org.apache.commons.math3.distribution
Class LevyDistribution

java.lang.Object
  org.apache.commons.math3.distribution.AbstractRealDistribution
      org.apache.commons.math3.distribution.LevyDistribution

All Implemented Interfaces:

Serializable, RealDistribution

public class LevyDistribution
extends AbstractRealDistribution

This class implements the Lévy distribution.

Since:

3.2

Version:

$Id: LevyDistribution.java 1456957 2013-03-15 13:54:52Z luc $

See Also:

Serialized Form

LevyDistribution

public LevyDistribution(RandomGenerator rng,
                        double mu,
                        double c)

Creates a LevyDistribution.

Parameters:

rng - random generator to be used for sampling

mu - location

c - scale parameter

density

public double density(double x)

Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, 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.

From Wikipedia: The probability density function of the Lévy distribution over the domain is

 f(x; μ, c) = √(c / 2π) * e-c / 2 (x - μ) / (x - μ)3/2
 

For this distribution, X, this method returns P(X < x). If x is less than location parameter μ, Double.NaN is returned, as in these cases the distribution is not defined.

Parameters:

x - the point at which the PDF is evaluated

Returns:

the value of the probability density function at point x

cumulativeProbability

public double cumulativeProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.

From Wikipedia: the cumulative distribution function is

 f(x; u, c) = erfc (√ (c / 2 (x - u )))
 

Parameters:

x - the 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

inverseCumulativeProbability

public double inverseCumulativeProbability(double p)
                                    throws OutOfRangeException

Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is

• inf{x in R | P(X<=x) >= p} for 0 < p <= 1,
• inf{x in R | P(X<=x) > 0} for p = 0.

The default implementation returns

• RealDistribution.getSupportLowerBound() for p = 0,
• RealDistribution.getSupportUpperBound() for p = 1.

Specified by:

inverseCumulativeProbability in interface RealDistribution

Overrides:

inverseCumulativeProbability in class AbstractRealDistribution

Parameters:

p - the cumulative probability

Returns:

the smallest p-quantile of this distribution (largest 0-quantile for p = 0)

Throws:

OutOfRangeException - if p < 0 or p > 1

getScale

public double getScale()

Get the scale parameter of the distribution.

Returns:

scale parameter of the distribution

getLocation

public double getLocation()

Get the location parameter of the distribution.

Returns:

location parameter of the distribution

getNumericalMean

public double getNumericalMean()

Use this method to get the numerical value of the mean of this distribution.

Returns:

the mean or Double.NaN if it is not defined

getNumericalVariance

public double getNumericalVariance()

Use this method to get the numerical value of the variance of this distribution.

Returns:

the variance (possibly Double.POSITIVE_INFINITY as for certain cases in TDistribution) or Double.NaN if it is not defined

getSupportLowerBound

public double getSupportLowerBound()

Access the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0). In other words, this method must return

inf {x in R | P(X <= x) > 0}.

Returns:

lower bound of the support (might be Double.NEGATIVE_INFINITY)

getSupportUpperBound

public double getSupportUpperBound()

Access the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1). In other words, this method must return

inf {x in R | P(X <= x) = 1}.

Returns:

upper bound of the support (might be Double.POSITIVE_INFINITY)

isSupportLowerBoundInclusive

public boolean isSupportLowerBoundInclusive()

Whether or not the lower bound of support is in the domain of the density function. Returns true iff getSupporLowerBound() is finite and density(getSupportLowerBound()) returns a non-NaN, non-infinite value.

Returns:

true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there

isSupportUpperBoundInclusive

public boolean isSupportUpperBoundInclusive()

Whether or not the upper bound of support is in the domain of the density function. Returns true iff getSupportUpperBound() is finite and density(getSupportUpperBound()) returns a non-NaN, non-infinite value.

Returns:

true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there

isSupportConnected

public boolean isSupportConnected()

Use this method to get information about whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support.

Returns:

whether the support is connected or not