# Discrete Distributions

• Nick T. Thomopoulos
Chapter

## Abstract

When the outcome of an independent experiment trial includes a specified set of values, usually integers, the outcome is a discrete variable. For example, the number of dots in a roll of two dice can take on only the integer numbers 2 to 12. When probabilities are assigned to each outcome and the sum over all possible outcomes is one, the variable, x, becomes a discrete random variable. The chapter describes the following six common discrete probability distributions: discrete uniform, binomial, geometric, Pascal, Poisson, and hyper geometric. For each of these, the probability function is listed, along with parameters of the function. Included, also are the mean, variance and applications on each distribution. When sample data is available, the analyst applies measures from the data to estimate the parameter values. The discrete uniform distribution applies when all the discrete integers between and including two limits (a,b) are possible outcomes and each can occur with an equal probability. The binomial occurs when n trials with a constant probability per trial yields zero to n successes. The geometric happens when n trials are needed to obtain the first success and the probability of a success is constant per trial. The Pascal is when n trials are needed to obtain k successes and the probability of a success is constant per trial. The Poisson occurs when the average rate of events occurring in specified duration is known. The hyper geometric happens with n trials are taken without replacement from a population of size N that has D defectives.

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