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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?

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    $\begingroup$ The result that you are trying to prove is false except when $\mu=0$. $\endgroup$ – Dilip Sarwate Dec 10 '16 at 15:21
  • $\begingroup$ Have you heard of the moment generating function? $\endgroup$ – Xi'an Dec 10 '16 at 15:35