Geometric distribution

	Geometric Distribution
	X∼Geo(p); for p the probability of each trials' success
	X=k means that the first failure occurred on the kth trial, k∈N≥1
Definition
Defined over	X may take values in N≥1={1,2,…}
p.m.f	P[X=k]:=(1−p)k−1p
c.d.f / c.m.f	P[X≤k]=1−(1−p)k
cor:	P[X≥k]=(1−p)k−1
Properties
Expectation:	E[X]=1p
Variance:	TODO: Unknown

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Notes

during proof of $\mathbb{P}[X\le k]$ 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 $S_n$ 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 $1-p$ as the parameter, giving $\mathbb{P}[X\eq k]:\eq (1-p)^{k-1}p$ along with the interpretation that allows 0

Definition

References

Notes

<cite_references_link_accessibility_label> ↑ Do we make this distinction for cumulative distributions?
<cite_references_link_accessibility_label> ↑ Due to different conventions on the definition of geometric (for example $X':\eq X-1$ for my $X$ and another's $X'\sim\text{Geo}(p)$ ) or even differing by using $1-p$ in place of $p$ in the $X$ and $X'$ just mentioned - I cannot be sure without working it out that it's $\frac{1-p}{p^2}$ - I record this value only for a record of what was once there with the correct expectation - DO NOT USE THIS EXPRESSION

[1] <cite_references_link_accessibility_label> ↑ Do we make this distinction for cumulative distributions?

[2] <cite_references_link_accessibility_label> ↑ Due to different conventions on the definition of geometric (for example $X':\eq X-1$ for my $X$ and another's $X'\sim\text{Geo}(p)$ ) or even differing by using $1-p$ in place of $p$ in the $X$ and $X'$ just mentioned - I cannot be sure without working it out that it's $\frac{1-p}{p^2}$ - I record this value only for a record of what was once there with the correct expectation - DO NOT USE THIS EXPRESSION

[Note 1]

[Note 2]

Definition
Geometric Distribution
$X\sim\text{Geo}(p)$ for $p$ the probability of each trials' success
$X\eq k$ means that the first failure occurred on the $k^\text{th}$ trial, $k\in\mathbb{N}_{\ge 1}$
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^{[Note 1]}	$\mathbb{P}[X\le k]\eq 1-(1-p)^k$
cor:	$\mathbb{P}[X\ge k]\eq (1-p)^{k-1}$
Properties
Expectation:	$\mathbb{E}[X]\eq\frac{1}{p}$
Variance:	TODO: Unknown ^{[Note 2]}

Geometric distribution

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Notes

Definition

References

Notes

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