 Negative multinomial distribution

Counts of several categories before a fixed number of stopping-category outcomes have a negative multinomial distribution. For one stopping outcome of probability $r=1-\alpha -\beta$, the two-count mass function is $r\left(\genfrac{}{}{0ex}{}{x+y}{x}\right){\alpha }^x{\beta }^y$. Its probability generating function is $r/(1-\alpha s-\beta t)$ and its means are $\alpha /r$ and $\beta /r$.

The negative multinomial distribution is a generalization of the negative binomial distribution and is used to model the number of trials needed to achieve a certain number of successes in a multinomial setting. This type of distribution is particularly useful when dealing with problems where outcomes can fall into more than two categories, as is the case with multinomial experiments.