Class TDistribution
- All Implemented Interfaces:
ContinuousDistribution
- Direct Known Subclasses:
TDistribution.NormalTDistribution
,TDistribution.StudentsTDistribution
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 \( x \in (-\infty, \infty) \).
- See Also:
-
Nested Class Summary
Nested ClassesModifier and TypeClassDescriptionprivate static class
Specialisation of the T-distribution used when there are infinite degrees of freedom.private static class
Implementation of Student's T-distribution.Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
-
Field Summary
FieldsModifier and TypeFieldDescriptionprivate final double
The degrees of freedom.(package private) static final NormalDistribution
A standard normal distribution used for calculations. -
Constructor Summary
Constructors -
Method Summary
Modifier and TypeMethodDescriptiondouble
Gets the degrees of freedom parameter of this distribution.abstract double
getMean()
Gets the mean of this distribution.double
Gets the lower bound of the support.double
Gets the upper bound of the support.abstract double
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)
.Methods inherited from class org.apache.commons.statistics.distribution.AbstractContinuousDistribution
createSampler, getMedian, inverseCumulativeProbability, isSupportConnected, probability
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
Methods inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
cumulativeProbability, density, logDensity
-
Field Details
-
STANDARD_NORMAL
A standard normal distribution used for calculations. This is immutable and thread-safe and can be used across instances. -
degreesOfFreedom
private final double degreesOfFreedomThe degrees of freedom.
-
-
Constructor Details
-
TDistribution
private TDistribution(double degreesOfFreedom) - Parameters:
degreesOfFreedom
- Degrees of freedom.
-
-
Method Details
-
of
Creates a Student's t-distribution.- Parameters:
degreesOfFreedom
- Degrees of freedom.- Returns:
- the distribution
- Throws:
IllegalArgumentException
- ifdegreesOfFreedom <= 0
-
getDegreesOfFreedom
public double getDegreesOfFreedom()Gets the degrees of freedom parameter of this distribution.- Returns:
- the degrees of freedom.
-
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
.
-
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:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \ge x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \ge x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
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
).
-
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.
-
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.
-
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
.
-
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
.
-