The **Irwin-Hall distribution** (*Uniform Sum Distribution*) is the distribution of the sum of *n *values taken from the uniform distribution U(0, 1). It is similar to the Bates distribution, which is the distribution of the *mean *of *n *values. The distribution is named after two proofs, given by Irwin (1927) and Hall (1927).

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 (Batsyn & Kalyagin, 2012).

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 (Marengo, 2017).

## PDF of the Irwin-Hall Distribution

The probability density function (PDF) of the Irwin-Hall distribution is given by (Florescu, 2014):

Where sign(x) is the sign function.

## Mean, Variance and Kurtosis

The mean of the Irwin-Hall distribution is *n */ 2 and the variance is *n */ 12. Irwin-Hall distributions are platykurtic; For large n, the kurtosis is near to 3 (Marengo, 2017).

## References

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.

Florescu, I. (2014). Probability and Stochastic Processes 1st Edition. Wiley.

Flury, B. A First Course in Multivariate Statistics (Springer Texts in Statistics) 1997 Edition.

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.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.

Biometrika, Vol. 19, No. 3/4. pp. 225–239.

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