Some Probability Distributions and Their Uses

Abstract

The range of all possible probability experiments covers a lot of ground, from the outcome of a weighted coin toss to next month’s price of a given stock, from the arrival time of the next phone call at a switchboard to the birth weight of a mother’s first child. To simulate these and other probability experiments we need to know how to convert uniformly distributed random numbers into a sample from whatever distribution underlies the experiment.

In this chapter we introduce some of the major probability distributions and show how to use the computer to draw a random sample from each of them. This means converting samples U drawn from the uniform [0, 1) distribution U(0, 1) into samples from the required density. We will present the probability density and cumulative distribution functions for each distribution along with their means and variances. We will show how the random samples can be used to calculate probabilities and solve problems.

The main techniques used for sampling are cdf inversion, simulation, composition, mapping, and rejection. Our objective is to introduce these sampling methodologies and to illustrate them. In many cases, better, but more involved, methods are available for sampling these distributions more efficiently should timing be a critical issue. See for example [Rip87, BFS83, Rub81].