< List of Probability Distributions

The** Bates Distribution** (or *rectangular mean distribution*) a probability distribution of the mean of a number of independent uniformly distributed random variables on the unit interval [1].

The Bates distribution resembles a normal distribution although it can also resemble several other distributions depending on how many items are in the sample.

- Triangular distribution: n = 2.
- Uniform distribution: n = 1.
- Unimodal (normal) distribution: n > 2.

More formally, it is the probability distribution of the mean of *n* independent standard uniform variates.

The distribution was is named after American mathematician Grace Bates [2] who tested the null hypothesis that a particular distribution is a uniform distribution [0, 1] with the alternate hypothesis that it is a truncated exponential distribution on [0, 1].

## Properties of the Bates distribution

The following formula shows the probability density function (PDF) for a Bates random variable X on the interval (0, 1):

where *sgn *denotes the sign function

Other properties include:

- Mean = ½(a + b)
- Variance 1/12n (b – a)
^{2} - Skewness = 0
- Kurtosis = – 6/5n

## Standardized Bates Distribution

The standardized Bates distribution is a single argument form [n] and equals the Bates distribution[n,{0,1}]. Found in many statistical software packages, it is characterized by [3]:

- Mean = 0;
- Standard deviation = 1.
- Sample size = 12.

An important historical use of the standardized Bates distribution was that it generated standard normal variables in computing [4].

## Distributions similar to the Bates distribution

- As noted above, when the sample size is 1, the Bates distribution is equal to the uniform distribution.
- For a sample size (n) of 2 it is equal to a triangular distribution.
- The PDF of a Bates distribution appears visually similar to the PDF of a normal distribution for larger values of
*n*. - The Bates distribution is sometimes confused with the Irwin-Hall distribution, but while the Irwin-Hall is the distribution of the sum, the Bates is the distribution of the mean.
- It is also closely related to the Uniform Sum Distribution, which represents the sum of statistically independent, uniformly distributed random variables — instead of their mean [5].

## References

Image: By Shiyu Ji – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=59354219

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

[2] 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.

[3] Kotz, S. & Dorp, J. (2004). Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications. World Scientific.

[4] Kotz, S. & Van Dorp, J. (2004). Beyond Beta. Other Continuous Families of Distributions with Bounded Support and Applications. World Scientific.

[5] Wolfram Research (2010), BatesDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BatesDistribution.html (updated 2016).