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:

:

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"

: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::
* For "n" = 2, "X" follows a triangular distribution::
* For "n" = 3,:
* For "n" = 4,:
* For "n" = 5,:

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.