# Proving that the Expected value of Z =0

• Mar 20th 2009, 10:20 PM
mibamars
Proving that the Expected value of Z =0
If $\displaystyle Z = (X - MEAN) / Standard Deviation$

Prove that the expected value of Z = 0

and that the variance of Z = 1
• Mar 20th 2009, 11:26 PM
Moo
Hello,
Quote:

Originally Posted by mibamars
If $\displaystyle Z = (X - MEAN) / Standard Deviation$

Prove that the expected value of Z = 0

and that the variance of Z = 1

Let $\displaystyle E(X)=m$ and $\displaystyle Var(X)=\sigma^2=v$
You want to prove that $\displaystyle Z=\frac{X-m}{\sigma}$ has an expected value of 0 and a variance of 1.

Use these properties :
(a is a constant)
$\displaystyle E(aX)=aE(X)$ (1)
$\displaystyle E(X+a)=E(X)+a$ (2)
$\displaystyle Var(aX)=a^2Var(X)$ (3)
$\displaystyle Var(X+a)=Var(X)$ (4)

$\displaystyle E(Z)=E \left(\tfrac{X-m}{\sigma}\right)=\frac{1}{\sigma} E(X-m)=\frac{1}{\sigma} [E(X)-m]$
But since $\displaystyle E(X)=m$, we can conclude that E(Z)=0.

Now try to do it with the variance. It's easy with the formulae I gave above !