Geometric distribution/Infobox

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

PAGE FOR TRANSCLUSION, YOU PROBABLY WANT GEOMETRIC DISTRIBUTION

Notes

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