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

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