Class FDistribution
- All Implemented Interfaces:
ContinuousDistribution
The probability density function of \( X \) is:
\[ \begin{aligned} f(x; n, m) &= \frac{1}{\operatorname{B}\left(\frac{n}{2},\frac{m}{2}\right)} \left(\frac{n}{m}\right)^{n/2} x^{n/2 - 1} \left(1+\frac{n}{m} \, x \right)^{-(n+m)/2} \\ &= \frac{n^{\frac n 2} m^{\frac m 2} x^{\frac{n}{2}-1} }{ (nx+m)^{\frac{(n+m)}{2}} \operatorname{B}\left(\frac{n}{2},\frac{m}{2}\right)} \end{aligned} \]
for \( n, m > 0 \) the degrees of freedom, \( \operatorname{B}(a, b) \) is the beta function, and \( x \in [0, \infty) \).
- 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
The denominator degrees of freedom.private final double
LogBeta(n/2, n/2) with n = numerator DF.private final double
Cached value for inverse probability function.private static final double
The minimum degrees of freedom for the denominator when computing the mean.private static final double
The minimum degrees of freedom for the denominator when computing the variance.private final double
n/2 * log(n) + m/2 * log(m) with n = numerator DF and m = denominator DF.private final double
The numerator degrees of freedom.private static final double
Support upper bound.private static final double
Support lower bound.private final double
Cached value for inverse probability function. -
Constructor Summary
ConstructorsModifierConstructorDescriptionprivate
FDistribution
(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom) -
Method Summary
Modifier and TypeMethodDescriptionprivate double
computeDensity
(double x, boolean log) Compute the density at point x.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
Gets the denominator degrees of freedom parameter of this distribution.double
getMean()
Gets the mean of this distribution.double
Gets the numerator degrees of freedom parameter 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 FDistribution
of
(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom) Creates an F-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
createSampler, getMedian, inverseCumulativeProbability, inverseSurvivalProbability, isSupportConnected, probability
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Field Details
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SUPPORT_LO
private static final double SUPPORT_LOSupport lower bound.- See Also:
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SUPPORT_HI
private static final double SUPPORT_HISupport upper bound.- See Also:
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MIN_DENOMINATOR_DF_FOR_MEAN
private static final double MIN_DENOMINATOR_DF_FOR_MEANThe minimum degrees of freedom for the denominator when computing the mean.- See Also:
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MIN_DENOMINATOR_DF_FOR_VARIANCE
private static final double MIN_DENOMINATOR_DF_FOR_VARIANCEThe minimum degrees of freedom for the denominator when computing the variance.- See Also:
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numeratorDegreesOfFreedom
private final double numeratorDegreesOfFreedomThe numerator degrees of freedom. -
denominatorDegreesOfFreedom
private final double denominatorDegreesOfFreedomThe denominator degrees of freedom. -
nHalfLogNmHalfLogM
private final double nHalfLogNmHalfLogMn/2 * log(n) + m/2 * log(m) with n = numerator DF and m = denominator DF. -
logBetaNhalfMhalf
private final double logBetaNhalfMhalfLogBeta(n/2, n/2) with n = numerator DF. -
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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FDistribution
private FDistribution(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom) - Parameters:
numeratorDegreesOfFreedom
- Numerator degrees of freedom.denominatorDegreesOfFreedom
- Denominator degrees of freedom.
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Method Details
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of
public static FDistribution of(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom) Creates an F-distribution.- Parameters:
numeratorDegreesOfFreedom
- Numerator degrees of freedom.denominatorDegreesOfFreedom
- Denominator degrees of freedom.- Returns:
- the distribution
- Throws:
IllegalArgumentException
- ifnumeratorDegreesOfFreedom <= 0
ordenominatorDegreesOfFreedom <= 0
.
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getNumeratorDegreesOfFreedom
public double getNumeratorDegreesOfFreedom()Gets the numerator degrees of freedom parameter of this distribution.- Returns:
- the numerator degrees of freedom.
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getDenominatorDegreesOfFreedom
public double getDenominatorDegreesOfFreedom()Gets the denominator degrees of freedom parameter of this distribution.- Returns:
- the denominator degrees of freedom.
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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.Returns the limit when
x = 0
:df1 < 2
: Infinitydf1 == 2
: 1df1 > 2
: 0
Where
df1
is the numerator degrees of freedom.- 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
.Returns the limit when
x = 0
:df1 < 2
: Infinitydf1 == 2
: 0df1 > 2
: -Infinity
Where
df1
is the numerator degrees of freedom.- 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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computeDensity
private double computeDensity(double x, boolean log) Compute the density at point x. Assumes x is within the support bound.- Parameters:
x
- Valuelog
- true to compute the log density- Returns:
- pdf(x) or logpdf(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 denominator degrees of freedom parameter \( m \), the mean is:
\[ \mathbb{E}[X] = \begin{cases} \frac{m}{m-2} & \text{for } m \gt 2 \\ \text{undefined} & \text{otherwise} \end{cases} \]
- Returns:
- the mean, or
NaN
if it is not defined.
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getVariance
public double getVariance()Gets the variance of this distribution.For numerator degrees of freedom parameter \( n \) and denominator degrees of freedom parameter \( m \), the variance is:
\[ \operatorname{var}[X] = \begin{cases} \frac{2m^2 (n+m-2)}{n (m-2)^2 (m-4)} & \text{for } m \gt 4 \\ \text{undefined} & \text{otherwise} \end{cases} \]
- Returns:
- the variance, or
NaN
if it is not defined.
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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 positive infinity.
- Returns:
positive infinity
.
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