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.

## 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(*nx* − *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 (PDF) of

Therefore, the PDF of the distribution is

## 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) + 0.5*U*(0,1) gives a trapezodial 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).

## See also

- Irwin–Hall distribution
- Normal distribution
- Central limit theorem
- Uniform distribution (continuous)
- Triangular distribution

## Notes

**^**Jonhson, N. L.; Kotz, S.; Balakrishnan (1995)*Continuous Univariate Distributions*, Volume 2, 2nd Edition, Wiley ISBN 0-471-58494-0(Section 26.9)**^**"The thing named "Irwin-Hall distribution" in d3.random is actually a Bates distribution · Issue #1647 · d3/d3".*GitHub*. Retrieved 2018-04-17.^{[permanent dead link]}

## 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