Beta distribution

Beta distribution

Probability distribution
name =Beta| type =density
pdf_

cdf_

parameters =$alpha > 0$ shape (real)
shape (real)
support =$x in \left[0; 1\right] !$
pdf =
{mathrm{B}(alpha,eta)}!
cdf =
mean =
median =
mode = for
variance =
skewness =
kurtosis ="see text"
entropy ="see text"
mgf =
char =
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 &beta;.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 distributionafter observing α − 1 independent events with probability "p" and &beta; − 1 with probability 1 − "p", if the priordistribution of "p" was uniform.

Characterization

Probability density function

The probability density function of the beta distribution is:

:

::

::

where $Gamma$ 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

The cumulative distribution function is

:

where is the incomplete beta function and is the regularized incomplete beta function.

Properties

Moments

The expected value and variance of a beta random variable "X" with parameters α and &beta; are given by the formulae:

:

The skewness is

:

The kurtosis excess is:

:

Quantities of information

Given two beta distributed random variables, "X" ~ Beta(α, &beta;) and "Y" ~ Beta(α', &beta;'), the information entropy of "X" is:where $psi$ is the digamma function.

The cross entropy is:

It follows that the Kullback-Leibler divergence between these two beta distributions is

:

hapes

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
* is the [uniform distribution (continuous)|uniform [0,1] distribution]
* or is strictly increasing (green plot)
** is strictly convex
** is a straight line
** is strictly concave
* is unimodal (purple & black plots)

Moreover, if then the density function is symmetric about 1/2 (red & purple plots).

=Parameter estimation= Let

:

be the sample mean and

:

be the sample variance. The method-of-moments estimates of the parameters are

:

:

If the distribution is required over an interval other than [0, 1] , say $scriptstyle \left[ell,h\right]$, then replace with and $v$ with $frac\left\{v\right\}\left\{\left(h-ell\right)^2\right\}$ in the above equations. [ [http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm Engineering Statistics Handbook] ] [ [http://www.brighton-webs.co.uk/distributions/beta.asp Brighton Webs Ltd. Data & Analysis Services for Industry & Education] ]

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" and "Y" are independently distributed Gamma(α, &theta;) and Gamma(&beta;, &theta;) respectively, then "X" / ("X" + "Y") is distributed Beta(α,&beta;).
* If "X" and "Y" are independently distributed Beta(α,&beta;) and "F"(2&beta;,2α) (Snedecor's F distribution with 2&beta; and 2α degrees of freedom), then Pr("X" &le; α/(α+x&beta;)) = 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 $X sim \left\{ m U\right\}\left(0, 1\right] ,$ has a uniform distribution, then $X^2 sim \left\{ m Beta\right\}\left(1/2,1\right)$ or for the 4 parameter case, $X^2 sim \left\{ m Beta\right\}\left(0,1,1/2,1\right)$ which is a special case of the Beta distribution called the power-function distribution.
* Binomial opinions in subjective logic are equivalent to Beta distributions.

Applications

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.

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.

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.

These approximations are exact only for particular values of α and β, specifically when [Grubbs, Frank E. (1962). Attempts to Validate Certain PERT Statistics or ‘Picking on PERT’. Operations Research 10(6), p. 912-915.] :

:$alpha = 3 - sqrt2$:

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 [Keefer, Donald L. and Verdini, William A. (1993). Better Estimation of PERT Activity Time Parameters. Management Science 39(9), p. 1986-1091.] [Keefer, Donald L. and Bodily, Samuel E. (1983). Three-point Approximations for Continuous Random variables. Management Science 29(5), p. 595-609.] [ [http://www.nps.edu/drmi/docs/1apr05-newsletter.pdf DRMI Newsletter, Issue 12, April 8, 2005] ]

References

* [http://demonstrations.wolfram.com/BetaDistribution/ "Beta Distribution"] by Fiona Maclachlan, The Wolfram Demonstrations Project, 2007.
* [http://www.xycoon.com/beta.htm Beta Distribution - Overview and Example] , xycoon.com
* [http://www.brighton-webs.co.uk/distributions/beta.asp Beta Distribution] , brighton-webs.co.uk

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Noncentral beta distribution — In probability theory and statistics, the noncentral beta distribution is a continuous probability distribution that is a generalization of the (central) beta distribution. Contents 1 Probability density function 2 Cumulative distribution… …   Wikipedia

• Beta — may refer to: *Beta (β), the second letter of the Greek alphabetIn finance: * Beta coefficient in Capital Asset Pricing ModelIn mathematics: * Beta function in mathematics * Beta distribution in statistics * False negative rate in statistics *… …   Wikipedia

• Beta function — This article is about Euler beta function. For other uses, see Beta function (disambiguation). In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by for The beta function was studied …   Wikipedia

• Beta-binomial model — In empirical Bayes methods, the Beta binomial model is an analytic model where the likelihood function L(x| heta) is specifed by a binomial distribution:L(x| heta) = operatorname{Bin}(x, heta),::: = {nchoose x} heta^x(1 heta)^{n x},and the… …   Wikipedia

• Beta prime distribution — Probability distribution name =Beta Prime| type =density pdf cdf parameters =alpha > 0 shape (real) eta > 0 shape (real) support =x > 0! pdf =f(x) = frac{x^{alpha 1} (1+x)^{ alpha eta{B(alpha,eta)}! cdf =frac{x^alpha cdot 2F 1(alpha,… …   Wikipedia

• Beta wavelet — Continuous wavelets of compact support can be built [1] , which are related to the beta distribution. The process is derived from probability distributions using blur derivative. These new wavelets have just one cycle, so they are termed unicycle …   Wikipedia

• Distribution De Gumbel — Gumbel Densité de probabilité / Fonction de masse Fonction de répartition …   Wikipédia en Français

• Distribution de gumbel — Gumbel Densité de probabilité / Fonction de masse Fonction de répartition …   Wikipédia en Français

• BETA ISRAEL — BETA ISRAEL, ethno religious group in Ethiopia which claims to be of Jewish origin and which is attached to a form   of the Jewish religion based on the Bible, certain books of the Apocrypha, and other post biblical Scripture; living in the… …   Encyclopedia of Judaism

• Distribution De Boltzmann — En physique, la distribution de Boltzmann prédit la fonction de distribution pour le nombre fractionnaire de particules Ni / N occupant un ensemble d états i qui ont chacun pour énergie Ei : où kB est la constante de Boltzmann, T est la… …   Wikipédia en Français