proof by induction to prove $E[X^{2n}]=(2n-1)(2n-3) . . . 5.3.1$

Question

X is standard normal, use proof by induction to prove for all $n\in N$ that $E[X^{2n}]=(2n-1)(2n-3) . . . 5.3.1$

I would appreciate any tips or help on how to prove this, if anyone has time. I assume I need to find a case of some E[x^y] where this is true then use induction to prove it for all? very confused and would appreciate any help!

Answer 1

Do a u-sub and you will see a gamma function. What can you do with this?

Answer 2

In response to your comment: integration by parts with $u=x^{2n+1}$ and $v = -e^{-x^2/2}$ yields

$$\int_{-\infty}^\infty x^{2(n+1)} e^{-x^2/2} \, dx = [-x^{2n+1} e^{-x^2/2}]_{x=-\infty}^\infty + (2n+1)\int_{-\infty}^\infty x^{2n} e^{-x^2/2}.$$