Derive the expected value and variance of $Z$

Suppose $X$ is a random variable with mean $\mu$ and variance $\sigma^2.$ Let $$Z = \frac{X-\mu}{\sigma}.$$

Derive the expected value and variance of $Z$. Remember to justify all non-algebraic steps.

I thought that I could just plug in the value of $Z$ and find the expected value, but that's not it I don't think. This is just stumping me.

1 Answer

This smells like homework so I won't post entire solution (although it is almost complete). \begin{align} E(Z)&=\dfrac{E(X-\mu )}{\sigma}\\ &=\dfrac{E(X)-\mu}{\sigma}\\ &=?\\ Var(Z)&=Var(\dfrac{X-\mu}{\sigma})\\ &=\dfrac{1}{\sigma^2}Var(X-\mu)\\ &=\dfrac{1}{\sigma^2}Var(X)\\ &=? \end{align}