Irwin–Hall distribution

The Irwin–Hall distribution is the continuous probability distribution for the sum of n independent and identically distributed U(0, 1) random variables: : X = \sum_{k=1}^n U_k. The probability density function (pdf) for 0\leq x\leq n is given by : f_X(x;n)=\frac{1}{(n-1)!}\sum_{k=0}^n (-1)^k{n \choose k} (x-k)_+^{n-1} where (x-k)_+ denotes the positive part of the expression: : (x-k)_+ = \begin{cases} x-k & x-k \geq 0 \\ 0 & x-k Since k is an integer, we have that (x-k)_+ = (x- k) if and only if k \leq \lfloor x \rfloor. Hence, a completely equivalent expression of the pdf for 0 \leq x \leq n is given by : 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) = \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 : a_j(k,n)=\begin{cases} 1&k=0, j=n-1\\ 0&k=0, j0\end{cases} The coefficients are also A188816 in OEIS. The coefficients for the cumulative distribution is A188668. For a geometric/combinatorial interpretation of the pdf in terms of a plane passing through a cube, please see here. The mean and variance are n/2 and n/12, respectively. ==Special cases==

• For n = 1, X follows a uniform distribution: :: f_X(x)= \begin{cases} 1 & 0\le x \le 1 \\ 0 & \text{otherwise} \end{cases} • For n = 2, X follows a triangular distribution: :: f_X(x)= \begin{cases} x & 0\le x \le 1\\ 2-x & 1\le x \le 2 \end{cases} • For n = 3, :: f_X(x)= \begin{cases} \frac{1}{2}x^2 & 0\le x \le 1\\ \frac{1}{2}(-2x^2 + 6x - 3)& 1\le x \le 2\\ \frac{1}{2}(3 - x)^2 & 2\le x \le 3 \end{cases} • For n = 4, :: f_X(x)= \begin{cases} \frac{1}{6}x^3 & 0\le x \le 1\\ \frac{1}{6}(-3x^3 + 12x^2 - 12x+4)& 1\le x \le 2\\ \frac{1}{6}(3x^3 - 24x^2 +60x-44) & 2\le x \le 3\\ \frac{1}{6}(4 - x)^3 & 3\le x \le 4 \end{cases} • For n = 5, :: f_X(x)= \begin{cases} \frac{1}{24}x^4 & 0\le x \le 1\\ \frac{1}{24}(-4x^4 + 20x^3 - 30x^2+20x-5)& 1\le x \le 2\\ \frac{1}{24}(6x^4-60x^3+210x^2-300x+155) & 2\le x \le 3\\ \frac{1}{24}(-4x^4+60x^3-330x^2+780x-655) & 3\le x \le 4\\ \frac{1}{24}(5 - x)^4 &4\le x\le5 \end{cases} == Approximating a normal distribution ==

By the central limit theorem, as n increases, the Irwin–Hall distribution more and more strongly approximates a normal distribution with mean \mu=n/2 and variance \sigma^2=n/12. To approximate the standard Normal distribution \phi(x)=\mathcal{N}(\mu=0, \sigma^2=1), the Irwin–Hall distribution can be centered by shifting it by its mean of n/2, and scaling the result by the square root of its variance: : \phi(x) \overset{n\gg 0}{\approx} \sqrt{\frac{n}{12}} f_X\left(x\sqrt{\frac{n}{12}}+\frac{n}{2};n \right ) This derivation leads to a computationally simple heuristic that removes the square root, whereby a standard normal distribution can be approximated with the sum of 12 uniform U(0, 1) draws like so: : \sum_{k=1}^{12}U_k -6 \sim f_X(x+6;12) \mathrel{\dot\sim} \phi(x) == Similar and related distributions ==

The Irwin–Hall distribution is similar to the Bates distribution, but still featuring only integers as parameter. An extension to real-valued parameters is possible by adding also a random uniform variable with N − trunc(N) as width. ==Extensions to the Irwin–Hall distribution==

When using the Irwin–Hall for data fitting purposes one problem is that the IH is not very flexible because the parameter n needs to be an integer. However, instead of summing n equal uniform distributions, we could also add e.g. U + 0.5U to address also the case n = 1.5 (giving a trapezoidal distribution). The Irwin–Hall distribution has an application to beamforming and pattern synthesis as shown in Figure 1 of reference. == See also ==