View source for Bates distribution

{{Short description|Propability distribution}}
{{more footnotes needed|date=June 2011}}
{{Probability distribution
  |name       = Bates
  |type       = density
  |pdf_image  = [[Image:batesPDF.svg|325px]]
  |cdf_image  = [[Image:batesCDF.svg|325px]]
  |parameters = <math>-\infty < a < b < \infty </math> <br> <math> n \geq 1 </math> integer
  |support    = <math>x \in [a,b]</math>
  |pdf        = see below
  |cdf        = 
  |mean       = <math>\tfrac{1}{2}(a+b)</math>
  |median     = 
  |mode       = 
  |variance   = <math>\tfrac{1}{12n}(b-a)^2</math>
  |skewness   = 0
  |kurtosis   = <math>-\tfrac{6}{5n}</math>
  |entropy    = 
  |mgf        = 
  |char       = <math>\left(-\frac{in (e^{\tfrac{ibt}{n}}-e^{\tfrac{iat}{n}}) }{(b-a)t}\right)^n</math> 
}}

In [[probability]] and business [[statistics]], the '''Bates distribution''', named after [[Grace Bates]], is a [[probability distribution]] of the [[mean]] of a number of [[statistically independent]]  [[continuous uniform distribution|uniformly distributed]] [[random variable]]s on the [[unit interval]].<ref>Jonhson, N. L.; Kotz, S.; Balakrishnan (1995) ''Continuous Univariate Distributions'', Volume 2, 2nd Edition, Wiley {{ISBN|0-471-58494-0}}(Section 26.9)</ref> This distribution is related to the [[Continuous uniform distribution|uniform]], the [[Triangular distribution|triangular]], and the [[Normal distributions|normal Gaussian distribution]], and has applications in [[broadcast engineering]] for signal enhancement. 

The Bates distribution is sometimes confused<ref>{{Cite web|url=https://github.com/d3/d3/issues/1647|archive-url=https://web.archive.org/web/20201212211649/https://github.com/d3/d3/issues/1647|url-status=dead|archive-date=December 12, 2020|title=The thing named "Irwin-Hall distribution" in d3.random is actually a Bates distribution · Issue #1647 · d3/d3|website=GitHub|language=en|access-date=2018-04-17}}</ref> 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&nbsp;1. If ''X'' has a Bates distribution on the unit interval, then ''n&nbsp;X'' has an Irwin-Hall distribution; when ''n''&nbsp;=&nbsp;1 they are both uniformly distributed. 

==Definition==
The Bates distribution is the continuous [[probability distribution]] of the [[mean]], ''X'', of ''n'' [[independence (probability theory)|independent]], [[continuous uniform distribution|uniformly distributed]], [[random variable]]s on the [[unit interval]], ''U<sub>k</sub>'':

:<math>
X = \frac{1}{n}\sum_{k=1}^n U_k.
</math>

The equation defining the probability density function of a Bates distribution random variable ''X'' is

: <math>
f_X(x;n)=\frac n {2(n-1)!} \sum_{k=0}^n (-1)^k {n \choose k} (nx-k)^{n-1} \sgn(nx-k)
</math>

for ''x'' in the interval (0,1), and zero elsewhere. Here sgn(''nx'' &minus; ''k'') denotes the [[sign function]]:

:<math> \sgn(nx-k) = \begin{cases} 
-1 &  nx < k \\
0 &  nx = k \\
1 &  nx > k. \end{cases}
</math>

More generally, the mean of ''n'' [[independence (probability theory)|independent]] [[continuous uniform distribution|uniformly distributed]] random variables on the interval [''a'',''b'']

:<math>
X_{(a,b)} = \frac{1}{n}\sum_{k=1}^n U_k(a,b).
</math>

would have the probability density function (PDF) of

:<math> g(x;n,a,b) = \frac{1}{b-a} f_X\left(\frac{x-a}{b-a};n\right) \text{ for } a \leq x \leq b </math>

==Extensions and Applications==
{{expand section|date=February 2020}}
With a few modifications, the Bates distribution encompasses the [[Continuous uniform distribution|uniform]], the [[Triangular distribution|triangular]], and, taking the limit as n goes to infinity, also the [[Normal distribution|normal Gaussian distribution]].

Replacing the term <math> \frac{1}{n} </math> when calculating the mean, ''X'', with <math>\frac{1}{\sqrt{n}}</math> will create a similar distribution with a constant variance, such as unity. Then, by subtracting the mean, the resulting mean of the distribution will be set at zero. Thus the parameter ''n'' would become a purely [[Shape parameter|shape-adjusting parameter]]. By also allowing ''n'' to be a non-integer, a highly flexible distribution can be created, for example, ''U''(0,1) + 0.5''U''(0,1) gives a [[trapezoidal distribution]].

The [[Student-t distribution]] provides a natural extension of the normal Gaussian distribution for modeling of [[long tail]] data. A Bates distribution that has been generalized as previously stated fulfills the same purpose for [[Short-tailed distribution|short tail]] data.

The Bates distribution has an application to [[beamforming]] and [[pattern synthesis]] in the field of [[electrical engineering]]. The distribution was found to increase the [[beamwidth]] of the [[main lobe]], representing an increase in the signal of the radiation pattern in a single direction, while simultaneously reducing the usually undesirable<ref>{{cite web|date=January 2018|title=Sidelobe behavior and bandwidth characteristics of distributed antenna arrays|url=https://ieeexplore.ieee.org/document/8299700/similar#similar|pages=1–2}}</ref> [[Side lobe|sidelobe]] levels.<ref>{{Cite web|title=Encyclopedia of Physical Science and Technology|url=https://www.sciencedirect.com/referencework/9780122274107/encyclopedia-of-physical-science-and-technology|access-date=2021-12-16|website=ScienceDirect}}</ref>{{page needed|date=August 2023}}

== See also ==
* [[Irwin–Hall distribution]]
* [[Normal distribution]]
* [[Central limit theorem]]
* [[Uniform distribution (continuous)]]
* [[Triangular distribution]]

== References ==
{{reflist}}

==Further reading==
*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&ndash;720
 
{{ProbDistributions|continuous-bounded}}

[[Category:Continuous distributions]]