Beta distribution

{\mathrm{B}(\alpha,\beta)}\!| cdf = $I_x(\alpha,\beta)\!$ | mean = $\frac{\alpha}{\alpha+\beta}\!$ | median =| mode = $\frac{\alpha-1}{\alpha+\beta-2}\!$ for $\alpha>1, \beta>1$ | 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 = ${}_1F_1(\alpha; \alpha+\beta; i\,t)\!$ | }} In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval 1:

f(x;\alpha,\beta) = \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;\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.

$B(i,j)$ with integer values of i and j is the distribution of the j-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 j-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.

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)}\!

Parameter estimation

When the expected value and variance of a beta random variable X are given, the parameters α and β are calculated by the formulae:

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

\beta = (1-\operatorname{E}(X)) \left( \frac{\operatorname{E}(X) (1 - \operatorname{E}(X))}{\operatorname{var}(X)} - 1 \right).

If the sample mean and sample variance are put in place of E(X) and var(X), then the result values of α and β are estimates of those parameters by the method of moments.

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.

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 $\mathrm{B}_x(\alpha,\beta)$ is the incomplete beta function and $I_x(\alpha,\beta)$ is the regularized incomplete beta function. For integer values of $\alpha$ and $\beta$ , this come to:

I_x(\alpha,\beta) = \sum_{j=\alpha}^{\alpha+\beta-1} {(\alpha+\beta-1)! \over \alpha!(\beta-1)!} x^\alpha (1-x)^{\beta-1}

which again shows the connection with the binomial distribution.

Shapes

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)
\alpha < 1,\ \beta \geq 1 or \alpha = 1,\ \beta > 1 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
\alpha = 1,\ \beta < 1 or \alpha > 1,\ \beta \leq 1 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 $\alpha = \beta$ then the density function is symmetric about 1/2 (red & purple plots).

Related distributions

The connection with the binomial distribution has been mentioned above.
$X \sim \mathrm{Uniform}(0,1)$ (that is, it follows a uniform distribution) if $X \sim \mathrm{Beta}(\alpha = 1, \beta = 1)$ .
If $X \sim \mathrm{Gamma}(\alpha, \theta)$ and $Y \sim \mathrm{Gamma}(\beta, \theta)$ are independent gamma variates, then $X/(X+Y) \sim \mathrm{Beta}(\alpha,\beta)$ .
If $X \sim \mathrm{Beta}(\alpha,\beta)$ is a beta variate and $Y \sim \mathrm{F}(2\beta,2\alpha)$ is an independent F variate, then $\Pr \leq \alpha/(\alpha+x\,\beta) = \Pr > x$ for all $x>0$ .
The Dirichlet distribution is the multivariate generalization of the beta distribution.
The Kumaraswamy distribution resembles the beta distribution.

Continuous distributions

Applications

Beta distributions are used extensively in Bayesian statistics, since beta distributions provide a family of conjugate prior distributions for binomial 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.

The c.d.f of the Beta distribution is used as a convenient way of obtaining the sum over a set of binomial outcomes.

External links

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