# Bates distribution

Template:Probability distribution

In probability and statistics, the **Bates distribution** 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, *U _{i}*:

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

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

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

would have the probability density function of

Template:Notability Template:Morefootnotes

## Notes

- ↑ 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

- REDIRECT Template:Probability distributions