# negative binomial random variable

$X$ 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. 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. 2.

$E[X]=r\frac{1-p}{p}$

3. 3.

$Var[X]=r\frac{1-p}{p^{2}}$

4. 4.

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

Title negative binomial random variable NegativeBinomialRandomVariable 2013-03-22 11:54:15 2013-03-22 11:54:15 bgins (4516) bgins (4516) 9 bgins (4516) Definition msc 62E15 msc 18E05 msc 18-00 negative binomial distribution