-a^3}{b-a} - \\frac14 (a+b)^2 = \\frac{(b-a)^2}{12}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a91aa7b8-fc96-4d2e-a1d3-54585fc63ce4", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Universality of Uniform distribution\n", + "\n", + "Let $X$ be an r.v. with CDF $F$. Find the distribution of r.v. $Y = F(X)$.\n", + "\n", + "$$\n", + "F_Y(y) = \\mathbb{P}\\left(Y \\leqslant y\\right) = \\mathbb{P}\\left(F(X) \\leqslant y\\right) = \\mathbb{P}\\left(X \\leqslant F^{-1}(y)\\right) = F(F^{-1}(y)) = y\n", + "$$\n", + "\n", + "What does it mean about distribution of $Y$?," + ] + }, + { + "cell_type": "markdown", + "id": "91f94c99-1101-44af-b499-f4b2b29374b5", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "fragment" + }, + "tags": [] + }, + "source": [ + "It means $Y \\sim U([0, 1])$" + ] + }, + { + "cell_type": "markdown", + "id": "965368f4", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Example 5\n", + "\n", + "Find the expectation and variance of normal distribution. Draw its CDF and PDF." + ] + }, + { + "cell_type": "markdown", + "id": "6f19bfc8", + "metadata": { + "editable": true, + "slideshow": { + "slide_type": "slide" + }, + "tags": [] + }, + "source": [ + "## Solution 5\n", + "\n", + "$Z \\sim \\mathcal{N}(0,1)$ means it has PDF $\\varphi(z) = \\frac{1}{\\sqrt{2 \\pi}} e^{-z^2/2}$. To prove that it is correct PDF:\n", + "\n", + "$$\n", + "\\left( \\int\\limits_{-\\infty}^\\infty \\varphi(x) dx\\right)\\left( \\int\\limits_{-\\infty}^\\infty \\varphi(y) dy\\right) = \\frac{1}{2\\pi} \\int\\limits_{-\\infty}^\\infty\\int\\limits_{-\\infty}^\\infty e^{-(x^2+y^2)} dxdy = \\frac{1}{2\\pi} \\int\\limits_0^\\infty \\int\\limits_0^{2\\pi} e^{-r^2} rdrd\\theta = \\int\\limits_0^\\infty e^{-t^2} dt = 1\n", + "$$\n", + "\n", + "$$\n", + "\\Phi(z) = \\frac{1}{\\sqrt{2\\pi}}\\int\\limits_{-\\infty}^z e^{-z^2/2} dz\n", + "$$\n", + "\n", + "$$\n", + "\\mathbb{E}\\left[Z^2\\right] =\\frac{1}{\\sqrt{2\\pi}} \\int\\limits_{-\\infty}^\\infty z^2 e^{-z^2/2} dz = \\frac{2}{\\sqrt{2\\pi}} \\int\\limits_0^\\infty z^2 e^{-z^2/2} dz = - \\frac{2}{\\sqrt{2\\pi}} \\left(\\left.ze^{-z^2/2}\\right|_0^\\infty - \\int\\limits_0^\\infty e^{-z^2/2} dz\\right) = 1\n", + "$$" + ] + } + ], + "metadata": { + "celltoolbar": "Slideshow", + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + 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