Bates distribution

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. 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 on ${\displaystyle [0,1]}$ and of parameter ${\displaystyle n}$ is sometimes confused with the Irwin–Hall distribution of parameter ${\displaystyle n}$, which is the distribution of the sum (not the mean) of ${\displaystyle n}$ independent random variables uniformly distributed on the unit interval ${\displaystyle [0,1]}$. More precisely, if ${\displaystyle X}$ has a Bates distribution on ${\displaystyle [0,1]}$, then ${\displaystyle nX}$ has an Irwin-Hall distribution of parameter ${\displaystyle n}$ (and support on ${\displaystyle [0,n]}$). For ${\displaystyle n=1}$, both the Bates distribution and the Irwin-Hall distribution coincide with the uniform distribution on the unit interval ${\displaystyle [0,1]}$.