A Geometric Derivation of the Irwin-Hall Distribution

James E Marengo,Lucas Stefanic,David L Farnsworth

doi:10.1155/2017/3571419

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.

Highlights

The simple continuous uniform or rectangular distribution Uniform(0, 1) with probability density function (PDF) f(x) = 1 for 0 < x < 1 and f(x) = 0 otherwise is very important
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])
Instead of characteristic functions, Gray and Odell [28] found the distribution of any linear combination of uniform random variables with different domains allowed

Summary

Introduction

The present goal is to derive the CDF and the PDF of the sum T = ∑ni=1 Xi, where Xi are independent identically distributed Uniform(0, 1) random variables for i = 1, 2, . 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]). Instead of characteristic functions, Gray and Odell [28] found the distribution of any linear combination of uniform random variables with different domains allowed. Since round-off errors for random variables that are rounded to the nearest integer are distributed Uniform(−1/2, 1/2), the sum of round-off errors is a linearly transformed Irwin-Hall distribution [12]. Linear combinations of independent random variables whose domains have an upper bound are given in [37]

Derivation of the Irwin-Hall Distribution

Discussion and a Generalization