Definition

$X \sim NB (r, p)$ where $r > 0$ is a number of successes, and $p$ is a probability in each experiment

The number of failures in a bernoulli process before the $r$ -th success

Properties

$f (y) = (y y + r - 1) p^{r} (1 - p)^{y}, x \in N_{0}$

$E (X) = \frac{r ( 1 - p )}{p}$

$Var (X) = \frac{r ( 1 - p )}{p ^{2}}$

$M_{Y} (t) = p^{r} [1 - (1 - p) e^{t}]^{- r} = (\frac{p}{1 - ( 1 - p ) e ^{t}})^{r}$