Negative Binomial

Definition:[Negative Binomial]
A random variable with the pmf:

$\begin{align*} p(x) &= \dbinom{x - 1}{r - 1}p^{r}(1 - p)^{x -r}, &0 < p < 1 &, x = r,r+1,... \end{align*}$

is called negative binomial with parameters $(r,p)$ . It has the following:

$\begin{align*} E[X] &= \frac{r}{p} \\ V[X] &= \frac{r(1-p)}{p^2} \\ \sigma_X &= \frac{\sqrt{r(1-p)}}{p} \end{align*}$