Chemistry - Beta distribution

In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function defined on the interval [0, 1]:

$f(x) = [\mbox{constant}]\cdot x^{\alpha-1}(1-x)^{\beta-1}.$

where $α$ and $β$ are parameters that must be greater than zero.

When the "constant" is included explicitly, the density looks like this:

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

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

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

where $Γ$ and $B$ are, respectively, the gamma function and the beta 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 $0 < \operatorname{var}(X) < \operatorname{E}(X) (1 - \operatorname{E}(X))$ .

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)

Categories: Probability distributions