Let $X$ be a random variable with mean $0$ and variance $\sigma ^2$, i.e. $X \sim \mathcal{N}(0, \sigma ^2)$.What is the distribution of $Y= X^n$, $n \in \mathbb {N}.$ ?
I know what distributribution has $Y=aX+b$, but I have not found anything for this case.
Revised version: In the end, only the expectation of $Y$ is needed, not its distribution.
"In the end, only the expectation of Y is needed, not its distribution." This a typical famous university exam question.
As Did already mentioned $E(X^n)=0$ for n being odd. Now for $n$ even ($2m$ say and WLOG assume $\sigma^2=1$), $$E(X^{2m})=2\int_0^{\infty} \frac{x^{2m}\exp(-x^{2}/2)}{\sqrt{2\pi}}dx$$ Substitute, $y=x^2/2$ to make the above as a $Gamma$ integral. The final answer is $\frac{2^m \times\Gamma(m+0.5)}{\sqrt\pi}$ which is $1\times3\times5...\times (2m-1)$