Question

If Z is a standard normal random variable, what is Cov(Z, Z²)?

Short answer

The covariance of $Z$and $Z^2$is$Cov(Z,Z^2)=0$.

Step-by-step solution

Given information 

Z is a standard normal random variable

Solution

Let $Z^2$be the square of the standard normal variable Z.

The covariance of Z and $Z^2$is calculated below,

$\mathrm{Cov}\left(Z,Z^2\right)=E\left(Z\cdot Z^2\right)-E\left(Z\right)E\left(Z^2\right)$

$=E\left(Z^3\right)-0\times E\left(Z^2\right)$

$=E\left(Z^3\right)\left(\begin{array}{l}\text{From the raw moments of}\\ \text{Standard normal distribution}{\mu }_3=0\end{array}\right)$

$=0$

Final answer

Therefore the covariance of $Z$and $Z^2$is $Cov\left(Z,Z^2\right)=0$.