Covariance of a Normal with its Square

Question

Assume there is a random variable distributed normal $X\sim N(\mu,\sigma^2)$. Is there an analytic expression for the covariance of $X$ with its square $X^2$?

$$\operatorname{Cov}(X,X^2)$$

I have done a extensive search but I haven't found any result on this. I conjecture that when $\mu=0$, the covariance is zero, but I don't know how to generalize it.

Answer 1

$Y:=\frac{X-\mu}{\sigma}\sim N(0,1)$, and \begin{align}\operatorname{Cov}(X,X^2)&=\operatorname{Cov}(\sigma Y+\mu,(\sigma Y+\mu)^2)\\ &=\operatorname{Cov}(\sigma Y+\mu,\sigma^2 Y^2+2\mu\sigma Y+\mu^2)\\ &=\sigma^3\operatorname{Cov}(Y,Y^2)+2\mu\sigma^2\operatorname{Cov}(Y,Y)\\ &=2\mu\sigma^2. \end{align}

Answer 2

You can also simulate the results to get the intuition. For example for $X ~N(0,1)$, $X$ and $X^2$ have a covariance of zero, but between $X$ and $X^3$ it's $3$. Here is the $R$ code:

x=rnorm(10^5)
y=x^2 # or x^3
z=cbind(x,y)
cov(z)