Enumeration Data and Probability Distributions

Introduction Many types of statistical applications are characterized by enumeration data in the form of counts. Examples are the number of lost-time accidents in a plant, the number of defective items in a sample, and the number of items in a sample that fall within several specified categories.

The sampling distribution of count data can be characterized through probability distributions. In many cases, count data are appropriately interpreted through their corresponding distributions. However, in other situations analysis is greatly facilitated through distributions which have been developed for measurement data. Examples of each will be illustrated in the following subsections.

Binomial Probability Distribution

Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probability of observing a specified number of successes x in n trials is defined by the binomial distribution.

The sequence of outcomes is called a Bernoulli process,

Nomenclature

n = total number of trials

x = number of successes in n trials

p = probability of observing a success on any one trial

ˆp = x/n, the proportion of successes in n trials

Probability Law

Geometric Probability Distribution

Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probability of observing the first success on the xth trial is defined by the geometric distribution.

Nomenclature

p = probability of observing a success on any one trial

x = the number of trials to obtain the first success

Probability Law

P(x) = p(1 - p)x - 1 x = 1, 2, 3, . . ..

Properties

E(x) = 1/p Var (x) = (1 - p)/p2

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