|
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:
For this reason it is also known as the uniform sum distribution. The probability density function (pdf) is given by
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
where the coefficients aj(k,n) may be found from a recurrence relation over k
The mean and variance are n/2 and n/12, respectively.
Special 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.
| |
