geometric random variable

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

$$f_X(x)=p{(1-p)}^x,x=0,1,2,\mathrm{\dots }$$

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

Notes:

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

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