If $\displaystyle Z = (X - MEAN) / Standard Deviation $

Prove that the expected value of Z = 0

and that the variance of Z = 1

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- Mar 20th 2009, 10:20 PMmibamarsProving 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 PMMoo
Hello,

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 !