The Foundations of Probability and Mathematical Statistics

These notes were written for an introductory course in probability and statistics at the post-calculus level that was presented during the fall term of 1974 to students in the Rand Graduate Institute. Most of the material is devoted to the basic concepts of probability theory that are prerequisite to learning mathematical statistics: probability models, random variables, expectation and variance, joint distributions, conditioning, correlation, and sampling theory. Among the distributions treated are the binomial, hypergeometric, Poisson, negative binomial, normal, gamma, lognormal, chi- square, and bivariate normal. The last section of the notes provides an introduction to some of the basic notions of parameter estimation: bias, efficiency, sufficiency, completeness, consistency, maximum likelihood, and least-squares estimation. Proofs of the Rao-Blackwell, Lehmann-Scheffe, and Gauss-Markov Theorems are included.