Difference between revisions of "Variance"

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m (Very embarrassing mistake)
(Definition: integrable r.v.)
 
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==Definition==
 
==Definition==
Given a [[Random variable|random variable]] {{M|X}} we define the '''variance''' of {{M|X}} as follows:
+
Given an [[Integral (measure theory)|integrable]] [[Random variable|random variable]] {{M|X}} we define the '''variance''' of {{M|X}} as follows:
 
* <math>\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]</math> where {{M|\mu}} is the ''mean'' or [[Expected value|expected value]] of {{M|X}}
 
* <math>\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]</math> where {{M|\mu}} is the ''mean'' or [[Expected value|expected value]] of {{M|X}}
 
  
 
==Other forms==
 
==Other forms==

Latest revision as of 19:45, 24 July 2016

Definition

Given an integrable random variable [ilmath]X[/ilmath] we define the variance of [ilmath]X[/ilmath] as follows:

  • [math]\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right][/math] where [ilmath]\mu[/ilmath] is the mean or expected value of [ilmath]X[/ilmath]

Other forms

Theorem: [math]\text{Var}(X)=\mathbb{E}[X^2]-(\mathbb{E}[X])^2[/math]


  • [math]\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right][/math]
[math]=\mathbb{E}\left[X^2-2X\mu+\mu^2\right][/math]
[math]=\mathbb{E}\left[X^2\right]-2\mu\mathbb{E}[X]+\mu^2[/math]
But! [math]\mu=\mathbb{E}[X][/math]
[math]=\mathbb{E}\left[X^2\right]-2\mu^2+\mu^2[/math]
[math]=\mathbb{E}\left[X^2\right]-\mu^2[/math]
[math]=\mathbb{E}\left[X^2\right]-(\mathbb{E}[X])^2[/math]

As required.


References