Irwin-Hall distribution

In probability theory and statistics, the Irwin-Hall distribution is a continuous probability distribution of the sum of "n" i.i.d. "U"(0,1) random variables:

:$$X = sum_{k=1}^n U_k.

For this reason it is also known as the uniform sum distribution. The probability density function (pdf) is given by

:$$f_X(x)=frac{1}{2left(n-1 ight)!}sum_{k=0}^{n}left(-1 ight)^k{n choose k}left(x-k ight)^{n-1}sgn(x-k)

where sgn("x − k") denotes the sign function:

:$$sgnleft(x-k ight) = left{ egin{matrix} -1 & : & x < k \0 & : & x = k \1 & : & x > k. end{matrix} ight.

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) = frac{1}{left(n-1 ight)!}sum_{j=0}^{n-1} a_j(k,n) x^j

where the coefficients "a_j(k,n)" may be found from a recurrence relation over "k"

:$$a_j(k,n)=egin{cases}I(j=n-1)&k=0\a_j(k-1,n) + left(-1 ight)^{n+k-j-1}{nchoose k}n-1}choose j}k^{n-j-1} &k>0end{cases}

The mean and variance are "n"/2 and "n"/12, respectively.

pecial cases

* For "n" = 1, "X" follows a uniform distribution::$$f_X(x)= egin{cases}1 & 0le x le 1end{cases}
* For "n" = 2, "X" follows a triangular distribution::$$f_X(x)= egin{cases}x & 0le x le 1\2-x & 1le x le 2end{cases}
* For "n" = 3,:$$f_X(x)= egin{cases}frac{1}{2}x^2 & 0le x le 1\frac{1}{2}left(-2x^2 + 6x - 3 ight)& 1le x le 2\frac{1}{2}left(x^2 - 6x +9 ight) & 2le x le 3end{cases}
* For "n" = 4,:$$f_X(x)= egin{cases}frac{1}{6}x^3 & 0le x le 1\frac{1}{6}left(-3x^3 + 12x^2 - 12x+4 ight)& 1le x le 2\frac{1}{6}left(3x^3 - 24x^2 +60x-44 ight) & 2le x le 3\frac{1}{6}left(-x^3 + 12x^2 -48x+64 ight) & 3le x le 4end{cases}
* For "n" = 5,:$$f_X(x)= egin{cases}frac{1}{24}x^4 & 0le x le 1\frac{1}{24}left(-4x^4 + 20x^3 - 30x^2+20x-5 ight)& 1le x le 2\frac{1}{24}left(6x^4-60x^3+210x^2-300x+155 ight) & 2le x le 3\frac{1}{24}left(-4x^4+60x^3-330x^2+780x-655 ight) & 3le x le 4\frac{1}{24}left(x^4-20x^3+150x^2-500x+625 ight) &4le xle5end{cases}

References

* Hall, Philip. (1927) "The Distribution of Means for Samples of Size N Drawn from a Population in which the Variate Takes Values Between 0 and 1, All Such Values Being Equally Probable". "Biometrika", Vol. 19, No. 3/4., pp. 240-245.
* Irwin, J.O. (1927) "On the Frequency Distribution of the Means of Samples from a Population Having any Law of Frequency with Finite Moments, with Special Reference to Pearson's Type II". "Biometrika", Vol. 19, No. 3/4., pp. 225-239.