International Journal of Mathematics and Mathematical Sciences

Volume 2017, Article ID 3571419, 6 pages

https://doi.org/10.1155/2017/3571419

## A Geometric Derivation of the Irwin-Hall Distribution

School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, USA

Correspondence should be addressed to David L. Farnsworth; ude.tir@amsfld

Received 23 May 2017; Revised 4 August 2017; Accepted 17 August 2017; Published 18 September 2017

Academic Editor: Shyam L. Kalla

Copyright © 2017 James E. Marengo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Irwin-Hall distribution is the distribution of the sum of a finite number of independent identically distributed uniform random variables on the unit interval. Many applications arise since round-off errors have a transformed Irwin-Hall distribution and the distribution supplies spline approximations to normal distributions. We review some of the distribution’s history. The present derivation is very transparent, since it is geometric and explicitly uses the inclusion-exclusion principle. In certain special cases, the derivation can be extended to linear combinations of independent uniform random variables on other intervals of finite length. The derivation adds to the literature about methodologies for finding distributions of sums of random variables, especially distributions that have domains with boundaries so that the inclusion-exclusion principle might be employed.

#### 1. Introduction

The simple continuous uniform or rectangular distribution Uniform(0, 1) with probability density function (PDF) for and otherwise is very important. Two applications arise in numerical simulation and Bayesian analysis of proportions. If is the cumulative distribution function (CDF) of the continuous random variable , then the random variable has a Uniform() distribution. The random variable can be simulated by first simulating and then letting . This is called the* inversion method* ([1, page 295], [2, pages 194–196]). The transformation is called the* probability integral transformation* ([3], [4, pages 203-204]). The uniform distribution is a Bayesian* noninformative prior distribution* for the distribution of a random variable defined on the unit interval, such as a beta distribution for a proportion ([2, page 33], [5, pages 82–90]). For other applications and generalizations of the uniform distribution, see [6–8].

The present goal is to derive the CDF and the PDF of the sum , where are independent identically distributed Uniform(0, 1) random variables for . The CDF and PDF arerespectively, where is the unit step functionThe derivation in Section 2 is geometric and explicitly uses the inclusion-exclusion principle.

Derivations of the distribution, which more recently acquired its name Irwin-Hall, go back to Lagrange and Laplace in the latter 18th century and the early 19th century. Lagrange used generating functions based on to obtain the distribution of* T* ([9, pages 603–612], [10, page 283]). Those generating functions are a predecessor of characteristic functions [10, page 286]. Laplace often revisited the problem of finding the distribution of and employed many methods ([9, pages 714-715], [10, pages 286–301]). The distribution is described in [1, pages 296–300], where it is called the* Irwin-Hall distribution*.

Some derivations employ characteristic functions in a variety of ways, since the characteristic function of a sum of independent random variables is the product of each summand’s characteristic function and the inverse transform is not intractable ([11, pages 188-189], [12–14], [15, pages 362-363], [16, 17]). Others utilize the convolution integral for sums and mathematical induction ([4, page 225], [11, pages 190-191 and 244–246], [18]). The distribution of the sum of uniform random variables that may have differing domains is found in [18–21]. Sums of dependent uniform random variables are examined in [22, 23].

Direct integration techniques can be used to obtain the distribution of a linear combination of Uniform(0, 1) random variables ([15, pages 358–360], [24, 25]). Similar techniques are used in [26] for uniform distributions whose domains are intervals with zero as their left endpoints. The distribution of the mean is obtained when all the constants are . In this case, the distribution is called the* Bates distribution* ([1, page 297], [27]), which can also be found by a simple transformation of the Irwin-Hall distribution ([15, page 359], [25, page 241]). Using moment generating functions, instead of characteristic functions, Gray and Odell [28] found the distribution of any linear combination of uniform random variables with different domains allowed. In Section 3, the present method or style of proof is extended to those cases giving the same distributions.

Because is a sum, the Irwin-Hall distribution approximates a normal distribution with a spline, since the Irwin-Hall distribution in (2) is composed of polynomials. The support of is the interval []; the mean, mode, and median of are ; and its variance is . By symmetry, all odd central moments are zero, including skewness. The kurtosis is [1, page 300]. This is the measure of kurtosis that is 3 for a normal distribution, so Irwin-Hall distributions are platykurtic, and the kurtosis is close to 3 for large . According to the Central Limit Theorem,([4, pages 280–283], [11, pages 213–218 and 245], [29, pages 220–222]). Figure 1 contains a normal distribution with mean and variance and its approximating Irwin-Hall distribution with . The approximation is very good even for this small value of [30]. The uniform error bound for the normal(0, 1) CDF is([31], [32, page 51]). Approximations with spline fitting can be useful with or without complete information about the distributional shape [33, 34].