Highlights of the Probability Theory Class - Assignment | MATH 407, Assignments of Probability and Statistics

Download Highlights of the Probability Theory Class - Assignment | MATH 407 and more Assignments Probability and Statistics in PDF only on Docsity! Highlights of our Probability Theory Class Problems. (1) Benford’s Law (distribution of the first significant digit in a set of “naturally occurring” numbers) (2) Birth day (3) Buffon’s needle (4) Coupon collecting (5) Gambler’s Ruin (6) Matching (7) Monty Hall (8) Runs in Bernoulli trials Methods. (1) Conditioning (2) Continuity correction (in the CLT) (3) Inclusion-exclusion principle (4) Probability tree (5) Table (6) Use of indicator functions (7) Venn diagram Tools. (1) Approximating hyper-geometric with binomial, binomial with Poisson or normal, and Poisson with normal (2) Bayes formula (3) Change of variables, especially in one and two dimensions. (4) Characteristics of a distribution (mean, variance, moment generating function, covari- ance, correlation coefficient, entropy, mutual information) (5) Convergence (almost sure, in probability, in distribution, in the mean) (6) Distributions • discrete: binomial, Poisson, hyper-geometric, geometric, negative binomial, Zipf. • continuous: uniform, exponential, normal (Gaussian) — one-, two-, and multi- dimensional, Gamma, Beta, Cauchy, Weibull. (7) Inequalities (Cauchy-Schwarz, Chebyshev, Chernoff, Hölder, Improved Chebyshev, Jensen, Lyapunov, Markov) (8) Limit Theorems (Law of Large Numbers — weak and strong, Central Limit Theorem, Law of iterated logarithm) (9) Markov chain and invariant distribution (10) Monte-Carlo simulations (11) Normal Correlation Theorem (12) Poisson process (13) Probabilistic methods (using probability to solve deterministic problems) (14) Producing a given continuous distribution from the uniform distribution on (0, 1) by inverse transformation (15) Stirling formula Other. (1) Bertrand Russel’ paradox (2) Beta function (3) Catalan numbers (4) Gamma function (5) Solution of a second-order finite-difference equation with constant coefficients 1