Geometric Distribution

$X \sim Geometric (p)$

A geometric random variable represents the number of trials until a first success where each trial is independent and has a probability $p$ of success
$p_{X} (x) = (1 - p)^{x - 1} p, x \in {1, 2, 3, \dots}$
$F_{X} (x) = 1 - (1 - p)^{x}$
$M_{X} (t) = \frac{p e ^{t}}{1 - ( 1 - p ) e ^{t}}, t < - ln (1 - p)$
$E [X] = \frac{1}{p}$
$var (X) = \frac{1 - p}{p ^{2}}$
A geometric distribution describes the number of trials until a success where each trial has a probability of $p$ of occurring