Geometric Distribution

A random variable $X$ has a geometric distribution $X \sim geometric (p)$ if and only if it has the probability distribution function

f (x; p) = p (1 - p)^{x - 1}

for $x = 1, 2, 3, \dots$

$f (x) \geq 0$
$\sum p (1 - p)^{x - 1} = 1$

If we are looking at the number of fails ( $x = 0$ is possible)

f (x) = p (1 - p)^{x}

Geometric distribution is equivalent to the Negative Binomial Distribution when only looking for the first success
$X \sim geometric (p) \equiv N B (1, p)$

E (X) = \frac{1}{p}

V a r (X) = \frac{1}{p} (\frac{1}{p} - 1) = \frac{1 - p}{p^{2}}