Part V — Common Probability Distributions

A reference catalog of the standard families. Each page gives the distribution's PMF or density, expectation, and variance, and — where written — a Monte Carlo simulation in R. Discrete families first, then continuous.

Topics

Discrete

Topic	Focus
Bernoulli Distribution	Indicator random variables, expectation, and variance
Binomial Distribution	PMF, moments, and Monte Carlo simulation of coin-toss counts
Geometric Distribution	PMF, memorylessness, and moments
Negative Binomial Distribution	PMF and moments of the trials-until-k-successes distribution
Hypergeometric Distribution	Sampling without replacement, and the contrast with the binomial
The Poisson Distribution	PMF, moments, and the Poisson limit of the binomial
Multinomial Distribution	Partitions and counts across more than two categories
Discrete Uniform Distribution	Equally likely outcomes on a finite range

Continuous

Topic	Focus
Continuous Uniform Distribution	The flat density on an interval
Exponential Distribution	Density, moments, and the continuous analog of geometric waiting times
Gamma Distribution	Sums of exponential waiting times and the gamma family
Beta Distribution	The conjugate family for a Bernoulli parameter on the unit interval
Chi-Square Distribution	Sums of squared standard normals and their role in variance tests
Student's t Distribution	Heavy-tailed sampling distribution of the standardized mean
F Distribution	Ratios of scaled chi-square variables, used to compare variances
The Gaussian Distribution	The normal family, standardization, and linear transformations
Lognormal Distribution	Multiplicative growth and the distribution of exponentiated normals
Weibull Distribution	Flexible failure-time distribution with shape-dependent hazard