Introduction to Probability Theory

Description: The mathematical theory underlying statistical methods. Probability spaces, distributions in one and several dimensions, generating functions, and limit theorems.

Textbook: Introduction to Probability and Mathematical Statistics, Second Edition, by Brain & Engelhardt.

Materials: Material: This material is mostly from Chapters 1-7 of the textbook

Probability, Random experiment, Set theory, Probabilistic model, Counting rules, Conditional probability, Independence, Probability Laws, Probability tree, Bayes’ rule

Discrete random variables, Probability distribution, Expected value ,Expected value of a function of a random variable, Special discrete distributions: Binomial, Negative Binomial distributions, Hypergeometric, Poisson distributions

Continuous random variables, CDF and PDF, Expected value, Special continuous distributions :Uniform, Normal distributions, Gamma, Beta distributions, Other expected values

Multivariate probability distributions, Marginal and conditional probability distributions,Independent random variables, Expectation of function of random variables, Covariance, Multinomial distribution

Distribution of a function of random variables, The method of distribution function, The method of transformation, Order statistics