A discrete, univariate distribution modelling the number of ${\rm Bernoulli}(p)$ trial successes until a specified number of failures occur.

Overview

The negative binomial is a discrete, univariate distribution modelling the number of ${\rm Bernoulli}(p)$ trials until a specified number of failures occur. It is parametrized by $r$, the number of failures in $k$ ${\rm Bernoulli}(p)$ trial successes.

A discrete random variable $X$ has a negative binomial distribution, indexed by parameters $p \in (0,1)$ and $r \in \mathbb{Z}$ if its probability mass function is

$$ \Pr(X = k) = {k+r-1 \choose k} (1-p)^r p^k $$

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