6 - Parametric distributions

Summary

Basic concepts

Parametric distributions are used extensively in classical statistics because mathematicians have manipulated their calculating formulas to devise test statistics whose predicted distributions are among the pre-calculated ones, some of which are listed in the backs of older statistics books. In addition, if a member of a parametric family adequately describes the variation in data of interest, these data can be used to estimate values for parameters; the name of the family together with values for its parameters make a useful summary description of that variation. Parametric distributions can also help define a hypothesized random process when you take a computational approach to statistical argument.

Binary distributions

Consider again a binary random variable b with possible values 1 and 0 and a distribution given by Pr(b = 1) = p. Specified in this way, the random variable, b, is chosen from a large class of random variables that differ with respect to the mechanisms that sample them, but always have only two possible values, 0 and 1. Although the random variables in the class to which b belongs may differ, their distributions will be the same if Pr(b = 1) is the same for both. We use the word, family, to refer to a collection of distributions that all have the same form, such as all the distributions for the random variable, b, but differ by the value of a number, such as p, which we call a parameter. Distributions that can be easily specified by designating the family to which they belong and specifying the value of a parameter (or sometimes the values of two or a few parameters), are called parametric distributions. Well-known families have names. The distribution of the random variable, b, belongs to a family named binary distributions.