Overview¶
In this chapter, we shall define and discuss several special families of distributions that are widely used in applications of probability and statistics. The distributions that will be presented here include discrete and continuous distributions of univariate, bivariate, and multivariate types. The discrete univariate distributions are the families of Bernoulli, binomial, hypergeometric, Poisson, negative binomial, and geometric distributions. The continuous univariate distributions are the families of normal, lognormal, gamma, exponential, and beta distributions. Other continuous univariate distributions (introduced in exercises and examples) are the families of Weibull and Pareto distributions. Also discussed is the multinomial family of multivariate discrete distributions, and the bivariate normal family of bivariate continuous distributions. We shall briefly describe how each of these families of distributions arise in applied problems and show why each might be an appropriate probability model for some experiment.
For each family, we shall present the form of the pmf or the pdf and discuss some of the basic properties of the distributions in the family. The list of distributions presented in this chapter, or in this entire text for that matter, is not intended to be exhaustive. These distributions are known to be useful in a wide variety of applied problems. In many real-world problems, however, one will need to consider other distributions not mentioned here. The tools that we develop for use with these distributions can be generalized for use with other distributions. Our purpose in providing in-depth presentations of the most popular distributions here is to give the reader a feel for how to use probablity to model the variation and uncertainty in applied problems as well as some of the tools that get used during probability modeling.