Bates distribution

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. 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 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.

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
Ex. kurtosis	${\displaystyle -{\tfrac {6}{5n}}}$
CF	${\displaystyle \left(-{\frac {in(e^{\tfrac {ibt}{n}}-e^{\tfrac {iat}{n}})}{(b-a)t}}\right)^{n}}$