Correlation coefficient between X~N(0,1) and $Y=X^{n}$

Question

Assume $X\sim N(0,1)$ and $Y = X^{n}$, where $n$ is a positive integer. Find $\rho_{xy}$ (correlation coefficient of $X$ and $Y$)

The only result that I have been able to find is: $$E[Y^{2n}] = \frac{(2n)!}{n!2^{n}}$$ But I can't proceed to finding the variance. If I could find the variance of Y, that is, the variance of the nth power of a standard normally distributed variable, then I could easily find the correlation.

I think it has something to do with moments and cumulants, but I can't make the exact connection.

The answer is:

$$ \rho_{xy} = \begin{cases} \frac{n!!}{\sqrt{(2n-1)!!}}, & \text{if $n$ is odd} \\ 0, & \text{if $n$ is even} \end{cases} $$

Thank you guys!

Answer 1

Start with the definition of the correlation coefficient,

$$\rho = {E(XY) - E(X) E(Y) \over SD(X) SD(Y)}$$

Since $Y = X^n$, this becomes

$$\rho = {E(X^{n+1}) - E(X) E(X^n) \over SD(X) SD(X^n)} $$

Now of course $E(X) = 0$ and $SD(X) = 1$ since $X$ is standard normal. So you have

$$ \rho = {E(X^{n+1}) \over SD(X^n)}$$

Now you just need to work out the numerator and denominator. Depending on whether $n$ is even or odd you'll get different results.