Negative multinomial distribution
In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(r,p)) to more than two outcomes.
Suppose we have an experiment that generates m+1≥2 possible outcomes, {X0,…,Xm}, each occurring with non-negative probabilities {p0,…,pm} respectively. If sampling proceeded until n observations were made, then {X0,…,Xm} would have been multinomially distributed. However, if the experiment is stopped once X0 reaches the predetermined value k0, then the distribution of the m-tuple {X1,…,Xm} is negative multinomial. These variables are not multinomially distributed because their sum X1+…+Xm is not fixed, being a draw from a negative binomial distribution.