What is the expectation of $ X^2$ where $ X$ is distributed normally?

Question

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.

Thanks

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

score 50 · Accepted Answer · edited Jan 14 '12 at 16:07

50

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.

edited Jan 14 '12 at 16:07

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answered Jan 14 '12 at 15:51

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What is the expectation of $ X^2$ where $ X$ is distributed normally?

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