Geometric distribution/Infobox

Geometric Distribution
$X\sim\text{Geo}(p)$ for $p$ the probability of each trials' success
$X\eq k$ means that the first success occurred on the $k^\text{th}$ trial, $k\in\mathbb{N}_{\ge 1}$
Definition
Defined over	$X$ may take values in $\mathbb{N}_{\ge 1}\eq\{1,2,\ldots\}$
p.m.f	$\mathbb{P}[X\eq k]:\eq (1-p)^{k-1}p$
c.d.f / c.m.f	$\mathbb{P}[X\le k]\eq 1-(1-p)^k$
cor:	$\mathbb{P}[X\ge k]\eq (1-p)^{k-1}$
Properties
Expectation:	$$\mathbb{E}[X]\eq\frac{1}{p}$$
Variance:	$$\text{Var}(X)\eq\frac{1-p}{p^2}$$

PAGE FOR TRANSCLUSION, YOU PROBABLY WANT GEOMETRIC DISTRIBUTION