ABSTRACT
The genesis of the bivariate negative binomial, as does that of its univariate counterpart, depends upon the underlying chance mechanism. The simplest derivation of the univariate negative binomial distribution involves waiting times. Recall that in Chapter 3 we discussed Bernoulli trials. Now consider an infinite sequence of Bernoulli trials with p as the probability of success and (1 − p) as the probability of a failure. Let the random variable X count the number of failures preceding the rth success; then X has the univariate negative binomial distribution with pf P { X = x } = ( r + x − 1 ) ! ( r − 1 ) ! x ! p r ( 1 − p ) x x = 0 , 1 , … https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315138480/5ae0c873-2684-4218-84a5-01e21f92e64f/content/eq706.tif"/>
