Class TDistribution
- java.lang.Object
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- org.apache.commons.statistics.distribution.AbstractContinuousDistribution
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- org.apache.commons.statistics.distribution.TDistribution
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- All Implemented Interfaces:
ContinuousDistribution
- Direct Known Subclasses:
TDistribution.NormalTDistribution
,TDistribution.StudentsTDistribution
public abstract class TDistribution extends AbstractContinuousDistribution
Implementation of Student's t-distribution.The probability density function of \( X \) is:
\[ f(x; v) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{\!-\frac{\nu+1}{2}} \]
for \( v > 0 \) the degrees of freedom, \( \Gamma \) is the gamma function, and \( x \in (-\infty, \infty) \).
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Nested Class Summary
Nested Classes Modifier and Type Class Description private static class
TDistribution.NormalTDistribution
Specialisation of the T-distribution used when there are infinite degrees of freedom.private static class
TDistribution.StudentsTDistribution
Implementation of Student's T-distribution.-
Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
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Field Summary
Fields Modifier and Type Field Description private double
degreesOfFreedom
The degrees of freedom.
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Constructor Summary
Constructors Constructor Description TDistribution(double degreesOfFreedom)
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Method Summary
All Methods Static Methods Instance Methods Abstract Methods Concrete Methods Modifier and Type Method Description double
getDegreesOfFreedom()
Gets the degrees of freedom parameter of this distribution.abstract 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.abstract double
getVariance()
Gets the variance of this distribution.double
inverseSurvivalProbability(double p)
Computes the inverse survival probability function of this distribution.static TDistribution
of(double degreesOfFreedom)
Creates a Student's t-distribution.double
survivalProbability(double x)
For a random variableX
whose values are distributed according to this distribution, this method returnsP(X > x)
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Methods inherited from class org.apache.commons.statistics.distribution.AbstractContinuousDistribution
createSampler, getMedian, inverseCumulativeProbability, isSupportConnected, probability
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Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
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Methods inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
cumulativeProbability, density, logDensity
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Method Detail
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of
public static TDistribution of(double degreesOfFreedom)
Creates a Student's t-distribution.- Parameters:
degreesOfFreedom
- Degrees of freedom.- Returns:
- the distribution
- Throws:
java.lang.IllegalArgumentException
- ifdegreesOfFreedom <= 0
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getDegreesOfFreedom
public double getDegreesOfFreedom()
Gets the degrees of freedom parameter of this distribution.- Returns:
- the degrees of freedom.
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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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inverseSurvivalProbability
public double inverseSurvivalProbability(double p)
Computes the inverse survival probability function of this distribution. For a random variableX
distributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
By default, this is defined as
inverseCumulativeProbability(1 - p)
, but the specific implementation may be more accurate.The default implementation returns:
ContinuousDistribution.getSupportLowerBound()
forp = 1
,ContinuousDistribution.getSupportUpperBound()
forp = 0
, or- the result of a search for a root between the lower and upper bound using
survivalProbability(x) - p
. The bounds may be bracketed for efficiency.
- Specified by:
inverseSurvivalProbability
in interfaceContinuousDistribution
- Overrides:
inverseSurvivalProbability
in classAbstractContinuousDistribution
- Parameters:
p
- Survival probability.- Returns:
- the smallest
(1-p)
-quantile of this distribution (largest 0-quantile forp = 1
).
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getMean
public abstract double getMean()
Gets the mean of this distribution.For degrees of freedom parameter \( v \), the mean is:
\[ \mathbb{E}[X] = \begin{cases} 0 & \text{for } v \gt 1 \\ \text{undefined} & \text{otherwise} \end{cases} \]
- Returns:
- the mean, or
NaN
if it is not defined.
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getVariance
public abstract double getVariance()
Gets the variance of this distribution.For degrees of freedom parameter \( v \), the variance is:
\[ \operatorname{var}[X] = \begin{cases} \frac{v}{v - 2} & \text{for } v \gt 2 \\ \infty & \text{for } 1 \lt v \le 2 \\ \text{undefined} & \text{otherwise} \end{cases} \]
- 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 negative infinity.
- Returns:
- negative infinity.
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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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