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I know that if $X$ were distributed as a standard normal, then $X^2$ would be distributed as chi-squared, and hence have expectation $1$, but I'm not sure about for a general normal.


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if $ Y \sim \mathcal{N}(\mu,\sigma^2)$ then you have $ Y = \sigma X +\mu $ where $ X \sim \mathcal{N}(0,1)$ – math Jan 14 '12 at 16:03

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up vote 17 down vote accepted

Use the identity $$ E(X^2)=\text{Var}(X)+[E(X)]^2 $$ and you're done.

Since you know that $X\sim N(\mu,\sigma)$, you know the mean and variance of $X$ already, so you know all terms on RHS.

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Haha, thanks. I didn't think of that. – maliky0_o Jan 14 '12 at 16:04

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