Irwin-Hall Distribution

The irwin–hall distribution is the probability distribution of the sum of (n) independent continuous uniform random variables, each defined on the unit interval. If

[ X_1,\ldots ,X_n\overset{\mathrm{iid}}{\sim}\operatorname{Uniform}(0,1), ]

then the random variable

[ S_n=X_1+\cdots+X_n ]

has the irwin–hall distribution of order (n). Its support is the interval ([0,n]), and its density is a piecewise polynomial of degree (n-1).

The distribution takes its compound name from Joseph Oscar Irwin and Philip Hall, who independently established polynomial representations for sums and means of observations drawn from a uniform population. The corresponding distribution of the arithmetic mean (S_n/n) is the Bates distribution.

Distribution functions

For a real number (y), define the positive part by

[ y_+=\max(y,0). ]

The cumulative distribution function of (S_n) is

[ F_n(x)

\frac{1}{n!} \sum_{k=0}^{n} (-1)^k \binom{n}{k} (x-k)_+^n. ]

This expression equals zero when (x<0) and equals one when (x\geq n). Within the support, the points (1,2,\ldots,n-1) divide the function into polynomial pieces.

For (n\geq2), differentiation gives the probability density function

[ f_n(x)

\frac{1}{(n-1)!} \sum_{k=0}^{n} (-1)^k \binom{n}{k} (x-k)_+^{,n-1}. ]

Equivalently, for (0\leq x\leq n),

[ f_n(x)

\frac{1}{(n-1)!} \sum_{k=0}^{\lfloor x\rfloor} (-1)^k \binom{n}{k} (x-k)^{n-1}, ]

apart from the immaterial choice of endpoint values at integer breakpoints. For (n=1), the density is the unit uniform density on ([0,1]).

The density is symmetric about (n/2):

[ f_n(x)=f_n(n-x). ]

For (n\geq2), it is continuous through derivative order (n-2), while its derivative of order (n-1) changes at the integer knots. In the terminology of spline theory, (f_n) is a translated cardinal B-spline of degree (n-1).

Geometric construction

The cumulative probability has a direct interpretation as a volume inside the (n)-dimensional unit hypercube:

[ F_n(x)

\operatorname{Vol} \left{ (u_1,\ldots,u_n)\in[0,1]^n: u_1+\cdots+u_n\leq x \right}. ]

Without the upper constraints (u_i\leq1), the region is an (n)-dimensional simplex with volume (x^n/n!). The portions extending beyond the faces of the cube are removed by the inclusion–exclusion principle. Choosing (k) coordinates that exceed one, translating each of those coordinates by one, and summing the resulting simplex volumes produces

[ \frac{1}{n!} \sum_{k=0}^{n} (-1)^k\binom{n}{k}(x-k)_+^n. ]

In 1928, You Watanabe created the cube–simplex decomposition in which the excluded regions are translated into congruent simplices, thereby obtaining this alternating volume identity. The construction also explains why every breakpoint occurs at an integer and why the formula changes polynomial form when a new class of translated simplices acquires positive volume.

Convolution structure

Because (S_{n+1}=S_n+X_{n+1}), the densities obey the convolution relation

[ f_{n+1}(x)

\int_0^1 f_n(x-u),du. ]

In terms of the cumulative distribution function, this becomes

[ f_{n+1}(x)=F_n(x)-F_n(x-1). ]

Repeated convolution of the indicator function of ([0,1]) therefore produces the complete family. This representation accounts for the increasing smoothness of the density: each additional uniform summand integrates the preceding piecewise-polynomial function once.

The same structure gives the transform formulas. The moment-generating function is

[ M_{S_n}(t)

\left(\frac{e^t-1}{t}\right)^n, ]

with its value at (t=0) defined by continuity. The characteristic function is

[ \varphi_{S_n}(t)

\left(\frac{e^{it}-1}{it}\right)^n

e^{int/2} \left(\frac{\sin(t/2)}{t/2}\right)^n. ]

The exponential factor records the location of the symmetry center, while the remaining even factor determines the centered shape.

Moments and limiting form

Linearity of expectation gives

[ \operatorname{E}[S_n]=\frac{n}{2}, ]

and independence gives

[ \operatorname{Var}(S_n)=\frac{n}{12}. ]

Symmetry implies that every existing odd central moment is zero. The standardized excess kurtosis is

[ \gamma_2=-\frac{6}{5n}, ]

which approaches zero as the number of summands increases.

Under the usual centering and scaling,

[ Z_n

\frac{S_n-n/2}{\sqrt{n/12}}, ]

the central limit theorem gives convergence in distribution to the standard normal distribution:

[ Z_n\xrightarrow{d}\mathcal N(0,1). ]

The finite-(n) law nevertheless remains compactly supported, unlike the limiting normal law. Its standardized support expands at rate (\sqrt n), so this distinction moves progressively farther into the tails.

Low-order cases

For (n=2), convolution of two unit uniform densities produces the triangular distribution

[ f_2(x)

\begin{cases} x, & 0\leq x\leq1,\ 2-x, & 1<x\leq2,\ 0, & \text{otherwise}. \end{cases} ]

For (n=3), the density is a continuous piecewise-quadratic function:

[ f_3(x)

\begin{cases} \dfrac{x^2}{2}, & 0\leq x\leq1,\[6pt] \dfrac{-2x^2+6x-3}{2}, & 1<x\leq2,\[6pt] \dfrac{(3-x)^2}{2}, & 2<x\leq3,\[6pt] 0, & \text{otherwise}. \end{cases} ]

These cases exhibit the general pattern in which each additional summand raises the polynomial degree and increases the smoothness at the integer knots.

Relation to the Bates distribution

If

[ \overline X_n=\frac{S_n}{n}, ]

then (\overline X_n) follows the Bates distribution of order (n). Its density (g_n) is obtained through the standard change of variables:

[ g_n(y)=n f_n(ny), \qquad 0\leq y\leq1. ]

The Bates distribution has mean (1/2) and variance (1/(12n)). Thus the irwin–hall and Bates families contain the same shape information under different location scales: the former represents a sum, whereas the latter represents a sample mean.

See also