Geometric distribution

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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 failure 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 0}\eq\{1,2,\ldots\} [/ilmath]
p.m.f	[ilmath]\mathbb{P}[X\eq k]:\eq p^{k-1}(1-p)[/ilmath]
c.d.f / c.m.f	[ilmath]\mathbb{P}[X\le k]\eq 1-p^k[/ilmath]
cor:	[ilmath]\mathbb{P}[X\ge k]\eq 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]

Notes

during proof of [ilmath]\mathbb{P}[X\le k][/ilmath] the result is obtained using a geometric series, however one has to align the sequences (not adjust the sum to start at zero, unless you adjust the [ilmath]S_n[/ilmath] formula too!)

Check the variance, I did part the proof, checked the MEI formula book and moved on, I didn't confirm interpretation.

Make a note that my Casio calculator uses [ilmath]1-p[/ilmath] as the parameter, giving [ilmath]\mathbb{P}[X\eq k]:\eq (1-p)^{k-1}p[/ilmath] along with the interpretation that allows 0