Beta distribution

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Beta
Probability density function
Cumulative distribution function
Parameters	$α > 0$ shape (real) $β > 0$ shape (real)
Support	$x \in [0; 1]\!$
Probability density function (pdf)	$\frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}\!$
Cumulative distribution function (cdf)	$I_x(\alpha,\beta)\!$
Mean	$\frac{\alpha}{\alpha+\beta}\!$
Median
Mode	$\frac{\alpha-1}{\alpha+\beta-2}\!$ for $α > 1,β > 1$
Variance	$\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\!$
Skewness	$\frac{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}$
Excess Kurtosis	see text
Entropy	$lnΒ(a, b) - (a - 1)ψ(a) - (b - 1)ψ(b) + (a + b - 2)ψ(a + b)$
mgf	$1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}$
Char. func.	${}_1F_1(\alpha; \alpha+\beta; i\,t)\!$

In probability theory and statistics, the beta distribution is a two-parameter family of continuous probability distributions defined on the interval [0, 1], with probability density function (pdf)

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

Parameters α and β must be greater than zero, and B is the beta function.

1 Properties
2 Shapes
3 Parameter estimation
4 Related distributions
5 Applications
6 External links

[edit] Properties

[edit] Normalization

The beta function appears as a normalization constant simply to ensure that the integral of the pdf is unity:

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

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

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

where Γ is the gamma function.

[edit] Moments

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

$\operatorname{E}(X) = \frac{\alpha}{\alpha+\beta}$

$\operatorname{Var}(X) = \frac{\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$

For any two numbers u, v such that 0 < u < 1 and 0 < v < u(1 − u) there is a beta distribution having expected value E(X) = u and variance Var(X) = v.

The skewness is

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

The kurtosis excess is:

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

[edit] 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.

[edit] Shapes

The beta density function (not to be confused with the Beta 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).

[edit] 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

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

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

[edit] Related distributions

The connection with the binomial distribution is mentioned below.
The B(1,1) distribution is identical to the standard uniform distribution.
If X and Y are independently distributed Γ(α, θ) and Γ(β, θ) respectively, then X / (X + Y) is distributed B(α,β).
If X and Y are independently distributed B(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.

[edit] Applications

B(i,j) with integer values of i and j is the distribution of the i-th 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 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, CPM and other project management / control systems to describe the time to completion of a task.

[edit] External links

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

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