Illustrating the Central Limit Theorem with Sums of Uniform and Exponential Random Variables
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The exact probability density function (PDF) of standardized sums of 
uniform 
or unit exponential variables is compared with the standard normal density. As 
, these PDFs converge to the standard normal PDF (central limit theorem).
Contributed by: Ian McLeod (March 2011)
Open content licensed under CC BY-NC-SA
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[1, p. 64] shows that the cumulative distribution function for the sum 
of 
independent uniform 
random variables, 
, is
.
Taking the derivative, we obtain the PDF of 
. In the case of the unit exponential, the PDF of 
is the gamma distribution with shape parameter 
and scale parameter 
. In each case we compare the standard normal PDF with the PDF of 
, where 
and 
are the mean and standard deviation of 
, respectively.
[1] N. L. Johnson and S. Kotz, Continuous Univariate Distributions, Vol. 2, Boston: Houghton Mifflin, 1970.
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