- Irwin-Hall distribution
In
probability theory andstatistics , the Irwin-Hall distribution is a continuousprobability 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 andvariance are "n"/2 and "n"/12, respectively.pecial cases
* For "n" = 1, "X" follows a uniform distribution::
* For "n" = 2, "X" follows atriangular 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.
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