en>Wikid77: 11 changes: trimmed 3 Google-Books footnote urls; for {cite book} put lowercase "oclc=".

2012-08-13T11:34:46Z

11 changes: trimmed 3 Google-Books footnote urls; for {cite book} put lowercase "oclc=".

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{{Probability distribution
|name = Bates
|type = density
|pdf_image = No image available
|cdf_image = No image available
|parameters = <math>-\infty < a < b < \infty \, </math> <br> <math> n \geq 1 </math> integer
|support = <math>x \in [a,b]</math>
|pdf =
|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 [[statistics]], the '''Bates distribution''', is a [[probability distribution]] of the [[mean]] of a number of [[statistically independent]] [[continuous uniform distribution|uniformly distributed]] random variables 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 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'' [[independence (probability theory)|independent]] [[continuous uniform distribution|uniformly distributed]] random variables on the [[unit interval]], ''U<sub>i</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\left(n-1\right)!}\sum_{k=0}^{n}\left(-1\right)^k{n \choose k}\left(nx-k\right)^{n-1}\sgn(nx-k)
</math>

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

:<math> \sgn\left(nx-k\right) = \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 of

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

{{notability|date=June 2011}}
{{morefootnotes|date=June 2011}}
==Notes==
{{reflist}}

==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

{{ProbDistributions|continuous-bounded}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]
{{probability-stub}}

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en>Wikid77: 11 changes: trimmed 3 Google-Books footnote urls; for {cite book} put lowercase "oclc=".