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:
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BetaDistribution beta1 = new BetaDistribution(1.5, 0.8);
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Dim beta1 As BetaDistribution = New BetaDistribution(1.5, 0.8)
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The second constructor takes two extra parameters 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]:
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BetaDistribution beta2 = new BetaDistribution(1.5, 0.8, 1, 4);
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Dim beta2 As BetaDistribution = New BetaDistribution(1.5, 0.8, 1, 4)
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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 parameter: a
NumericalVariable whose distribution is to be estimated.
This constructor estimates a standard beta distribution, with lower bound and upper bound equal to 0 and 1,
respectively.
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 variate using a user-supplied uniform random number generator.
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MersenneTwister random = new MersenneTwister();
double variate = BetaDistribution.GetRandomVariate(random, 1.5, 0.8);
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Dim random As MersenneTwister = New MersenneTwister()
Dim variate As Double = BetaDistribution.GetRandomVariate(random, 1.5, 0.8)
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For details of the properties and methods common to all continuous distribution classes, see the topic on
class.