How to proof that the $k$th moments of random variable $X\backsim N(0, \sigma^2)$ which is:

$$ E(X^k) = \begin{cases} \sigma^k(k!!) & \text{for k even;}\\ \ 0 & \text{for k odd} \end{cases} $$

I started from the basic definition of the $k$th moments:

$$E(X^k) =\int_{-\infty}^\infty x^k\frac{1}{\sqrt{2\pi}\sigma}e^{- \frac{x^2}{2\sigma^2}}dx$$

What kind of integration techniques that could solve this equation?

  • 2
    $\begingroup$ The result that you are trying to prove is false except when $\mu=0$. $\endgroup$ Commented Dec 10, 2016 at 15:21
  • $\begingroup$ Have you heard of the moment generating function? $\endgroup$
    – Xi'an
    Commented Dec 10, 2016 at 15:35