$X$ is a negative binomial random variable with parameters $r$ and $p$ if

$f_X(x) ={r+x-1 \choose x} p^r (1-p)^x$ , $x=\{0,1,...\}$

Parameters:

$\star$

$r > 0$

$\star$

$p \in [0,1]$

Syntax:

$X\sim NegBin(r,p)$

Notes:

1. If $r \in \mathbb{N}$ , $X$ represents the number of failed Bernoulli trials before the $r$ th success. Note that if $r=1$ the variable is a geometric random variable.
2. $E[X] = r \frac{1-p}{p}$
3. $Var[X] = r \frac{1-p}{p^2}$
4. $M_X(t) = (\frac{p}{1 - (1-p)e^t})^r$