Q.7.54 - A First Course in Probability

Question

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

Step-by-Step Solution

Answer

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

Step 1: Given information

Z is a standard normal random variable 

Step 2: Solution

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

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

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

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

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

$=0$

Step 3: Final answer

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