Variance

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]

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• [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.