Beta Distribution

Information about Beta Distribution

Not to be confused with Beta function.

Beta
Probability density function Probability density function for the Beta distribution
Cumulative distribution function Cumulative distribution function for the Beta distribution
Parameters	$\text{[math]}$ shape (real) $\text{[math]}$ shape (real)
Support	$\text{[math]}$
Probability density function (pdf)	$\text{[math]}$
Cumulative distribution function (cdf)	$\text{[math]}$
Mean	$\text{[math]}$
Median
Mode	$\text{[math]}$ for $\text{[math]}$
Variance	$\text{[math]}$
Skewness	$\text{[math]}$
Excess kurtosis	see text
Entropy	see text
Moment-generating function (mgf)	$\text{[math]}$
Characteristic function	$\text{[math]}$

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

$\text{[math]}$

: $\text{[math]}$

where $\text{[math]}$ 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

$\text{[math]}$

where $\text{[math]}$ is the incomplete beta function and $\text{[math]}$ 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:

$\text{[math]}$

The skewness is

$\text{[math]}$

The kurtosis excess is:

$\text{[math]}$

Quantities of information

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

$\text{[math]}$

where $\text{[math]}$ is the digamma function.

The cross entropy is

$\text{[math]}$

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

$\text{[math]}$

Shapes

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

$\text{[math]}$ is U-shaped (red plot)
$\text{[math]}$ or $\text{[math]}$ is strictly decreasing (blue plot)
$\text{[math]}$ is strictly convex
$\text{[math]}$ is a straight line
$\text{[math]}$ is strictly concave
$\text{[math]}$ is the uniform distribution
$\text{[math]}$ or $\text{[math]}$ is strictly increasing (green plot)
$\text{[math]}$ is strictly convex
$\text{[math]}$ is a straight line
$\text{[math]}$ is strictly concave
$\text{[math]}$ is unimodal (purple & black plots)

Moreover, if $\text{[math]}$ then the density function is symmetric about 1/2 (red & purple plots).

Parameter estimation

Let

$\text{[math]}$

be the sample mean and

$\text{[math]}$

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

$\text{[math]}$

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 $\text{[math]}$ has a uniform distribution, then $\text{[math]}$ or for the 4 parameter case, $\text{[math]}$ 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:

$\text{[math]}$

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

	Probability distributions [ • • [ edit] ]
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lvy • Lvy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	bimodal • Cantor • conditional • equilibrium • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular

beta function, also called the Euler integral of the first kind, is a special function defined by

for Re(x), Re(y) > 0.

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.
..... Click the link for more information.

In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions.

Definition

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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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In mathematics, a support of a function f from a set X to the real numbers R is a subset Y of X such that f (x) is zero for all x in X and outside Y.
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In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.

Formally, a probability distribution has density f, if f
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In probability theory, the cumulative distribution function (CDF), also called probability distribution function or just distribution function,^[1] completely describes the probability distribution of a real-valued random variable X.
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expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
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median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking
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In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. The term is applied both to probability distributions and to collections of experimental data.
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variance of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.
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skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.

Introduction

Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side.
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kurtosis (from the Greek word kurtos, meaning bulging) is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent
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Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable.

Shannon entropy quantifies the information contained in a piece of data: it is the minimum average message length, in bits (if using base-2 logarithms), that must
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In probability theory and statistics, the moment-generating function of a random variable X is

wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.
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In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where X is any random variable with the distribution in question:

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Probability theory is the branch of mathematics concerned with analysis of random phenomena.^[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
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Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities.
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probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied.
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In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions.

Definition

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Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. For a complex number z with positive real part it is defined by

..... Click the link for more information.

beta function, also called the Euler integral of the first kind, is a special function defined by

for Re(x), Re(y) > 0.

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.
..... Click the link for more information.

beta function, also called the Euler integral of the first kind, is a special function defined by

for Re(x), Re(y) > 0.

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.
..... Click the link for more information.

beta function, also called the Euler integral of the first kind, is a special function defined by

for Re(x), Re(y) > 0.

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.
..... Click the link for more information.

A random variable is an abstraction of the intuitive concept of chance into the theoretical domains of mathematics, forming the foundations of probability theory and mathematical statistics.
..... Click the link for more information.

skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.

Introduction

Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side.
..... Click the link for more information.

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Beta Distribution

Information about Beta Distribution

Characterization

Probability density function

Cumulative distribution function

Properties

Moments

Quantities of information

Shapes

Parameter estimation

Related distributions

Applications

External links

Definition

Introduction

Definition

Introduction