negative binomial random variable

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

$f_X(x)=\left(\genfrac{}{}{0pt}{}{r+x-1}{x}\right)p^r{(1-p)}^x$, $x=\{0,1,\mathrm{\dots }\}$

Parameters:

• $r>0$

• $p\in [0,1]$

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. $M_X(t)={(\frac{p}{1-(1-p)e^t})}^r$

Title	negative binomial random variable
Canonical name	NegativeBinomialRandomVariable
Date of creation	2013-03-22 11:54:15
Last modified on	2013-03-22 11:54:15
Owner	bgins (4516)
Last modified by	bgins (4516)
Numerical id	9
Author	bgins (4516)
Entry type	Definition
Classification	msc 62E15
Classification	msc 18E05
Classification	msc 18-00
Synonym	negative binomial distribution