Difference between revisions of "Variance of the geometric distribution"

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m (Adding definition note and confirming the convention.)
 
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{{Stub page|grade=A|msg=There's work to do, not just writing out the entire proof [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 15:04, 16 January 2018 (UTC) }}
 
{{ProbMacros}}{{M|\newcommand{\d}[0]{\mathrm{d} } }}{{M|\newcommand{\ddq}[1]{ \frac{\d}{\d q}{\left[{#1}\middle]\right\vert_q} }  }}__TOC__
 
{{ProbMacros}}{{M|\newcommand{\d}[0]{\mathrm{d} } }}{{M|\newcommand{\ddq}[1]{ \frac{\d}{\d q}{\left[{#1}\middle]\right\vert_q} }  }}__TOC__
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==Notes==
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==Statement==
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[[File:MostOfGeometricDistributionVarianceComputations.JPG|thumbnail|Workings so far]]It's {{MM|\frac{1-p}{p^2} }} I believe - I did this quickly on a sticky note starting from the warning below. My "formal attempt" (here) though I messed up the final step. I'll fix it later today. DO NOT USE RESULT unless you complete the proof, everything is fine until you hit the red and highly visible WARNING marker below.
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Let {{M|X\sim}}[[Geometric distribution|{{M|\text{Geo} }}]]{{MM|(p) }} - as defined on the [[geometric distribution]] page<ref group="Note">So using ''our'' convention, to say it explicitly</ref> - then the [[variance]] of {{M|X}} is:
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==Final steps==
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* {{MM|\Var{X}\eq \frac{1-p}{p^2} }} for {{M|p\in(0,1)}} with easy extension (as per ''[[expectation of the geometric distribution]]'') to {{M|p\in(0,1]}}
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==Proof==
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[[File:MostOfGeometricDistributionVarianceComputations.JPG|thumbnail|Workings so far]]
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===Final steps===
 
Recall {{M|q:\eq 1-p}}
 
Recall {{M|q:\eq 1-p}}
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===Computing {{MM|\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3} q^k\middle]\right\vert_q}}===
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====Computing {{MM|\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3} q^k\middle]\right\vert_q}}====
 
We leave the bottom of the paper workings with:
 
We leave the bottom of the paper workings with:
 
* {{MM|\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3} q^k\middle]\right\vert_q}}
 
* {{MM|\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3} q^k\middle]\right\vert_q}}
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** This yields:
 
** This yields:
 
*** {{MM|\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3} q^k\middle]\right\vert_q \eq 2\left(\frac{1}{p^3}-1\right)}}
 
*** {{MM|\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3} q^k\middle]\right\vert_q \eq 2\left(\frac{1}{p^3}-1\right)}}
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===Computing {{M|\E{X^2} }}===
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====Computing {{M|\E{X^2} }}====
 
Recall:
 
Recall:
 
* {{M|q:\eq 1-p}}
 
* {{M|q:\eq 1-p}}
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Now we substitute this all in to  {{M|\E{X^2}\eq \alpha-\beta+\E{X}+pq\left(\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3}q^k\middle]\right\vert_q\right)}} and:
 
Now we substitute this all in to  {{M|\E{X^2}\eq \alpha-\beta+\E{X}+pq\left(\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3}q^k\middle]\right\vert_q\right)}} and:
 
* {{MM|\E{X^2}\eq \P{X\eq 1}+4\P{X\eq 2}-\P{X\eq 1}-2\P{X\eq 2}+\frac{1}{p}+2(1-p)\left(\frac{1}{p^2}-p\right)}}
 
