Beta distribution

**Beta**
Probability density function Missing image Beta_distribution_pdf.png Probability density function for the Beta distribution
Cumulative distribution function Missing image Beta_distribution_cdf.png Cumulative distribution function for the Beta distribution
Parameters	$α > 0$ shape (real) $β > 0$ shape (real)
Support	$x \in [0; 1]\!$
pdf	$\frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}\!$
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	$\frac{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}$
Kurtosis	see text
Entropy
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 continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]:

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

where $α$ and $β$ are parameters that must be greater than zero and $B$ is the beta function.

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

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

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

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

where $Γ$ is the gamma function.

The special case of the beta distribution when $α = 1$ and $β = 1$ is the standard uniform distribution.

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

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

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

The kurtosis excess is:

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

On the other hand, with the expected value and variance of a beta random variable $X$ given, the parameters $α$ and $β$ are calculated by the formulae:

$\alpha = \operatorname{E}(X) \left( \frac{\operatorname{E}(X)}{\operatorname{var}(X)} \left[ 1 - \operatorname{E}(X) \right] - 1 \right),$

$\beta = \alpha \frac{1-\operatorname{E}(X)}{\operatorname{E}(X)}$

where $0 < \operatorname{E}(X) < 1$ and $0 < \operatorname{var}(X) < \operatorname{E}(X) (1 - \operatorname{E}(X))$ .

Contents

1 Cumulative distribution function

2 Shapes

3 Related distributions

4 Applications

Cumulative distribution function

The cumulative distribution function is

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

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

Shapes

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

$α = β = 1$ is the uniform distribution
$α = β$ is symmetric about 1/2 (red & purple plots)
$\alpha < 1,\ \beta > 1$ is U-shaped (red plot)
$\alpha > 1,\ \beta = 1$ is strictly increasing (green plot)
$\alpha = 1,\ \beta > 1$ is strictly decreasing (blue plot)
$\alpha > 1,\ \beta > 1$ is unimodal (purple & black plots)

Related distributions

$X \sim \mathrm{Uniform}(0,1)$ is a uniform distribution if $X \sim \mathrm{Beta}(\alpha = 1, \beta = 1)$ .

Applications

Beta distributions are used extensively in Bayesian statistics, since the beta distribution is the conjugate prior distribution to the binomial distribution.

Beta distribution

Cumulative distribution function

Shapes

Related distributions

Applications

See also: Beta distribution, Bayesian statistics, Beta function, Binomial distribution, Characteristic function, Conjugate prior distribution, Cumulative distribution function, Expected value