probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

and business

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, the Bates distribution, named after

Grace Bates Grace Elizabeth Bates (13 August 1914 – 19 November 1996) was an American mathematician and one of few women in the United States to be granted a Ph.D. in mathematics in the 1940s. She became an emeritus professor at Mount Holyoke College. Bate ...

, is a

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

of the

mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...

of a number of

statistically independent Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...

uniformly distributed

random variables A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

on the

unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...

. This distribution is related to the

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

, the

triangular A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- collinea ...

, and the normal Gaussian distribution, and has applications in

broadcast engineering Broadcast engineering is the field of electrical engineering, and now to some extent computer engineering and information technology, which deals with radio and television broadcasting. Audio engineering and RF engineering are also essential par ...

for signal enhancement. The Bates distribution is sometimes confused with the

Irwin–Hall distribution In probability and statistics, the Irwin–Hall distribution, named after Joseph Oscar Irwin and Philip Hall, is a probability distribution for a random variable defined as the sum of a number of independent random variables, each having a unifo ...

, which is the distribution of the sum (not the mean) of ''n'' independent random variables uniformly distributed from 0 to 1. Thus, the two distributions are simply ''versions'' of each other as they only differ in scale.

Definition

The Bates distribution is the continuous

of the

, ''X'', of ''n''

independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...

, uniformly distributed,

on the

, ''U_k'': :

X = \frac\sum_^n U_k.

The equation defining the probability density function of a Bates distribution random variable ''X'' is :

f_X(x;n)=\frac n  \sum_^n (-1)^k  (nx-k)^ \sgn(nx-k)

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

sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avoi ...

: :

\sgn(nx-k) = \begin 
-1 &  nx < k \\
0 &  nx = k \\
1 &  nx > k. \end

More generally, the mean of ''n''

uniformly distributed random variables on the interval 'a'',''b'' :

X_ = \frac\sum_^n U_k(a,b).

would have the probability density function (PDF) of :

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

Extensions and Applications

With a few modifications, the Bates distribution encompasses the

, the

and, taking the limit as n goes to infinity, also the normal Gaussian distribution. Replacing the term

\frac

when calculating the mean, ''X'', with

\frac

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) + 0.5''U''(0,1) gives a

trapezoidal distribution In probability theory and statistics, the trapezoidal distribution is a continuous probability distribution whose probability density function graph resembles a trapezoid. Likewise, trapezoidal distributions also roughly resemble mesas or plateau ...

. The

Student-t distribution In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in sit ...

provides a natural extension of the normal Gaussian distribution for modeling of

long tail In statistics and business, a long tail of some probability distribution, distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involv ...

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 The beam diameter or beam width of an electromagnetic beam is the diameter along any specified line that is perpendicular to the beam axis and intersects it. Since beams typically do not have sharp edges, the diameter can be defined in many differ ...

of the

main lobe In a radio antenna's radiation pattern, the main lobe, or main beam, is the lobe containing the higher power. This is the lobe that exhibits the greater field strength. The radiation pattern of most antennas shows a pattern of "''lobes''" at v ...

, representing an increase in the signal of the radiation pattern in a single direction, while simultaneously reducing the, usually undesirable,

sidelobe In antenna engineering, sidelobes are the lobes (local maxima) of the far field radiation pattern of an antenna or other radiation source, that are not the ''main lobe''. The radiation pattern of most antennas shows a pattern of "''lobes''" ...

levels.

Definition

Extensions and Applications

See also

References

Further reading