Difference between revisions of "Variance"
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==Other forms== | ==Other forms== | ||
{{Begin Theorem}} | {{Begin Theorem}} | ||
| − | Theorem: <math>\text{Var}(X)=\mathbb{E}[X^2] | + | Theorem: <math>\text{Var}(X)=\mathbb{E}[X^2]-(\mathbb{E}[X])^2</math> |
{{Begin Proof}} | {{Begin Proof}} | ||
* <math>\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]</math> | * <math>\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]</math> | ||
Revision as of 13:23, 24 July 2016
Definition
Given a 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.
