|Probability density function
|Cumulative distribution function
|Parameters||n ∈ N0|
In probability and statistics, the Irwin–Hall distribution, named after Joseph Oscar Irwin and Philip Hall, is probability distribution for a random variable defined as sum of a number of independent random variables, each having a uniform distribution. For this reason it is also known as the uniform sum distribution.
The generation of pseudo-random numbers having an approximately normal distribution is sometimes accomplished by computing the sum of a number of pseudo-random numbers having a uniform distribution; usually for the sake of simplicity of programming. Rescaling the Irwin–Hall distribution provides the exact distribution of the random variates being generated.
This distribution is sometimes confused with the Bates distribution, which is the mean (not sum) of n independent random variables uniformly distributed from 0 to 1.
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
Special 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,
- Jonhson, N.L.; Kotz, S.; Balakrishnan (1995) Continuous Univariate Distributions, Volume 2, 2nd Edition, Wiley ISBN 0-471-58494-0(Section 26.9)
- 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.