* {{MM|\E{X^2}\eq \P{X\eq 1}+4\P{X\eq 2}-\P{X\eq 1}-2\P{X\eq 2}+\frac{1}{p}+2(1-p)\left(\frac{1}{p^2}-p\right)}}
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*: {{MM|\eq 2\P{X\eq 2}+\frac{1}{p}+2(1-p)\left(\frac{1}{p^2}-\frac{p^2}{p^2}\right)}} - {{Warning|Error is here, the {{MM|\frac{p^2}{p^2} }} should be {{MM|\frac{p^3}{p^2} }} instead!}} - I'm just saving my work, I scrutinised everything and the error was here!
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*: {{MM|\eq 2\P{X\eq 2}+\frac{1}{p}+2(1-p)\left(\frac{1}{p^2}-\frac{p^3}{p^2}\right)}}
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*: {{MM|\eq 2(1-p)p+\frac{1}{p}+2(1-p)\frac{1-p^2}{p^2} }}
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*: {{MM|\eq 2(1-p)p+\frac{1}{p}+2(1-p)\frac{1-p^3}{p^2} }}
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*: {{MM|\eq \frac{1}{p}+2(1-p)\left(p+\frac{1-p^2}{p^2}\right)}}
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*: {{MM|\eq \frac{1}{p}+2(1-p)\left(p+\frac{1-p^3}{p^2}\right)}}
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*: {{MM|\eq \frac{1}{p}+2(1-p)\left(\frac{p^3}{p^2}+\frac{1-p^2}{p^2}\right)}}
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*: {{MM|\eq \frac{1}{p}+2(1-p)\left(\frac{p^3}{p^2}+\frac{1-p^3}{p^2}\right)}}
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*: {{MM|\eq \frac{1}{p}+2(1-p)\frac{p^3-p^2+1}{p^2} }}
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*: {{MM|\eq \frac{1}{p}+2(1-p)\frac{1}{p^2} }}
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*: {{MM|\eq \frac{1}{p} +\frac{2}{p^2} - \frac{2}{p} }}
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*: {{MM|\eq \frac{2}{p^2}-\frac{1}{p} }}
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*: {{MM|\eq \frac{2}{p^2}-\frac{p}{p^2} }} - it's probably easier in this form
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Given we'll need to subtract {{M|\frac{1}{p^2} }} there's no point in proceeding any further
 +
 
 +
====Computing {{M|\Var{X} }}====
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Lastly:
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* {{M|\Var{X}\eq\E{X^2}-(\E{X})^2}}, so
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** {{MM|\Var{X}\eq \frac{2}{p^2}-\frac{p}{p^2} - \frac{1}{p^2} }}
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**: {{MM|\eq\frac{1}{p^2}-\frac{p}{p^2} }}
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**: {{MM|\eq\frac{1-p}{p^2} }}
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Thus
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* {{MM|\Var{X}\eq\frac{1-p}{p^2} }}
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==Notes==
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<references group="Note"/>
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==References==
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<references/>
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{{Theorem Of|Probability|Statistics|Elementary Probability}}
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[[Category:Variance Calculations]]

Latest revision as of 15:08, 16 January 2018

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There's work to do, not just writing out the entire proof Alec (talk) 15:04, 16 January 2018 (UTC)
[ilmath]\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} }[/ilmath]
[ilmath]\newcommand{\E}[1]{ {\mathbb{E}{\left[{#1}\right]} } } [/ilmath][ilmath]\newcommand{\Mdm}[1]{\text{Mdm}{\left({#1}\right) } } [/ilmath][ilmath]\newcommand{\Var}[1]{\text{Var}{\left({#1}\right) } } [/ilmath][ilmath]\newcommand{\ncr}[2]{ \vphantom{C}^{#1}\!C_{#2} } [/ilmath][ilmath]\newcommand{\d}[0]{\mathrm{d} } [/ilmath][ilmath]\newcommand{\ddq}[1]{ \frac{\d}{\d q}{\left[{#1}\middle]\right\vert_q} } [/ilmath]

Statement

Let [ilmath]X\sim[/ilmath][ilmath]\text{Geo} [/ilmath][math](p) [/math] - as defined on the geometric distribution page[Note 1] - then the variance of [ilmath]X[/ilmath] is:

Proof

Workings so far

Final steps

Recall [ilmath]q:\eq 1-p[/ilmath]

Computing [math]\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3} q^k\middle]\right\vert_q[/math]

We leave the bottom of the paper workings with:

