A Geometric Derivation of the Irwin-Hall Distribution

Received May 23, 2017; Revised Aug 4, 2017; Accepted Aug 17, 2017

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1. Introduction

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. Two applications arise in numerical simulation and Bayesian analysis of proportions. If F is the cumulative distribution function (CDF) of the continuous random variable X, then the random variable Y=F(X) has a Uniform(0,1) distribution. The random variable X can be simulated by first simulating Y and then letting X=F–1(Y). 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 T=∑i=1nXi, where Xi are independent identically distributed Uniform(0, 1) random variables for i=1,2,…,n. The CDF and PDF are(1)Ft=∑i=0n-1init-inn!sit,(2)ft=∑i=0n-1init-in–1n-1!sit,respectively, where sa(t) is the unit step function(3)sat=0t<a1a≤t.The...