Difference between revisions of "Geometric distribution"

	Geometric Distribution
	X∼Geo(p); for p the probability of each trials' success
	X=k means that the first success 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:	Var(X)=1−pp2

Revision as of 03:22, 16 January 2018

Stub grade: A*

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It's crap, look at it.

Dire notice: "The content is iffy, and uses a weird convention where geometric is time to first failure. New page content at Geometric distribution2"
- Dire notice removed Alec (talk) 03:35, 15 January 2018 (UTC)
  - Deleted Geometric distribution2 as no longer needed Alec (talk) 03:22, 16 January 2018 (UTC)
Partial expectation proof to be found at Geometric distribution2 page.
- Turns out there was an Expectation of the geometric distribution page - I feel silly Alec (talk) 03:22, 16 January 2018 (UTC)

$\newcommand{\P}[2][]{\mathbb{P}#1{\left[{#2}\right]} } \newcommand{\Pcond}[3][]{\mathbb{P}#1{\left[{#2}\!\ \middle\vert\!\ {#3}\right]} } \newcommand{\Plcond}[3][]{\Pcond[#1]{#2}{#3} } \newcommand{\Prcond}[3][]{\Pcond[#1]{#2}{#3} }$

$\newcommand{\E}[1]{ {\mathbb{E}{\left[{#1}\right]} } }$

$\newcommand{\Mdm}[1]{\text{Mdm}{\left({#1}\right) } }$

$\newcommand{\Var}[1]{\text{Var}{\left({#1}\right) } }$

$\newcommand{\ncr}[2]{ \vphantom{C}^{#1}\!C_{#2} }$

Definition

Consider a potentially infinite sequence of $\text{Borv}$ variables, $({ X_i })_{ i = 1 }^{ n }$ , each independent and identically distributed (i.i.d) with $X_i\sim$ $\text{Borv}$ $(p)$ , so $p$ is the probability of any particular trial being a "success".

The geometric distribution models the probability that the first success occurs on the $k^\text{th}$ trial, for $k\in\mathbb{N}_{\ge 1}$ .

As such:

$\P{X\eq k} :\eq (1-p)^{k-1}p$ - pmf / pdf - Claim 1 below
P[X≤k]=1−(1−p)k - cdf - Claim 2 below
- $\mathbb{P}[X\ge k]\eq (1-p)^{k-1}$ - an obvious extension.

Convention notes

Grade: A**

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If

$X\sim\text{Geo}(p)$ is defined as above then there are 3 other conventions I've seen:

$X_1\sim\text{Geo}(1-p)$ in our terminology, they would write $\text{Geo}(p)$ , which measures "trials until first failure" instead of success as we do
$X_2:\eq X-1$ - the number of trials BEFORE first success
$X_3:\eq X_1-1$ - the number of trials BEFORE first failure

Document and explain Alec (talk) 03:17, 16 January 2018 (UTC)

Warning:That grade doesn't exist!

Properties

For $p\in[0,1]\subseteq\mathbb{R}$ and $X\sim\text{Geo}(p)$ we have the following results about the geometric distribution:

E[X]=1p for p∈(0,1] and is undefined or tentatively defined as +∞ if p=0
- Proof: Expectation of the geometric distribution

To do:

Proof of claims

Claim 1: $\P{X\eq k}\eq (1-p)^{k-1} p$

TODO: This requires improvement, it was copy and pasted from some notes

P[X=k]:=(1−p)k−1p - which is derived as folllows:
- P[X=k]:=(P[X1=0]×P[X2=0 | X1=0]×⋯×P[Xk−1=0 | X1=0∩X2=0∩⋯∩Xk−2=0])×P[Xk=1 | X1=0∩X2=0∩⋯∩Xk−1=0]
  - Using that the Xi are independent random variables we see:
    - $\P{X\eq k}\eq \left(\prod^{k-1}_{i\eq 1}\P{X_i\eq 0}\right)\times \P{X_k\eq 1}$
      $\eq (1-p)^{k-1} p$ as they all have the same distribution, namely $X_i\sim\text{Borv}(p)$

Claim 2: $\mathbb{P}[X\le k]\eq 1-(1-p)^k$

Grade: A**

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Trivial to do, direct application of Geometric series result Alec (talk) 03:17, 16 January 2018 (UTC)

Notes

Jump up ↑ Do we make this distinction for cumulative distributions?

