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).


For the uniform case, convergence occurs quickly. In the case of the exponential, convergence is less rapid.


Contributed by: Ian McLeod (March 2011)
Open content licensed under CC BY-NC-SA



[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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