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

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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).

[more]
Contributed by: Ian McLeod (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

[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.

## Permanent Citation