# Bates distribution

In probability and 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 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.

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

## Definition

The Bates distribution is the continuous probability distribution of the mean, X, of n independent uniformly distributed random variables on the unit interval, Ui:

${\displaystyle X={\frac {1}{n}}\sum _{k=1}^{n}U_{k}.}$

The equation defining the probability density function of a Bates distribution random variable X is

${\displaystyle f_{X}(x;n)={\frac {n}{2(n-1)!}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(nx-k)^{n-1}\operatorname {sgn}(nx-k)}$

for x in the interval (0,1), and zero elsewhere. Here sgn(nx k) denotes the sign function:

${\displaystyle \operatorname {sgn}(nx-k)={\begin{cases}-1&nxk.\end{cases}}}$

More generally, the mean of n independent uniformly distributed random variables on the interval [a,b]

${\displaystyle X_{(a,b)}={\frac {1}{n}}\sum _{k=1}^{n}U_{k}(a,b).}$

would have the probability density function (PDF) of

${\displaystyle g(x;n,a,b)=f_{X}\left({\frac {x-a}{b-a}};n\right){\text{ for }}a\leq x\leq b}$

Therefore, the PDF of the distribution is

${\displaystyle f(x)={\begin{cases}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}\left({\frac {x-a}{b-a}}-k/n\right)^{n-1}\operatorname {sgn} \left({\frac {x-a}{b-a}}-k/n\right)&{\text{if }}x\in [a,b]\\0&{\text{otherwise}}\end{cases}}}$

## Extensions to the Bates distribution

Instead of dividing by n we can also use n to create a similar distribution with a constant variance (like unity). By subtracting the mean we can set the resulting mean to zero. This way the parameter n would become a purely shape-adjusting parameter, and we obtain a distribution which covers the uniform, the triangular and, in the limit, also the normal Gaussian distribution. By allowing also non-integer n a highly flexible distribution can be created (e.g. U(0,1) + 0.5U(0,1) gives a trapezoidal distribution). Actually the Student-t distribution provides a natural extension of the normal Gaussian distribution for modeling of long tail data. And such generalized Bates distribution is doing so for short tail data (kurtosis < 3).