Thm:         Given independent and identically distributed random variables Xi for i = 1 to n, we define the normalized sample average, Z, as follows:

where .

Then as n approaches infinity, the probability density function for Z approaches a standard normal (i.e., standard gaussian) distribution, (see [1]):

 

 

 

 

 

 

 

 

 

Ref:    [1] Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, and Keying Ye, Probability and Statistics for Engineers and Scientists, 8th Ed., Upper Saddle River, NJ: Prentice Hall, 2007.