Probability Distributions Review for Exam MAS-I

Discrete Distributions

Exam MAS-I requires thorough knowledge of discrete distributions. The Bernoulli and Binomial model fixed-trial successes. The Poisson models rare event counts with E[X] = Var(X) = lambda. The Negative Binomial arises as a gamma mixture of Poissons, with variance exceeding the mean (overdispersion). The Geometric is the special case of Negative Binomial with r=1. The Hypergeometric models sampling without replacement. Know each distribution's PMF, mean, variance, MGF, and the relationships connecting them (e.g., Binomial approaches Poisson as n grows large and p approaches 0 with np = lambda).

Continuous Distributions

Key continuous distributions include the Normal (foundation for CLT and inference), Exponential (memoryless, hazard rate 1/theta), Gamma (sum of exponentials), Beta (conjugate prior for binomial proportion), Lognormal (multiplicative processes), and Chi-squared (sum of squared normals). The t-distribution arises in small-sample inference, and the F-distribution in ANOVA and regression. For each, know the PDF, CDF (when closed-form), moments, and key properties. Exam MAS-I integrates these distributions into statistical methods throughout the syllabus.