Bates distribution

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Probability density function
Cumulative distribution function
PDF see below
Skewness 0
Ex. kurtosis

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. Thus, the two distributions are simply versions of each other as they only differ in scale.


The Bates distribution is the continuous probability distribution of the mean, X, of n independent uniformly distributed random variables on the unit interval, Ui:

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(nxk) 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

Extensions to the Bates distribution[edit]

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) + 0.5U(0,1) gives a trapezoidal 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).

The Bates distribution has an application to beamforming and pattern synthesis in Figure 1 of reference [3] [4]

See also[edit]


  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. ^ "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]
  3. ^
  4. ^


  • 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