Illustrating the Central Limit Theorem with Sums of Uniform and Exponential Random Variables

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.

[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.