Difference between revisions of "Variance"
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Revision as of 23:55, 29 April 2015
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[/math]
- [math]=\mathbb{E}\left[X^2\right]-\mathbb{E}[X][/math]
As required.
