Irwin-Hall distribution

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In probability theory and statistics, the Irwin-Hall distribution is a continuous probability distribution corresponding to the 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}{2\left(n-1\right)!}\sum_{k=0}^{n}\left(-1\right)^k{n \choose k}\left(x-k\right)^{n-1}\sgn(x-k)

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

\sgn\left(x-k\right) = \left\{ \begin{matrix} -1 & : & x < k \\ 0 & : & x = k \\ 1 & : & x > k. \end{matrix} \right.

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\right)!}\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}I(j=n-1)&k=0\\ a_j(k-1,n) + \left(-1\right)^{n+k-j-1}{n\choose k}{{n-1}\choose j}k^{n-j-1} &k>0\end{cases}

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

[edit] Special cases

f_X(x)= \begin{cases} \frac{1}{2}x^2 & 0\le x \le 1\\ \frac{1}{2}\left(-2x^2 + 6x - 3 \right)& 1\le x \le 2\\ \frac{1}{2}\left(x^2 - 6x +9 \right) & 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}\left(-3x^3 + 12x^2 - 12x+4 \right)& 1\le x \le 2\\ \frac{1}{6}\left(3x^3 - 24x^2 +60x-44 \right) & 2\le x \le 3\\ \frac{1}{6}\left(-x^3 + 12x^2 -48x+64 \right) & 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}\left(-4x^4 + 20x^3 - 30x^2+20x-5 \right)& 1\le x \le 2\\ \frac{1}{24}\left(6x^4-60x^3+210x^2-300x+155 \right) & 2\le x \le 3\\ \frac{1}{24}\left(-4x^4+60x^3-330x^2+780x-655 \right) & 3\le x \le 4\\ \frac{1}{24}\left(x^4-20x^3+150x^2-500x+625\right) &4\le x\le5 \end{cases}

[edit] 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.