Bates distribution

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Bates
Parameters -\infty < a < b < \infty \,
 n \geq 1 integer
Support x \in [a,b]
Mean \tfrac{1}{2}(a+b)
Variance \tfrac{1}{12n}(b-a)^2
Skewness 0
Ex. kurtosis -\tfrac{6}{5n}
CF \left(-\frac{in (e^{\tfrac{ibt}{n}}-e^{\tfrac{iat}{n}}) }{(b-a)t}\right)^n

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[edit]

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


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


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)

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

 \sgn\left(nx-k\right) = \begin{cases} 
-1 &  nx < k \\
0 &  nx = k \\
1 &  nx > k. \end{cases}

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


X_{(a,b)} = \frac{1}{n}\sum_{k=1}^n U_k(a,b).

would have the probability density function of

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

Notes[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)

References[edit]

  • 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