Class BetaDistribution
- java.lang.Object
-
- org.apache.commons.statistics.distribution.AbstractContinuousDistribution
-
- org.apache.commons.statistics.distribution.BetaDistribution
-
- All Implemented Interfaces:
ContinuousDistribution
public final class BetaDistribution extends AbstractContinuousDistribution
Implementation of the beta distribution.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.
-
-
Nested Class Summary
-
Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
-
-
Field Summary
Fields Modifier and Type Field Description private double
alpha
First shape parameter.private double
beta
Second shape parameter.private double
logBeta
Normalizing factor used in log density computations.private double
mean
Cached value for inverse probability function.private double
variance
Cached value for inverse probability function.
-
Constructor Summary
Constructors Modifier Constructor Description private
BetaDistribution(double alpha, double beta)
-
Method Summary
All Methods Static Methods Instance Methods Concrete Methods Modifier and Type Method Description ContinuousDistribution.Sampler
createSampler(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
getSupportLowerBound()
Gets the lower bound of the support.double
getSupportUpperBound()
Gets the upper bound of the support.double
getVariance()
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
-
-
-
-
Field Detail
-
alpha
private final double alpha
First shape parameter.
-
beta
private final double beta
Second shape parameter.
-
logBeta
private final double logBeta
Normalizing factor used in log density computations. log(beta(a, b)).
-
mean
private final double mean
Cached value for inverse probability function.
-
variance
private final double variance
Cached value for inverse probability function.
-
-
Method Detail
-
of
public static BetaDistribution of(double alpha, double beta)
Creates a beta distribution.- Parameters:
alpha
- First shape parameter (must be positive).beta
- Second shape parameter (must be positive).- Returns:
- the distribution
- Throws:
java.lang.IllegalArgumentException
- ifalpha <= 0
orbeta <= 0
.
-
getAlpha
public double getAlpha()
Gets the first shape parameter of this distribution.- Returns:
- the first shape parameter.
-
getBeta
public double getBeta()
Gets the second shape parameter of this distribution.- Returns:
- the second shape parameter.
-
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 the CDF. 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
.
-
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
.
-
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
.
-
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
.
-
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.
-
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.
-
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.
-
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.
-
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.
-
-