251probex2 12/01/06 (Open this document in 'Page Layout' view!)
Standardized Random Variables
A standardized random variable (often called a z-score) is gotten from any other random variable by the
x
formula z 
. You should know what the expected value (mean) and variance of such a variable will

always be. This is explained below.
1
1 
A standardized random variable can be written in the form z  ax  b    x    , where a 
and

   
b

.


1
So, from the formulas in the outline  z  Eax  b  aEx   b       
 
 
 
 
 1 
 
1
       0 , and Varz    z2  Varax  b  a 2Varx      2   2   2  1 . (  z  1 .)



   
 
 
So a standardized random variable always has a mean of zero and a standard deviation of 1.
2