Basic Discrete Distributions

Abstract

In this chapter we study some distributions that arise frequently in applications. Two broad areas of application are to be distinguished. One is sampling, performed for the purpose of drawing inferences about the population from which the sample is taken. As examples of such sampling, we would include consumer polls on the one hand, and scientific measurements on the other. In consumer polls, we wish, say, to determine the percentage of people in a certain city who approve of a certain manufactured product. When a complete poll of all persons is prohibitively expensive in time or effort, we sample; and from the percentage observed in the sample, we draw conclusions about the numerical value of the percentage in the entire population. In scientific experiments, we measure the value of an observable variable, usually getting somewhat different numerical values on each measurement. From a few measurements, which in reality constitute a sample, we attempt to establish or estimate the true value. The variation in measured values may be caused by errors of measurement or by inherent randomness in the process being observed.

We define the art of conjecture, or stochastic art, as the art of evaluating as exactly as possible the probabilities of things. Jakob Bernoulli (1654–1705)