# Bates distribution

Parameters $-\infty < a < b < \infty \,$ $n \geq 1$ integer $x \in [a,b]$ $\begin{cases} \sum_{k=0}^n (-1)^k \binom{n}{k} \left( \frac{x-a}{b-a}-k/n \right)^{n-1} \sgn\left( \frac{x-a}{b-a}-k/n \right) & \text{if } x\in[a,b]\\ 0 & \text{otherwise} \end{cases}$ $\tfrac{1}{2}(a+b)$ $\tfrac{1}{12n}(b-a)^2$ 0 $-\tfrac{6}{5n}$ $\left(-\frac{in (e^{\tfrac{ibt}{n}}-e^{\tfrac{iat}{n}}) }{(b-a)t}\right)^n$

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 with the Irwin–Hall distribution, which is the distribution of the sum (not mean) of n independent random variables uniformly distributed from 0 to 1.

## 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:

$X = \frac{1}{n}\sum_{k=1}^n U_k.$

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

$f_X(x;n)=\frac{n}{2\left(n-1\right)!}\sum_{k=0}^{n}\left(-1\right)^k{n \choose k}\left(nx-k\right)^{n-1}\sgn(nx-k)$

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

$\sgn\left(nx-k\right) = \begin{cases} -1 & nx < k \\ 0 & nx = k \\ 1 & nx > k. \end{cases}$

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

$X_{(a,b)} = \frac{1}{n}\sum_{k=1}^n U_k(a,b).$

would have the probability density function of

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

## Notes

1. ^ Jonhson, N.L.; Kotz, S.; Balakrishnan (1995) Continuous Univariate Distributions, Volume 2, 2nd Edition, Wiley ISBN 0-471-58494-0(Section 26.9)

## References

• Bates,G.E. (1955) "Joint distributions of time intervals for the occurrence of successive accidents in a generalized Polya urn scheme", Annals of Mathematical Statistics, 26, 705–720