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 $X^2$?

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.
