exam questions list

Exam-questions.md

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+# MSAI Probability Exam Questions
+
+1. 
+    - Sample spaces.
+    - Union/intersection, complements to events, mutually exclusive events, implication, de Morgan's laws, partition of the sample space
+
+2. 
+    - Naïve definition of probability.
+    - Counting rules: multiplication rule, sampling with and without replacement, permutations and factorials.
+    - The Birthday problem/paradox
+
+3. 
+    - Binomial coefficients, Binomial theorem.
+    - Bose-Einstein statistic, “stars and bars”
+
+4.
+    - Non-naïve definition of probability: probability spaces, properties of probability (additivity).
+    - Frequentist and Bayesian view
+
+5. 
+    - Inclusion-exclusion formula.
+    - de Montmort’s matching problem
+
+6. 
+    - Conditional probability: definition, prior/posterior.
+    - Prosecutor’s fallacy.
+    - Frequentist interpretation.
+    - Martin Gardner's "Two children" puzzle
+
+7. 
+    - Bayes' rule and the Law of Total Probability (LOTP).
+    - Testing for diseases.
+    - Conditional probabilities are probabilities
+
+8. 
+    - Independence of events.
+    - Independence of 3 events, pairwise independence.
+    - Conditional independence
+
+9. 
+    - Random variables. Indicator r.v.s.
+    - Discrete and continuous r.v.s.
+    - Distribution, probability mass function (PMF), its properties
+    - Cumulative distribution function (CDF)
+
+10. 
+    - Bernoulli and Binomial distributions
+    - Hypergeometric distribution
+
+11. 
+    - Functions of random variables.
+    - Independence of random variables
+
+12. 
+    - Expectation. 
+    - Linearity of expectation.
+    - Expectation of Binomial and Hypergeometric distr-s
+
+13. 
+    - Geometric and Negative Binomial distributions.
+    - Coupon collector problem.
+
+14. 
+    - Indicator r.v.s and the fundamental bridge.
+    - Expectation in the card matching problem.
+    - Boole, Bonferroni, inclusion-exclusion
+
+15. 
+    - Law of the unconscious statistician (LOTUS).
+    - St. Petersburg paradox
+
+16. 
+    - Variance and standard deviation.
+    - Variance of Binomial, Geometric, Negative Binomial distributions
+
+17. 
+    - Continuous random variables.
+    - Probability density function (PDF).
+    - Expectation of a continuous r.v.
+    - Logistic and Rayleigh distributions.
+
+18. 
+    - Continuous uniform distribution, its expectation and variance.
+    - Location-scale transformations, universality of uniform distribution
+
+19. 
+    - The Normal distribution, its expectation and variance. Its symmetry. 
+    - Poisson integral (normalization of the normal). Standardized normal.
+    - The 68-95-99 rule.
+
+20. 
+    - Measures of central tendency: mean, median, mode.
+    - What do they minimize.
+
+21. 
+    - Moments. Their interpretation.
+    - Skewness and Kurtosis.
+
+22. 
+    - Sample moments.
+    - Proof that sample std with 1/n is biased.
+
+23. Joint, marginal, conditional distributions: discrete case.
+
+24. 
+    - Joint, marginal, conditional distributions: continuous case.
+    - Unit 2D circle distribution
+
+25. 
+    - 2D LOTUS, expected distance between 2 uniforms.
+    - Covariance and correlation, their properties
+
+26. 
+    - Change of variables. 
+    - Log-normal distribution.
+    - Chi-squared distribution
+
+27. 
+    - Change of multiple variables, the Jacobian matrix.
+    - Box-Muller method
+
+28. 
+    - Convolutions.
+    - Uniforms and exponentials convolutions
+
+29.
+    - Conditional expectation given an event.
+    - Two-envelope paradox.
+    - Time until HH vs HT
+
+30. 
+    - Conditional expectation given an r.v.
+    - Stick breaking.
+    - Properties of conditional expectation.
+
+31. 
+    - Cauchy-Schwarz inequality.
+    - Second moment method
+
+32. 
+    - Jensen’s inequality.
+    - Relation to St. Petersburg paradox, bias of sample std.
+    - Entropy, KL-divergence.
+
+33. 
+    - Markov, Chebyshev, Chernoff inequalities.
+    - Bounds on Normal tail probability
+
+34. 
+    - Law of large numbers: weak and strong.
+    - Running proportion of heads, Monte-Carlo idea
+
+35. The Central Limit Theorem.
+
+36. Chi-squared and Students’ t-distributions.
+