Simulating a Normal Process from Sums of Uniform Distributions
A convenient simulation of a random normal process comes from a sum of random uniform variables. The probability density function (pdf) of sums of random variables is the convolution of their pdfs. Sums of uniform random variables can be seen to approach a Gaussian distribution.
This simulation compares the pdf resulting from a chosen number of uniform pdfs to a normal distribution. The top plot shows the probabilities for a simulated sample. The bottom graphic is a quantile plot of the sample compared to the normal distribution. For a sum of 12 uniform random variables, the distribution is approximately normal with a standard deviation near 1.
Apart from the parameter for the number of uniform distributions to be summed (the slider), you can control the discretization of the distribution and the sample size of the simulation. The uniform distribution is discrete in all cases in this simulation on a grid with points for a few choices of . The sample size varies between 10 and 5000 points.
The statistics of distributions that are the sums of elementary distributions is detailed in most statistics textbooks. For example:
A. M. Mood, F. A. Graybill, and D. C. Boes, Introduction to the Theory of Statistics, New York: McGraw-Hill, 1974.