In probability and statistics, the Irwin–Hall distribution, named after Joseph Oscar Irwin and Philip Hall, is a probability distribution for a random variable defined as the sum of a number of independent random variables, each having a uniform distribution. For this reason it is also known as the uniform sum distribution. The generation of pseudo-random numbers having an approximately normal distribution is sometimes accomplished by computing the sum of a number of pseudo-random numbers having a uniform distribution; usually for the sake of simplicity of programming. Rescaling the Irwin–Hall distribution provides the exact distribution of the random variates being generated. This distribution is sometimes confused with the Bates distribution, which is the mean (not sum) of n independent random variables uniformly distributed from 0 to 1.
Definition The Irwin–Hall distribution is the continuous probability distribution for the sum of n independent and identically distributed U(0, 1) random variables:
X = ∑ k = 1 n U k . {\displaystyle X=\sum _{k=1}^{n}U_{k}.}
The probability density function (pdf) for 0 ≤ x ≤ n {\displaystyle 0\leq x\leq n} is given by
f X ( x ; n ) = 1 ( n − 1 ) ! ∑ k = 0 n ( − 1 ) k ( n k ) ( x − k ) + n − 1 {\displaystyle f_{X}(x;n)={\frac {1}{(n-1)!}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x-k)_{+}^{n-1}}
where ( x − k ) + {\displaystyle (x-k)_{+}} denotes the positive part of the expression:
( x − k ) + = { x − k x − k ≥ 0 0 x − k < 0. {\displaystyle (x-k)_{+}={\begin{cases}x-k&x-k\geq 0\\0&x-k<0.\end{cases}}}
Since k {\displaystyle k} is an integer, we have that ( x − k ) + = ( x − k ) {\displaystyle (x-k)_{+}=(x-k)} if and only if k ≤ ⌊ x ⌋ {\displaystyle k\leq \lfloor x\rfloor } . Hence, a completely equivalent expression of the pdf for 0 ≤ x ≤ n {\displaystyle 0\leq x\leq n} is given by
f X ( x ; n ) = 1 ( n − 1 ) ! ⋅ ∑ k = 0 ⌊ x ⌋ ( − 1 ) k ( n k ) ( x − k ) n − 1 . {\displaystyle f_{X}(x;n)={\frac {1}{(n-1)!}}\cdot \sum _{k=0}^{\lfloor x\rfloor }(-1)^{k}{\binom {n}{k}}(x-k)^{n-1}.}
Thus the pdf is a spline (piecewise polynomial function) of degree n − 1 over the knots 0, 1, ..., n. In fact, for x between the knots located at k and k + 1, the pdf is equal to
f X ( x ; n ) = 1 ( n − 1 ) ! ∑ j = 0 n − 1 a j ( k , n ) x j {\displaystyle f_{X}(x;n)={\frac {1}{(n-1)!}}\sum _{j=0}^{n-1}a_{j}(k,n)x^{j}}
where the coefficients aj(k,n) may be found from a recurrence relation over k
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