# geometric random variable

A with parameter $p\in(0,1]$ is one whose density distribution function is given by

 $f_{X}(x)=p(1-p)^{x},\qquad x=0,1,2,\ldots$

This is denoted by $X\sim Geo(p)$.

Notes:

1. 1.

A standard application of geometric random variables is where $X$ represents the number of failed Bernoulli trials before the first success.

2. 2.

The expected value of a geometric random variable is given by $E[X]=\frac{1-p}{p}$, and the variance by $Var[X]=\frac{1-p}{p^{2}}$

3. 3.

The moment generating function of a geometric random variable is given by $M_{X}(t)=\frac{p}{1-(1-p)e^{t}}$.

Title geometric random variable GeometricRandomVariable 2013-03-22 11:54:06 2013-03-22 11:54:06 mathcam (2727) mathcam (2727) 14 mathcam (2727) Definition msc 62-00 msc 60-00 msc 92-01 msc 92B05 geometric distribution