Bates distribution

Bates

Probability density function No image available
Cumulative distribution function No image available
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. 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, U_i:

$X = \frac{1}{n}\sum_{k=1}^n U_k.$

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

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

References

• Bates,G.E. (1955) "Joint distributions of time intervals for the occurrence of succesive accidents in a generalized Polya urn scheme", Annals of Mathematical Statistics, 26, 705–720