Class CauchyDistribution
- All Implemented Interfaces:
ContinuousDistribution
The probability density function of \( X \) is:
\[ f(x; x_0, \gamma) = { 1 \over \pi \gamma } \left[ { \gamma^2 \over (x - x_0)^2 + \gamma^2 } \right] \]
for \( x_0 \) the location, \( \gamma > 0 \) the scale, and \( x \in (-\infty, \infty) \).
- 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 location of this distribution.private final double
The scale of this distribution.private final double
Density factor (scale^2).private final double
Density factor (scale / pi). -
Constructor Summary
Constructors -
Method Summary
Modifier and TypeMethodDescriptionprivate static double
cdf
(double x) Compute the CDF of the Cauchy distribution with location 0 and scale 1.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
Gets the location parameter of this distribution.double
getMean()
Gets the mean of this distribution.(package private) double
Gets the median.double
getScale()
Gets the scale 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
inverseCumulativeProbability
(double p) Computes the quantile function of this distribution.double
inverseSurvivalProbability
(double p) Computes the inverse survival probability function of this distribution.static CauchyDistribution
of
(double location, double scale) Creates a Cauchy 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
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
logDensity
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Field Details
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location
private final double locationThe location of this distribution. -
scale
private final double scaleThe scale of this distribution. -
scaleOverPi
private final double scaleOverPiDensity factor (scale / pi). -
scale2
private final double scale2Density factor (scale^2).
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Constructor Details
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CauchyDistribution
private CauchyDistribution(double location, double scale) - Parameters:
location
- Location parameter.scale
- Scale parameter.
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Method Details
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of
Creates a Cauchy distribution.- Parameters:
location
- Location parameter.scale
- Scale parameter.- Returns:
- the distribution
- Throws:
IllegalArgumentException
- ifscale <= 0
.
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getLocation
public double getLocation()Gets the location parameter of this distribution.- Returns:
- the location parameter.
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getScale
public double getScale()Gets the scale parameter of this distribution.- Returns:
- the scale parameter.
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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.- Parameters:
x
- Point at which the PDF is evaluated.- Returns:
- the value of the probability density function at
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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cdf
private static double cdf(double x) Compute the CDF of the Cauchy distribution with location 0 and scale 1.- Parameters:
x
- Point at which the CDF is evaluated- Returns:
- CDF(x)
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inverseCumulativeProbability
public double inverseCumulativeProbability(double p) Computes the quantile function of this distribution. For a random variableX
distributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]
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.
Returns
Double.NEGATIVE_INFINITY
whenp == 0
andDouble.POSITIVE_INFINITY
whenp == 1
.- Specified by:
inverseCumulativeProbability
in interfaceContinuousDistribution
- Overrides:
inverseCumulativeProbability
in classAbstractContinuousDistribution
- Parameters:
p
- Cumulative probability.- Returns:
- the smallest
p
-quantile of this distribution (largest 0-quantile forp = 0
).
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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:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \ge x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \ge x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
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.
Returns
Double.NEGATIVE_INFINITY
whenp == 1
andDouble.POSITIVE_INFINITY
whenp == 0
.- 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 double getMean()Gets the mean of this distribution.The mean is always undefined.
- Returns:
NaN
.
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getVariance
public double getVariance()Gets the variance of this distribution.The variance is always undefined.
- Returns:
NaN
.
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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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getMedian
double getMedian()Gets the median. This is used to determine if the arguments to theAbstractContinuousDistribution.probability(double, double)
function are in the upper or lower domain.The default implementation calls
AbstractContinuousDistribution.inverseCumulativeProbability(double)
with a value of 0.5.- Overrides:
getMedian
in classAbstractContinuousDistribution
- Returns:
- the median
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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.
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