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The Bates distribution is sometimes confused [2] with the Irwin–Hall distribution, which is the distribution of the sum (not the mean) of n independent random variables uniformly distributed from 0 to 1. If X has a Bates distribution on the unit interval, then n X has an Irwin-Hall distribution; when n = 1 they are both uniformly distributed.
If X has a standard uniform distribution, then Y = X n has a beta distribution with parameters (1/n,1). As such, The Irwin–Hall distribution is the sum of n i.i.d. U(0,1) distributions. The Bates distribution is the average of n i.i.d. U(0,1) distributions. The standard uniform distribution is a special case of the beta distribution, with ...
The Bates distribution is the distribution of the mean of n independent random variables, each of which having the uniform distribution on [0,1]. The logit-normal distribution on (0,1). The Dirac delta function , although not strictly a probability distribution, is a limiting form of many continuous probability functions.
In probability and statistics, the Irwin–Hall distribution, named after Joseph Oscar Irwin and Philip Hall, is a probability distribution for a random variable defined as the sum of a number of independent random variables, each having a uniform distribution. [1] For this reason it is also known as the uniform sum distribution.
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