Bates Distribution: Definition

Probability Distributions > Bates Distribution

What is the Bates Distribution?

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

The distribution resembles can look like several other distributions depending on how many items are in the sample. For larger samples, the distribution starts looking more and more like a bell-shaped curve and will approach a normal distribution when > 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].

bates distribution
PDF for the Bates distribution.

 

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):

bates distribution pdf

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

  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).

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