Difference between revisions of "Geometric distribution"
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|header1=Definition | |header1=Definition | ||
|label1=Defined over | |label1=Defined over | ||
| − | |data1={{M|X}} may take values in {{M|\mathbb{N}_{\ge | + | |data1={{M|X}} may take values in {{M|\mathbb{N}_{\ge 1}\eq\{1,2,\ldots\} }} |
|label2=[[probability mass function|p.m.f]] | |label2=[[probability mass function|p.m.f]] | ||
| − | |data2={{nowrap|{{M|\mathbb{P}[X\eq k]:\eq p^{k-1} | + | |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>}} | |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-p^k}} | + | |data3={{M|\mathbb{P}[X\le k]\eq 1-(1-p)^k}} |
|label4=''[[corollary|cor:]]'' | |label4=''[[corollary|cor:]]'' | ||
| − | |data4={{M|\mathbb{P}[X\ge k]\eq p^{k-1} }}<!-- | + | |data4={{M|\mathbb{P}[X\ge k]\eq (1-p)^{k-1} }}<!-- |
| Line 24: | Line 24: | ||
|data10={{MM|\mathbb{E}[X]\eq\frac{1}{p} }} | |data10={{MM|\mathbb{E}[X]\eq\frac{1}{p} }} | ||
|label11=[[Variance]]: | |label11=[[Variance]]: | ||
| − | |data11={{ | + | |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>}} |
}} | }} | ||
__TOC__ | __TOC__ | ||
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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 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] |
| Variance: | TODO: Unknown [Note 2]
|
Contents
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
Definition
References
Notes
- ↑ Do we make this distinction for cumulative distributions?
- ↑ Due to different conventions on the definition of geometric (for example [ilmath]X':\eq X-1[/ilmath] for my [ilmath]X[/ilmath] and another's [ilmath]X'\sim\text{Geo}(p)[/ilmath]) or even differing by using [ilmath]1-p[/ilmath] in place of [ilmath]p[/ilmath] in the [ilmath]X[/ilmath] and [ilmath]X'[/ilmath] just mentioned - I cannot be sure without working it out that it's [math]\frac{1-p}{p^2} [/math] - I record this value only for a record of what was once there with the correct expectation - DO NOT USE THIS EXPRESSION
