# 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).

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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.