Geometric distribution/Infobox

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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[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]
PAGE FOR TRANSCLUSION, YOU PROBABLY WANT GEOMETRIC DISTRIBUTION

Notes

  1. Do we make this distinction for cumulative distributions?

References

  1. See Expectation of the geometric distribution
  2. See Variance of the geometric distribution