Bates distribution

Propability distribution

Bates
Probability density function
Cumulative distribution function
Parameters	$-\infty <a<b<\infty$ $n\geq 1$ integer
Support	$x\in [a,b]$
PDF	see below
Mean	${\tfrac {1}{2}}(a+b)$
Variance	${\tfrac {1}{12n}}(b-a)^{2}$
Skewness	0
Excess kurtosis	$-{\tfrac {6}{5n}}$
CF	$\left(-{\frac {in(e^{\tfrac {ibt}{n}}-e^{\tfrac {iat}{n}})}{(b-a)t}}\right)^{n}$

In probability and business statistics, the Bates distribution, named after Grace Bates, 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 related to the uniform, the triangular, and the normal Gaussian distribution, and has applications in broadcast engineering for signal enhancement. The Bates distribution is sometimes confused^[2] with the Irwin–Hall distribution, which is the distribution of the sum (not the 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_k:

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(n-1)!}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(nx-k)^{n-1}\operatorname {sgn}(nx-k)

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

\operatorname {sgn}(nx-k)={\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 (PDF) of

g(x;n,a,b)={\frac {1}{b-a}}f_{X}\left({\frac {x-a}{b-a}};n\right){\text{ for }}a\leq x\leq b

Extensions and Applications

With a few modifications, the Bates distribution encompasses the uniform, the triangular, and, taking the limit as n goes to infinity, also the normal Gaussian distribution.

Replacing the term ${\frac {1}{n}}$ when calculating the mean, X, with ${\frac {1}{\sqrt {n}}}$ will create a similar distribution with a constant variance, such as unity. Then, by subtracting the mean, the resulting mean of the distribution will be set at zero. Thus the parameter n would become a purely shape-adjusting parameter. By also allowing n to be a non-integer, a highly flexible distribution can be created, for example, U(0,1) + 0.5U(0,1) gives a trapezoidal distribution.

The Student-t distribution provides a natural extension of the normal Gaussian distribution for modeling of long tail data. A Bates distribution that has been generalized as previously stated fulfills the same purpose for short tail data.

The Bates distribution has an application to beamforming and pattern synthesis in the field of electrical engineering. The distribution was found to increase the beamwidth of the main lobe, representing an increase in the signal of the radiation pattern in a single direction, while simultaneously reducing the usually undesirable^[3] sidelobe levels.^[4]^{[page needed]}

References

^ Jonhson, N. L.; Kotz, S.; Balakrishnan (1995) Continuous Univariate Distributions, Volume 2, 2nd Edition, Wiley ISBN 0-471-58494-0(Section 26.9)
^ "The thing named "Irwin-Hall distribution" in d3.random is actually a Bates distribution · Issue #1647 · d3/d3". GitHub. Archived from the original on December 12, 2020. Retrieved 2018-04-17.
^ "Sidelobe behavior and bandwidth characteristics of distributed antenna arrays". January 2018. pp. 1–2.
^ "Encyclopedia of Physical Science and Technology". ScienceDirect. Retrieved 2021-12-16.

with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric negative Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic mixed Poisson negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous
univariate

supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi chi-squared noncentral inverse scaled Dagum Davis Erlang hyper exponential hyperexponential hypoexponential logarithmic F noncentral folded normal Fréchet gamma generalized inverse gamma/Gompertz Gompertz shifted half-logistic half-normal Hotelling's T-squared inverse Gaussian generalized Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal log-t Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice truncated normal type-2 Gumbel Weibull discrete Wilks's lambda
supported on the whole real line	Cauchy exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t Tracy–Widom variance-gamma Voigt
with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda