Beta

Probability density function

Cumulative distribution function

parameters:

no closed form

for α > 1,β >

1

see text

In probability theory and statistics, the beta distribution is a

family of continuous probability distributions defined on

the interval (0, 1) parameterized by two positive shape parameters, typically denoted by α and

β. It is the special case of the Dirichlet distribution with only two

parameters. Since the Dirichlet distribution is the conjugate prior of the multinomial distribution, the

beta distribution is the

conjugate prior of the binomial distribution. In Bayesian statistics,

it can be seen as the posterior

distribution of the parameter p of a binomial

distribution after observing

α???1 independent events with

probability p and β???1

with probability 1???p, if

there is no other information regarding the distribution of

p.

Characterization

Probability density

function

The probability density function of

the beta distribution is:

where Γ is the gamma function. The beta function, B, appears as a normalization

constant to ensure that the total probability integrates to

unity.

Cumulative distribution

function

Properties

The expected value (μ), first central moment, variance (second central moment), skewness (third central moment), and kurtosis excess (forth

central moment) of a Beta distribution random variable X with parameters α

and β are:

In general, the kth raw moment is given by

where (x)k

is a Pochhammer symbol representing rising

factorial. It can also be written in a recursive form as

One can also show that

Quantities

of information

Given two beta distributed random variables, X ~ Beta(α,

β) and Y ~ Beta(α', β'), the information entropy

of X is

It follows that the Kullback–Leibler divergence

between these two beta distributions is

Shapes

The beta density function can take on different shapes depending

on the values of the two parameters:

is U-shaped (red plot)

or

is strictly decreasing (blue plot)

is strictly convex

is a straight line

is strictly concave

or

is strictly increasing (green plot)

is strictly convex

is a straight line

is strictly concave

is unimodal (purple & black

plots)

Moreover, if α = β then the density

function is symmetric about 1/2 (red & purple

plots).

Parameter

estimation

Let

be the sample variance.

The method-of-moments

estimates of the parameters are

When the distribution is required over an interval other than

[0,?1], say

, then replace

with

and

with

in the above equations.

Related

distributions

If X has a beta distribution, then

T?=?X/(1???X)

has a "beta distribution of the second kind", also called the

beta prime distribution.

The connection with the binomial distribution is mentioned

below.

The Beta(1,1) distribution is identical to the standard

uniform

distribution.

If X has the Beta(3/2,3/2) distribution and R

> 0 is a real parameter, then

Y

If X and Y are independently distributed

Gamma(α,?θ) and

Gamma(β,?θ) respectively, then

X?/?(X?+?Y)

is distributed Beta(α,?β).

If X and Y are independently distributed

Beta(α,β) and F(2β,?2α)

(Snedecor's F distribution with 2β and

2α degrees of freedom), then

Pr(X?≤?α/(α?+?xβ))

=?Pr(Y?>?x)

for all

x?>?0.

The beta distribution is a special case of the Dirichlet distribution for only two

parameters.

The Kumaraswamy distribution resembles

the beta distribution.

If

has a uniform distribution, then

, which is a special case of the Beta distribution called the

power-function

distribution.

Binomial opinions in subjective logic are equivalent to Beta

distributions.

Beta(1/2,1/2) is the Jeffreys prior for a proportion and is

equivalent to arcsine distribution.

Beta(i,?j) with integer values of

i and j is the distribution of the i-th order

statistic (the i-th smallest value) of a sample of

i?+?j???1

independent random variables uniformly distributed

between 0 and 1. The cumulative probability from 0 to x is

thus the probability that the i-th smallest value is less

than x, in other words, it is the probability that at least

i of the random variables are less than x, a

probability given by summing over the binomial distribution with its p

parameter set to x. This shows the intimate connection

between the beta distribution and the binomial distribution.

Applications

Rule of

succession

A classic application of the beta distribution is the rule of succession, introduced in the 18th

century by Pierre-Simon Laplace in the course of

treating the sunrise problem. It states that, given

s successes in n conditionally independent Bernoulli

trials with probability p, that p should be estimated

as

. This estimate may be regarded as the expected value of the

posterior distribution over p, namely

Beta(s?+?1,?n???s?+?1),

which is given by Bayes' rule if one assumes a uniform prior

over p (i.e., Beta(1,?1)) and then observes

that p generated s successes in n trials.

Bayesian

statistics

Beta distributions are used extensively in Bayesian statistics,

since beta distributions provide a family of conjugate prior

distributions for binomial (including Bernoulli) and geometric distributions. The Beta(0,0)

distribution is an improper prior and

sometimes used to represent ignorance of parameter values.

Task duration

modeling

The beta distribution can be used to model events which are

constrained to take place within an interval defined by a minimum

and maximum value. For this reason, the beta distribution — along

with the triangular distribution — is used

extensively in PERT, critical path method (CPM) and other

project management / control systems to

describe the time to completion of a task. In project management,

shorthand computations are widely used to estimate the mean and

standard deviation of the beta distribution:

where a is the minimum, c is the maximum, and

b is the most likely value.

Using this set of approximations is known as three-point estimation and are exact

only for particular values of α and β, specifically when

or vice versa.

These are notably poor approximations for most other beta

distributions exhibiting average errors of 40% in the mean and 549%

in the variance

Information

theory

All or part of this section may

be confusing or

unclear.

Please help talk page. (March

2010)

This section Please help improve this article by

adding citations to reliable

sources. Unsourced material may be challenged and removed. (March

2010)

We introduce one exemplary use of beta distribution in

information theory, particularly for the information theoretic

performance analysis for a communication system. In sensor array

systems, the distribution of two vector production is used for the

performance estimation in frequent. Assume that s and

v are vectors the

(M???1)-dimensional

nullspace of h with isotropic i.i.d. where s,

v and h are in CM and the

elements of h are i.i.d complex Gaussian random values.

Then, the production of s and v with absolute of the

result |sHv| is

beta(1,?M???2)

distributed.

Four parameters

A beta distribution with the two shape parameters α and

β is supported on the range [0,1]. It is possible to alter

the location and scale of the distribution by introducing two

further parameters representing the minimum and maximum values of

the distribution.

The probability density function of

the four parameter beta distribution is given by

The standard form can be obtained by letting

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