References

[1] Jump up ↑ Do we make this distinction for cumulative distributions?

[2] Jump up ↑ See Expectation of the geometric distribution

[3] Jump up ↑ See Variance of the geometric distribution

[Note 1]

[1]

[2]

@@ Line 2: / Line 2: @@
 * Dire notice: "The content is iffy, and uses a weird convention where geometric is time to first failure. New page content at [[Geometric distribution2]]"
 ** Dire notice removed [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 03:35, 15 January 2018 (UTC)
-* Partial expectation proof to be found at [[Geometric distribution2]] page.}}
+*** Deleted [[Geometric distribution2]] as no longer needed [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 03:22, 16 January 2018 (UTC)
-{{Infobox
+* {{strike|Partial expectation proof to be found at [[Geometric distribution2]] page.}}
-|style=max-width:25em;
+** Turns out there was an [[Expectation of the geometric distribution]] page - I feel silly [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 03:22, 16 January 2018 (UTC)}}
-|title=Geometric Distribution
+{{/Infobox}}
-|above=<span style="font-size:1.5em;">{{M|X\sim\text{Geo}(p)}}</span><br/>
+{{ProbMacros}}
-<span style="font-size:0.75em;">''for {{M|p}} the [[probability (event)|probability]] of each trials' success''</span>
-|subheader={{M|X\eq k}} means that the first success occurred on the {{M|k^\text{th} }} trial, {{M|k\in\mathbb{N}_{\ge 1} }}
-|header1=Definition
-|label1=Defined over
-|data1={{M|X}} may take values in {{M|\mathbb{N}_{\ge 1}\eq\{1,2,\ldots\} }}
-|label2=[[probability mass function|p.m.f]]
-|data2={{nowrap|{{M|\mathbb{P}[X\eq k]:\eq (1-p)^{k-1}p}}}}
-|label3={{nowrap|[[cumulative density function|c.d.f]] / [[cumulative mass function|c.m.f]]<ref group="Note">Do we make this distinction for cumulative distributions?</ref>}}
-|data3={{M|\mathbb{P}[X\le k]\eq 1-(1-p)^k}}
-|label4=''[[corollary|cor:]]''
-|data4={{M|\mathbb{P}[X\ge k]\eq (1-p)^{k-1} }}<!--
-EXP and Var
--->
-|header10=Properties
-|label10=[[Expectation]]:
-|data10={{MM|\mathbb{E}[X]\eq\frac{1}{p} }}<ref>See ''[[Expectation of the geometric distribution]]''</ref>
-|label11=[[Variance]]:
-|data11={{Nowrap|{{XXX|Unknown}}<ref group="Note">Due to different conventions on the definition of geometric (for example {{M|X':\eq X-1}} for my {{M|X}} and another's {{M|X'\sim\text{Geo}(p)}}) or even differing by using {{M|1-p}} in place of {{M|p}} in the {{M|X}} and {{M|X'}} just mentioned - I cannot be sure without working it out that it's {{MM|\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</ref>}}
-}}{{ProbMacros}}
 __TOC__
 ==Definition==

Difference between revisions of "Geometric distribution"

Revision as of 03:22, 16 January 2018

Contents

Definition

Convention notes

Properties

To do:

Proof of claims

Claim 1: $\P{X\eq k}\eq (1-p)^{k-1} p$

Claim 2: $\mathbb{P}[X\le k]\eq 1-(1-p)^k$

See also

Distributions

Notes

References

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Definition
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, $k\in\mathbb{N}_{\ge 1}$
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:	$\mathbb{E}[X]\eq\frac{1}{p}$ ^[1]
Variance:	$\text{Var}(X)\eq\frac{1-p}{p^2}$ ^[2]

Difference between revisions of "Geometric distribution"

Revision as of 03:22, 16 January 2018

Contents

Definition

Convention notes

Properties

To do:

Proof of claims

Claim 1: P[X=k]=(1−p)k−1p\P{X\eq k}\eq (1-p)^{k-1} p

Claim 2: P[X≤k]=1−(1−p)k\mathbb{P}[X\le k]\eq 1-(1-p)^k

See also

Distributions

Notes

References

Navigation menu

Search

Claim 1: $\P{X\eq k}\eq (1-p)^{k-1} p$

Claim 2: $\mathbb{P}[X\le k]\eq 1-(1-p)^k$