Bates distribution

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In probability and statistics, the Bates distribution 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 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 independent uniformly distributed random variables on the unit interval, Ui:

${\displaystyle X={\frac {1}{n}}\sum _{k=1}^{n}U_{k}.}$

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

${\displaystyle f_{X}(x;n)={\frac {n}{2\left(n-1\right)!}}\sum _{k=0}^{n}\left(-1\right)^{k}{n \choose k}\left(nx-k\right)^{n-1}\operatorname {sgn} (nx-k)}$

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

${\displaystyle \operatorname {sgn} \left(nx-k\right)={\begin{cases}-1&nxk.\end{cases}}}$

More generally, the mean of n independent uniformly distributed random variables on the interval [a,b]

${\displaystyle X_{(a,b)}={\frac {1}{n}}\sum _{k=1}^{n}U_{k}(a,b).}$

would have the probability density function of

${\displaystyle g(x;n,a,b)=f_{X}\left({\frac {x-a}{b-a}};n\right){\text{ for }}a\leq x\leq b\,}$

Notes

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

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
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