- Extreme Optimization
- Documentation
- Statistics Library User's Guide
- Continuous Distributions
- Continuous Distributions
- The Beta Distribution
- The Cauchy Distribution
- The Chi Square Distribution
- The Erlang Distribution
- The Exponential Distribution
- The F Distribution
- The Gamma Distribution
- The Generalized Pareto Distribution
- The Gumbel Distribution
- The Laplace Distribution
- The Logistic Distribution
- Log-Logistic Distribution
- The Lognormal Distribution
- The Non-central Beta Distribution
- The Non-central Chi Square Distribution
- The Non-central F Distribution
- The Non-central Student t distribution
- The Normal Distribution
- The Pareto Distribution
- The Rayleigh Distribution
- Student's t Distribution
- The Transformed Beta Distribution
- The Transformed Gamma Distribution
- The Triangular Distribution
- The Continuous Uniform Distribution
- The Weibull Distribution

- The Beta Distribution

## The Beta Distribution | Extreme Optimization Numerical Libraries for .NET Professional |

The Beta distribution is often used to model random variables with a finite range. The Beta distribution is also used in Bayesian analysis.

The Beta distribution has two shape parameters, usually denoted by the Greek letters α and β. Its probability density function (PDF) is:

Unlike most other distributions, location and scale parameters are not usually used to specify the general form of the Beta distribution. Instead, the lower and upper bounds of the definition interval are used.

For certain specific values of the parameters α and β, the beta distribution is equivalent to a simpler distribution. For α = β = 1, the beta distribution is equivalent to the uniform distribution. For α = 1 and β = 2, and α = 2 and β = 1, the beta distribution reduces to a triangular distribution. For α and β very large, the beta distribution approximates to the normal distribution.

The beta distribution is implemented by the BetaDistribution class. It has three constructors. The first constructor takes the two shape parameters, α and β as arguments. The following constructs a beta distribution with α = 1.5 and β = 0.8:

The second constructor takes two extra arguments that specify the lower and upper bound of the interval on which the beta distribution is defined. The default is a lower bound of 0 and an upper bound of 1. The following constructs a beta distribution with α = 1.5 and β = 0.8 over the interval [1, 4]:

If a variable is assumed to have a beta distribution, then the parameters of the distribution can be estimated
using the method of matching moments. The third constructor performs this calculation. It takes one argument: a
Vector

Note that parameter estimation says nothing about how well the estimated distribution fits the variable's distribution. Use one of the goodness-of-fit tests to verify the appropriateness of the choice of distribution.

The BetaDistribution class has four specific properties that correspond to the parameters of the distribution. The Alpha and Beta properties return the shape parameters, α and β. The LowerBound and UpperBound properties return the bounds of the interval on which the beta distribution is defined.

BetaDistribution has one static (*Shared* in Visual Basic) method, Sample, which generates
a random sample using a user-supplied uniform random number generator.

var random = new MersenneTwister(); double sample = BetaDistribution.Sample(random, 1.5, 0.8);

For details of the properties and methods common to all continuous distribution classes, see the topic on continuous distributions..

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