Class TDistribution
- java.lang.Object
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- org.apache.commons.statistics.distribution.TDistribution
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- All Implemented Interfaces:
ContinuousDistribution
public abstract class TDistribution extends Object
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
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Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
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Method Summary
All Methods Static Methods Instance Methods Abstract Methods Concrete Methods Modifier and Type Method Description ContinuousDistribution.SamplercreateSampler(org.apache.commons.rng.UniformRandomProvider rng)Creates a sampler.doublegetDegreesOfFreedom()Gets the degrees of freedom parameter of this distribution.abstract doublegetMean()Gets the mean of this distribution.doublegetSupportLowerBound()Gets the lower bound of the support.doublegetSupportUpperBound()Gets the upper bound of the support.abstract doublegetVariance()Gets the variance of this distribution.doubleinverseCumulativeProbability(double p)Computes the quantile function of this distribution.doubleinverseSurvivalProbability(double p)Computes the inverse survival probability function of this distribution.static TDistributionof(double degreesOfFreedom)Creates a Student's t-distribution.doubleprobability(double x0, double x1)For a random variableXwhose values are distributed according to this distribution, this method returnsP(x0 < X <= x1).doublesurvivalProbability(double x)For a random variableXwhose values are distributed according to this distribution, this method returnsP(X > x).-
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:
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 variableXwhose 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 variableXdistributed 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:
inverseSurvivalProbabilityin interfaceContinuousDistribution- 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
NaNif 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
NaNif 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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probability
public double probability(double x0, double x1)For a random variableXwhose values are distributed according to this distribution, this method returnsP(x0 < X <= x1). The default implementation uses the identityP(x0 < X <= x1) = P(X <= x1) - P(X <= x0)- Specified by:
probabilityin interfaceContinuousDistribution- Parameters:
x0- Lower bound (exclusive).x1- Upper bound (inclusive).- Returns:
- the probability that a random variable with this distribution
takes a value between
x0andx1, excluding the lower and including the upper endpoint.
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inverseCumulativeProbability
public double inverseCumulativeProbability(double p)
Computes the quantile function of this distribution. For a random variableXdistributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]
The default implementation returns:
ContinuousDistribution.getSupportLowerBound()forp = 0,ContinuousDistribution.getSupportUpperBound()forp = 1, or- the result of a search for a root between the lower and upper bound using
cumulativeProbability(x) - p. The bounds may be bracketed for efficiency.
- Specified by:
inverseCumulativeProbabilityin interfaceContinuousDistribution- Parameters:
p- Cumulative probability.- Returns:
- the smallest
p-quantile of this distribution (largest 0-quantile forp = 0). - Throws:
IllegalArgumentException- ifp < 0orp > 1
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createSampler
public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
Creates a sampler.- Specified by:
createSamplerin interfaceContinuousDistribution- Parameters:
rng- Generator of uniformly distributed numbers.- Returns:
- a sampler that produces random numbers according this distribution.
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