Probability Distributions > Irwin-Hall Distribution
What is the Irwin-Hall Distribution?
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.
A related distribution is the Bates distribution — which is the distribution of mean values of n — instead of the sum. The two rectangulars added distribution is connected to the Irwin-Hall distribution, but its exact properties remain a mystery.
This distribution has many applications in applied mathematics, in part because of its simplicity. For example, sums of random variables need to be calculated in problems dealing with:
- Aggregating scaled values with varying significant figures,
- Change point analysis,
- Data drawn from measurements with different levels of precision [3].
For small n, the distribution is very simple and follows a uniform distribution (n = 1) or triangular distribution (n = 2). It can give spline approximations to normal distributions [4].
Irwin-Hall distribution properties
The Irwin-Hall distribution is the distribution of the sum of n values taken from the uniform distribution on the interval (0, 1).
The probability density function (PDF) of the Irwin-Hall distribution is given by [5]:
Where sign(x) is the sign function.
Mean: n/2
Variance: n/12
Kurtosis: ≈ 3 , or more precisely: 3 – (6/5n)
Irwin-Hall distributions are platykurtic; For large n, the kurtosis is near to 3 [4].
Two rectangulars added distribution
The “Two rectangulars added distribution” seems to be lost to history, although it is likely connected to the Irwin-Hall distribution.
The entry for “two rectangulars Added” in Haight’s 1958 Index to the Distributions of Mathematical Statistics [1] refers to two rectangular distributions added together. Haight’s book does not have a formula. Instead the entry states: Two rectangulars added: [n]8-3:74. This notation refers to an article published in the journal Metron in 1930 by British statistician Joseph Oscar Irwin titled “On the Frequency distributions of means, etc.” in which Irwin gave a distribution of arithmetic means of samples of size n from a rectangular universe.
Three years earlier, Irwin had published “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,” which led to the development of the Irwin-Hall distribution (IHD), which is the sum of n independent random variables uniformly distributed from 0 to 1. According to Craig [7], Irwin extended his method of integral equations to samples from Pearson Type I and VII curves. As integrals are mentioned, it’s possible “two rectangulars added” may be related to the Irwin-Hall distribution, which is continuous.
However, as Volume 8 of Metron isn’t anywhere to be found (except perhaps, in an uncatalogued basement in Rome), the original formula isn’t available. Therefore, it’s impossible to say for sure that the “two rectangulars added” is another name for the Irwin-Hall distribution. The fact that [1] also contains a separate entry for the IHD suggests that they are different distributions.
If anyone has access to a copy of Volume 8 of Metron, please let me know.
Irwin is well-known for other contributions to mathematics. For example, he independently developed an exact probability test for 2×2 contingency tables which we now call Fisher’s exact probability test.
References
- Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.
- Irwin, J.O. (1930). On the Frequency distributions of means, etc. Metron Vol 8, issue 3, pp 58-105.
- Batsyn, M. & Kalyagin, V. (2012). An Analytical Expression for the Distribution of the Sum of Random Variables with a Mixed Uniform Density and Mass Function.
- Marengo, J et al. (2017). A Geometric Derivation of the Irwin-Hall Distribution. International Journal of Mathematics and Mathematical Sciences.
- Florescu, I. (2014). Probability and Stochastic Processes 1st Edition. Wiley.
- Image: Thomasda|Wikimedia Commons. CC 4.0.
- Craig, A. T. (1932). On the Distributions of Certain Statistics. American Journal of Mathematics, 54(2), 353–366. https://doi.org/10.2307/2371000