# Correlation Coefficient between these two random variables

Suppose that $X$ is real-valued normal random variable with mean $\mu$ and variance $\sigma^2$. What is the correlation coefficient between $X$ and $X^2$?

-
May I ask, if $\mu=0$, are $X$ and $X^2$ independent? I know their covariance is $0$, but this doesn't suffice. – Julie Oct 12 '11 at 1:06
Nope. They are not independent. See here. – cardinal Oct 12 '11 at 1:26
@Zoe: Since $X^2$ is determined by $X$, they can't be independent unless $X$ is constant. – Michael Hardy Oct 12 '11 at 2:15
@Michael: That's not entirely true. You actually need $X^2$ to be constant, not $X$. :) – cardinal Oct 12 '11 at 2:20
Also, I was mistaken in another respect: Adding a constant to $X$ has the effect of adding something other than a constant to $X^2$. (And now I've deleted that comment.) – Michael Hardy Oct 12 '11 at 3:48

Here's an efficient way to deal with the numerator in the fraction that defines the correlation. $$\operatorname{cov}(X,X^2) = \operatorname{cov}\Big((X-\mu)+\mu,\ \ (X-\mu)^2 + 2\mu(X-\mu) + \mu^2\Big).$$ Now we can throw away the "${}+ \mu$" and "${}+ \mu^2$" at the end and we have $$\operatorname{cov}\Big((X-\mu),\ \ (X-\mu)^2 + 2\mu(X-\mu)\Big).$$ Then use bilinearity of covariances and this becomes: $$\operatorname{cov}(X-\mu, (X-\mu)^2) + 2\mu\operatorname{cov}(X-\mu,X-\mu)).$$ This is $$0 + 2\mu\sigma^2.$$ The first term is $0$ because the expected value of $X-\mu$ is $0$ and the distribution is symmetric about $0$.

Summary: $\operatorname{cov}(X,X^2) = 2\mu\sigma^2$.

-

Hint: You are trying to find: $$\frac{E\left[\left(X^2-E\left[X^2\right]\right)\left(X-E\left[X\right]\right)\right]}{\sqrt{E\left[\left(X^2-E\left[X^2\right]\right)^2\right]E\left[\left(X-E\left[X\right]\right)^2\right]}}$$

For a normal distribution the raw moments are

• $E\left[X^1\right] = \mu$
• $E\left[X^2\right] = \mu^2+\sigma^2$
• $E\left[X^3\right] = \mu^3+3\mu\sigma^2$
• $E\left[X^4\right] = \mu^4+6\mu^2\sigma^2+3\sigma^4$

so multiply out, substitute and simplify.

-
I see the problem. When I was calculating the covariance, I mistook $E(X^2)=\sigma^2$, which led me to the wrong solution. – Julie Oct 12 '11 at 0:19
Therefore, the covariance should be $2\mu\sigma^2$. – Julie Oct 12 '11 at 0:30
Shouldn't the last term by $3\sigma^4$? – Michael Hardy Oct 12 '11 at 4:07
@Michael: Indeed it should - edited – Henry Oct 12 '11 at 6:09
And I think the final answer should be $\mu\sqrt{\dfrac{2}{2\mu^2+\sigma^2}}$ – Henry Oct 12 '11 at 7:24