  • [math]\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3} q^k\middle]\right\vert_q[/math]
    [math]\eq\frac{\d}{\d q}\left[\frac{1}{(1-q)^2}-1-2q\middle]\right\vert_q [/math]
    [math]\eq -2+\ddq{(1-q)^{-2} } [/math]
    [math]\eq -2+(-2)(1-q)^{-3}\cdot\ddq{1-q} [/math]
    [math]\eq -2+\frac{-2}{(1-q)^3}\cdot (-1)[/math]
    [math]\eq\ 2\left(\frac{1}{(1-q)^3}-1\right)[/math]
  • We may now substitute [ilmath]q\eq 1-p[/ilmath] (as [ilmath]q:\eq 1-p[/ilmath] so [ilmath]p\eq 1-q[/ilmath] follows)
    • This yields:
      • [math]\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3} q^k\middle]\right\vert_q \eq 2\left(\frac{1}{p^3}-1\right)[/math]

Computing [ilmath]\E{X^2} [/ilmath]

Recall:

  • [ilmath]q:\eq 1-p[/ilmath]
  • [ilmath]\alpha:\eq \P{X\eq 1}+4\P{X\eq 2} [/ilmath]
  • [ilmath]\beta:\eq \P{X\eq 1}+2\P{X\eq 2} [/ilmath]

The previous step yielded:

  • [math]\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3} q^k\middle]\right\vert_q \eq\ 2\left(\frac{1}{p^3}-1\right)[/math]

and we got as far as:

  • [math]\E{X^2}\eq \alpha-\beta+\E{X}+pq\left(\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3}q^k\middle]\right\vert_q\right)[/math]

So:

  • [math]pq\left(\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3}q^k\middle]\right\vert_q\right)[/math]
    [math]\eq 2pq\left(\frac{1}{p^3}-1\right)[/math]
    [math]\eq 2q\left(\frac{1}{p^2}-p\right)[/math]
    [math]\eq 2(1-p)\left(\frac{1}{p^2}-p\right)[/math]


Now we substitute this all in to [ilmath]\E{X^2}\eq \alpha-\beta+\E{X}+pq\left(\frac{\d^2}{\d q^2}\left[\sum^\infty_{k\eq 3}q^k\middle]\right\vert_q\right)[/ilmath] and:

  • [math]\E{X^2}\eq \P{X\eq 1}+4\P{X\eq 2}-\P{X\eq 1}-2\P{X\eq 2}+\frac{1}{p}+2(1-p)\left(\frac{1}{p^2}-p\right)[/math]
    [math]\eq 2\P{X\eq 2}+\frac{1}{p}+2(1-p)\left(\frac{1}{p^2}-\frac{p^3}{p^2}\right)[/math]
    [math]\eq 2(1-p)p+\frac{1}{p}+2(1-p)\frac{1-p^3}{p^2} [/math]
    [math]\eq \frac{1}{p}+2(1-p)\left(p+\frac{1-p^3}{p^2}\right)[/math]
    [math]\eq \frac{1}{p}+2(1-p)\left(\frac{p^3}{p^2}+\frac{1-p^3}{p^2}\right)[/math]
    [math]\eq \frac{1}{p}+2(1-p)\frac{1}{p^2} [/math]
    [math]\eq \frac{1}{p} +\frac{2}{p^2} - \frac{2}{p} [/math]
    [math]\eq \frac{2}{p^2}-\frac{1}{p} [/math]
    [math]\eq \frac{2}{p^2}-\frac{p}{p^2} [/math] - it's probably easier in this form

Given we'll need to subtract [ilmath]\frac{1}{p^2} [/ilmath] there's no point in proceeding any further

Computing [ilmath]\Var{X} [/ilmath]

Lastly:

  • [ilmath]\Var{X}\eq\E{X^2}-(\E{X})^2[/ilmath], so
    • [math]\Var{X}\eq \frac{2}{p^2}-\frac{p}{p^2} - \frac{1}{p^2} [/math]
      [math]\eq\frac{1}{p^2}-\frac{p}{p^2} [/math]
      [math]\eq\frac{1-p}{p^2} [/math]

Thus

  • [math]\Var{X}\eq\frac{1-p}{p^2} [/math]

Notes

  1. ↑ So using our convention, to say it explicitly

References

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