Class BetaDistribution
- All Implemented Interfaces:
ContinuousDistribution
The probability density function of \( X \) is:
\[ f(x; \alpha, \beta) = \frac{1}{ B(\alpha, \beta)} x^{\alpha-1} (1-x)^{\beta-1} \]
for \( \alpha > 0 \), \( \beta > 0 \), \( x \in [0, 1] \), and the beta function, \( B \), is a normalization constant:
\[ B(\alpha, \beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} \]
where \( \Gamma \) is the gamma function.
\( \alpha \) and \( \beta \) are shape parameters.
- See Also:
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Nested Class Summary
Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
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Field Summary
FieldsModifier and TypeFieldDescriptionprivate final double
First shape parameter.private final double
Second shape parameter.private final double
Normalizing factor used in log density computations.private final double
Cached value for inverse probability function.private final double
Cached value for inverse probability function. -
Constructor Summary
Constructors -
Method Summary
Modifier and TypeMethodDescriptioncreateSampler
(org.apache.commons.rng.UniformRandomProvider rng) Creates a sampler.double
cumulativeProbability
(double x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X <= x)
.double
density
(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified pointx
.double
getAlpha()
Gets the first shape parameter of this distribution.double
getBeta()
Gets the second shape parameter of this distribution.double
getMean()
Gets the mean of this distribution.double
Gets the lower bound of the support.double
Gets the upper bound of the support.double
Gets the variance of this distribution.double
logDensity
(double x) Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx
.static BetaDistribution
of
(double alpha, double beta) Creates a beta distribution.double
survivalProbability
(double x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X > x)
.Methods inherited from class org.apache.commons.statistics.distribution.AbstractContinuousDistribution
getMedian, inverseCumulativeProbability, inverseSurvivalProbability, isSupportConnected, probability
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Field Details
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alpha
private final double alphaFirst shape parameter. -
beta
private final double betaSecond shape parameter. -
logBeta
private final double logBetaNormalizing factor used in log density computations. log(beta(a, b)). -
mean
private final double meanCached value for inverse probability function. -
variance
private final double varianceCached value for inverse probability function.
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Constructor Details
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BetaDistribution
private BetaDistribution(double alpha, double beta) - Parameters:
alpha
- First shape parameter (must be positive).beta
- Second shape parameter (must be positive).
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Method Details
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of
Creates a beta distribution.- Parameters:
alpha
- First shape parameter (must be positive).beta
- Second shape parameter (must be positive).- Returns:
- the distribution
- Throws:
IllegalArgumentException
- ifalpha <= 0
orbeta <= 0
.
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getAlpha
public double getAlpha()Gets the first shape parameter of this distribution.- Returns:
- the first shape parameter.
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getBeta
public double getBeta()Gets the second shape parameter of this distribution.- Returns:
- the second shape parameter.
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density
public double density(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified pointx
. In general, the PDF is the derivative of theCDF
. If the derivative does not exist atx
, then an appropriate replacement should be returned, e.g.Double.POSITIVE_INFINITY
,Double.NaN
, or the limit inferior or limit superior of the difference quotient.The density is not defined when
x = 0, alpha < 1
, orx = 1, beta < 1
. In this case the limit of infinity is returned.- Parameters:
x
- Point at which the PDF is evaluated.- Returns:
- the value of the probability density function at
x
.
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logDensity
public double logDensity(double x) Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx
.The density is not defined when
x = 0, alpha < 1
, orx = 1, beta < 1
. In this case the limit of infinity is returned.- Parameters:
x
- Point at which the PDF is evaluated.- Returns:
- the logarithm of the value of the probability density function
at
x
.
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cumulativeProbability
public double cumulativeProbability(double x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X <= x)
. In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.- Parameters:
x
- Point at which the CDF is evaluated.- Returns:
- the probability that a random variable with this
distribution takes a value less than or equal to
x
.
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survivalProbability
public double survivalProbability(double x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X > x)
. In other words, this method represents the complementary cumulative distribution function.By default, this is defined as
1 - cumulativeProbability(x)
, but the specific implementation may be more accurate.- Parameters:
x
- Point at which the survival function is evaluated.- Returns:
- the probability that a random variable with this
distribution takes a value greater than
x
.
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getMean
public double getMean()Gets the mean of this distribution.For first shape parameter \( \alpha \) and second shape parameter \( \beta \), the mean is:
\[ \frac{\alpha}{\alpha + \beta} \]
- Returns:
- the mean.
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getVariance
public double getVariance()Gets the variance of this distribution.For first shape parameter \( \alpha \) and second shape parameter \( \beta \), the variance is:
\[ \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \].
- Returns:
- the variance.
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getSupportLowerBound
public double getSupportLowerBound()Gets the lower bound of the support. It must return the same value asinverseCumulativeProbability(0)
, i.e. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).The lower bound of the support is always 0.
- Returns:
- 0.
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getSupportUpperBound
public double getSupportUpperBound()Gets the upper bound of the support. It must return the same value asinverseCumulativeProbability(1)
, i.e. \( \inf \{ x \in \mathbb R : P(X \le x) = 1 \} \).The upper bound of the support is always 1.
- Returns:
- 1.
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createSampler
public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng) Creates a sampler.- Specified by:
createSampler
in interfaceContinuousDistribution
- Overrides:
createSampler
in classAbstractContinuousDistribution
- Parameters:
rng
- Generator of uniformly distributed numbers.- Returns:
- a sampler that produces random numbers according this distribution.
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