Bates distribution

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Bates
Probability density function No image available
Cumulative distribution function No image available
Parameters	$-\infty < a < b < \infty \,$ n>1 real
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

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

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

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Notes

^ 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 occurence of succesive accidents in a generalized Polya urn scheme", Annals of Mathematical Statistics, 26, 705–720

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding-Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial multivariate Pólya negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse-Wishart matrix normal Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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