I have the following problem: given a random variable $X \sim \mathcal{N} \left( 0, 1 \right)$ (i.e. given a standard normally distributed random variable), I want to calculate the correlation between $X$ and $X^{2}$. Therefore i proceed in the following way:
\begin{equation} \varrho_{X, X^{2}} = \frac{cov \left( X, X^{2} \right) }{\sigma_{X} \sigma_{X^2}} = \frac{\mathbb{E} \left[ (X - 0) (X^{2} - 1) \right]}{1 \cdot 2} = \frac{\mathbb{E} \left( X^{3} - X \right)}{2}. \end{equation}
Can anyone help me further?
Thanks a lot in advance
Recall $\mathbb{E} [X^3-X] = \mathbb{E}[X^3]-\mathbb{E}[X]$. Now $\mathbb{E}[X] = 0$, while $$ \mathbb{E}[X^3] = \frac{1}{\sqrt{2\pi}} \int_\mathbb{R} x^3 e^{-\frac{x^2}{2}} dx.$$ Integrating by parts, it turns out that $\mathbb{E}[X^3] = 0$, and hence $ \varrho_{X,X^2} = 0. $