Bates distribution

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Bates

Probability density function
Cumulative distribution function
Parameters	${\displaystyle -\infty <a<b<\infty }$ ${\displaystyle n\geq 1}$ integer
Support	${\displaystyle x\in [a,b]}$
PDF	see below
Mean	${\displaystyle {\tfrac {1}{2}}(a+b)}$
Variance	${\displaystyle {\tfrac {1}{12n}}(b-a)^{2}}$
Skewness	0
Excess kurtosis	${\displaystyle -{\tfrac {6}{5n}}}$
CF	${\displaystyle \left(-{\frac {in(e^{\tfrac {ibt}{n}}-e^{\tfrac {iat}{n}})}{(b-a)t}}\right)^{n}}$

In probability and business statistics, the Bates distribution, named after Grace Bates, is a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval.^[1] This distribution is related to the uniform, the triangular, and the normal Gaussian distribution, and has applications in broadcast engineering for signal enhancement.

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.