Moment of normal distribution (Iproof by induction)

Question

I want to show $$E(Z^{2k})=\frac{(2k)!}{2^kk!} \text{ for } Z\sim N(0,1).$$

I showed $$E(Z^{2k})=(2k-1)\cdot E(Z^{2k-2})$$ and guessed by trying (since $E(Z^2)=1$): $$E(Z^{2k})=(2k-1)!!$$ How do I prove this by induction?

$k=1$ is true: $1=E(Z^2)=(2-1)!!=1$. Now $k\mapsto k+1$:

$$E(Z^{2(k+1)})=E(Z^{2k+2})=...?$$

Thanks for any help!

Answer 1

You are nearly done. $$(2k)!=(2k)!!(2k-1)!!=2^kk!(2k-1)!!$$