# Bates distribution

Parameters |
integer |
---|---|

Support | |

see below | |

Mean | |

Variance | |

Skewness | 0 |

Ex. kurtosis | |

CF |

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, *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 (PDF) of

Therefore, the PDF of the distribution is

## Extensions to the Bates Distribution

Instead of dividing by n we can also use sqrt(n) and create this way a similar distribution with a constant variance (like unity) can be created. With subtraction of the mean we can set the resulting mean of 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 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).

## 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