Third Moment of Expectation?

Question

I'm trying to find the $E[X^3]$ in terms of $\mu$ and $\sigma^2$. I found online somewhere that it's

$\mu^3 + 3\mu\sigma^2$.

But there's no proof or explanation anywhere for it. Could someone please explain?

Answer 1

For a general distribution, $\mu,\,\sigma^2$ doesn't determine $E[X^3]$.

For a Normal distribution, write $X=\mu+\sigma Z$ with $Z\sim N(0,\,1)$ so$$E[X^3]=E[(\mu+\sigma Z)^3]=E[\mu^3+3\mu^2\sigma Z+3\mu\sigma^2Z^2+\sigma^3Z^3]=\mu^3+3\mu\sigma^2.$$More generally, with $Z:=(X-\mu)/\sigma$ we call $\Bbb EZ^3$ the skewness of $X$, and denote it $\tilde{\mu}_3$. Then $\Bbb EZ^3=\mu^3+3\mu\sigma^2+\sigma^3\tilde{\mu}_3$.