Beta distribution

Beta
	Probability density function;
	Cumulative distribution function; Image:Beta distribution cdf.png
Parameters	α > 0 shape (real); β > 0 shape (real)
Support
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean
Median
Mode	for α > 1,β > 1
Variance
Skewness
Excess kurtosis	see text
Entropy	see text
Moment-generating function (mgf)
Characteristic function

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Overview

Not to be confused with Beta function.

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two non-negative shape parameters, typically denoted by α and β.

Characterization

Probability density function

The probability density function of the beta distribution is

$f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{\int_0^1 u^{\alpha-1} (1-u)^{\beta-1}\, du} \!$

$= \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!$

$= \frac{1}{\mathrm{B}(\alpha,\beta)}\, x ^{\alpha-1}(1-x)^{\beta-1}\!$

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

The cumulative distribution function is

$F(x;\alpha,\beta) = \frac{\mathrm{B}_x(\alpha,\beta)}{\mathrm{B}(\alpha,\beta)} = I_x(\alpha,\beta) \!$

where $B x (α,β)$ is the incomplete beta function and $I x (α,β)$ is the regularized incomplete beta function.

Properties

Moments

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

$\begin{align} \operatorname{E}(X) = & \frac{\alpha}{\alpha+\beta} \\ \operatorname{Var}(X) = & \frac{\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)} \end{align}$

The skewness is

$\frac{2 (\beta - \alpha) \sqrt{\alpha + \beta + 1} } {(\alpha + \beta + 2) \sqrt{\alpha \beta}}. \,\!$

The kurtosis excess is:

$6\,\frac{\alpha^3-\alpha^2(2\beta-1)+\beta^2(\beta+1)-2\alpha\beta(\beta+2)} {\alpha \beta (\alpha+\beta+2) (\alpha+\beta+3)}.\,\!$

Quantities of information

Given two beta distributed random variables, X ~ Beta(α, β) and Y ~ Beta(α', β'), the information entropy of X is

$\begin{align} H(X) &= \ln\mathrm{B}(\alpha,\beta)-(\alpha-1)\psi(\alpha)-(\beta-1)\psi(\beta)+(\alpha+\beta-2)\psi(\alpha+\beta) \end{align} \,$

where $ψ$ is the digamma function.

The cross entropy is

$H(X,Y) = \ln\mathrm{B}(\alpha',\beta')-(\alpha'-1)\psi(\alpha)-(\beta'-1)\psi(\beta)+(\alpha'+\beta'-2)\psi(\alpha+\beta).\,$

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

$D_{\mathrm{KL}}(X,Y) = \ln\frac{\mathrm{B}(\alpha',\beta')} {\mathrm{B}(\alpha,\beta)} - (\alpha'-\alpha)\psi(\alpha) - (\beta'-\beta)\psi(\beta) + (\alpha'-\alpha+\beta'-\beta)\psi(\alpha+\beta)$

Shapes

The beta density function can take on different shapes depending on the values of the two parameters:

$\alpha < 1,\ \beta < 1$ is U-shaped (red plot)
or is strictly decreasing (blue plot)
- $\alpha = 1,\ \beta > 2$ is strictly convex
- $\alpha = 1,\ \beta = 2$ is a straight line
- $\alpha = 1,\ 1 < \beta < 2$ is strictly concave
$\alpha = 1,\ \beta = 1$ is the uniform distribution
or is strictly increasing (green plot)
- $\alpha > 2,\ \beta = 1$ is strictly convex
- $\alpha = 2,\ \beta = 1$ is a straight line
- $1 < \alpha < 2,\ \beta = 1$ is strictly concave
$\alpha > 1,\ \beta > 1$ is unimodal (purple & black plots)

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

Parameter estimation

Let

$\bar{x} = \frac{1}{N}\sum_{i=1}^N x_i$

be the sample mean and

$v = \frac{1}{N}\sum_{i=1}^N (x_i - \bar{x})^2$

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

$\alpha = \bar{x} \left(\frac{\bar{x} (1 - \bar{x})}{v} - 1 \right),$

$\beta = (1-\bar{x}) \left(\frac{\bar{x} (1 - \bar{x})}{v} - 1 \right).$

Related distributions

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, 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 $X \sim {\rm U}(0, 1]\,$ has a uniform distribution, then $X^2 \sim {\rm Beta}(1/2,1) \$ or for the 4 parameter case, $X^2 \sim {\rm Beta}(0,1,1/2,1) \$ 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

B(i, j) with integer values of i and j is the distribution of the ith-highest 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 highest 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:

$\begin{align} \mathrm{mean}(X) & {} = E(X)= \frac{a + 4b + c}{6}, \\ \mathrm{s.d.}(X) & {} = \frac{c-a}{6}, \end{align}$

where a is the minimum, c is the maximum, and b is the most likely value.

External links

Beta Distribution, wolfram.com
Beta Distribution - Overview and Example, xycoon.com
Beta Distribution, brighton-webs.co.uk

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Acknowledgement and Attribution Regarding Sources of Content

Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .