# 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}

• Thanks so much, I thought that was what I had to do. I just needed to be reminded about the properties of expected values. – Caty Feb 13 '13 at 7:34
• @Caty: If I may point to another feature of this site, if you liked a question or answer, you can upvote it (top left next to the answer/question); and also accept an answer that you liked best. The person having submitted such question/answer will be rewarded with some reputation (which, for many, is part of the fun :) ). For accepting an answer, you also get a few points. – gnometorule Feb 13 '13 at 7:46
• how did the omega squared term appear in the second line of your Var(Z) solution? – sweetmusicality Sep 11 '17 at 23:27