Irwin–Hall Distribution.

< List of probability distributions

The Irwin-Hall distribution, also known as the Uniform Sum Distribution, is a powerful mathematical tool with many practical applications. Named after proofs provided by Irwin and Hall in 1927 [1,2], it helps to determine sums of random variables within problems such as statistics or probability distributions. As one of its most useful features, this type of distribution offers simplicity for mathematicians tackling various issues across disciplines like data science and finance.

More precisely, the Irwin-Hall distribution is the distribution of the sum of values taken from the uniform distribution on the interval (0, 1).

A related distribution is the Bates distribution — which is the distribution of mean values of n — instead of the sum.

Irwin-Hall distribution properties

Probability density function (PDF):

irwin-hall distribution PDF

Mean: n/2

Variance: n/12

Kurtosis: ≈ 3 [3]

Or, more precisely: 3 – (6/5n)

Irwin-Hall distribution usage

Many problems in applied mathematics require the calculation of the distribution of the sum of independent random uniform variables, including aggregating scaled values, change point analysis and working with data drawn from measurements with different precision levels.

The Irwin-Hall distribution is also used to approximate a normal distribution and demonstrate the central limit theorem: as n increases, the distribution will rapidly start to look like a normal distribution.

As n increases, the distribution starts to look like the normal distribution. Credit: Thomasda: Wikimedia Commons

The generation of pseudo-random numbers  having an approximately normal distribution is sometimes accomplished by computing the sum of a number of pseudo-random numbers having a uniform distribution; usually for the sake of simplicity of programming. Rescaling the Irwin–Hall distribution provides the exact distribution of the random variates being generated.


Image: Thomasda | Wikimedia Commons CC BY-SA 3.0 File:Irwin-hall-pdf.svg

[1] Hall, P. (1927). The distribution of means for samples of size n drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika, Vol. 19, No. 3/4. pp. 240–245.

[2] Irwin J.(1927). On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II.

[3] Marengo, J et al. (2017). A Geometric Derivation of the Irwin-Hall Distribution. International Journal of Mathematics and Mathematical Sciences.