Geometric

Definition[Geometric Random Variable]
The probability mass function given by

$\begin{align*} p(x) &= \begin{cases} (1 - p)^{x - 1}p & 0 < p < 1, x = 1,2,3,... \\ 0 & \text{elsewhere} \end{cases} \end{align*}$

is called geometric. It has the following:

$\begin{align*} E[X] &= \frac{1}{p} \\ V[X] &= \frac{1 - p}{p^2} \\ \sigma_X &= \frac{\sqrt{1 - p}}{p} \end{align*}$