# Lectures on Probability Theory and Mathematical Statistics

Marco Taboga

The book is a compilation of 80 concise, self-contained lectures that cover the majority of the material typically covered in intermediate probability theory and mathematical statistics courses. There are countless instances, completed tasks, and thorough derivations of significant findings. The book is simple to understand and excellent for independent study due to its step-by-step format. One of the book’s main goals is to save the reader time. To that end, it offers a number of conclusions and proofs, particularly those related to probability distributions, that are difficult to locate in standard references and are dispersed among more specialized publications. These are the subjects that are covered in the book. Set theory, permutations, combinations, partitions, sequences, and limits, a review of differentiation and integration rules, and the Gamma and Beta functions are some of the mathematical tools covered in Part 1. Events, probability, independence, conditional probability, Bayes’ rule, random variables and random vectors, expected value, variance, covariance, correlation, covariance matrix, conditional distributions and conditional expectation, independent variables, indicator functions—all of these concepts are covered in Part 2: THE FUNDAMENTALS OF PROBABILITY. Part 3 of the probability theory covers additional topics such as probabilistic inequalities, probability distribution construction and transformation, moments and cross-moments, moment generating functions, and characteristic functions. Bernoulli, binomial, Poisson, uniform, exponential, normal, Chi-square, gamma, Student’s t, F, multinomial, multivariate normal, multivariate Student’s t, and Wishart are some of the probability distributions covered in Part 4. Part 5: Additional information on the normal distribution, including partitions, linear combinations, and quadratic forms. Sequences of random vectors and random variables, pointwise convergence, almost certain convergence, convergence in probability, mean-square convergence, convergence in distribution, relationships between modes of convergence, Laws of Large Numbers, Central Limit Theorems, Continuous Mapping Theorem, and Slutsky’s Theorem are all covered in Part 6 of Asymptotic Theory. The statistical inference, point estimation, set estimation, hypothesis testing, statistical inferences about the mean, and statistical inferences about the variance are all covered in Part 7 of the Fundamentals of Statistics.