LevyDistribution (Apache Commons Math 3.6.1 API)

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

See Also:
Serialized Form

Field Summary
- Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution
  random, randomData, SOLVER_DEFAULT_ABSOLUTE_ACCURACY

Constructor Summary

Constructors
Constructor and Description
`LevyDistribution(double mu, double c)` Build a new instance.
`LevyDistribution(RandomGenerator rng, double mu, double c)` Creates a LevyDistribution.

Method Summary

Methods
Modifier and Type	Method and Description
`double`	`cumulativeProbability(double x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
`double`	`density(double x)` Returns the probability density function (PDF) of this distribution evaluated at the specified point `x`.
`double`	`getLocation()` Get the location parameter of the distribution.
`double`	`getNumericalMean()` Use this method to get the numerical value of the mean of this distribution.
`double`	`getNumericalVariance()` Use this method to get the numerical value of the variance of this distribution.
`double`	`getScale()` Get the scale parameter of the distribution.
`double`	`getSupportLowerBound()` Access the lower bound of the support.
`double`	`getSupportUpperBound()` Access the upper bound of the support.
`double`	`inverseCumulativeProbability(double p)` Computes the quantile function of this distribution.
`boolean`	`isSupportConnected()` Use this method to get information about whether the support is connected, i.e.
`boolean`	`isSupportLowerBoundInclusive()` Whether or not the lower bound of support is in the domain of the density function.
`boolean`	`isSupportUpperBoundInclusive()` Whether or not the upper bound of support is in the domain of the density function.
`double`	`logDensity(double x)` Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point `x`.

Methods inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution
cumulativeProbability, getSolverAbsoluteAccuracy, probability, probability, reseedRandomGenerator, sample, sample

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - LevyDistribution
```
public LevyDistribution(double mu,
                double c)
```
    Build a new instance.
    Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see AbstractRealDistribution.sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead.
    
    Parameters:
    mu - location parameter
    c - scale parameter
    Since:
    
    3.4
  - 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
- Method Detail
  - 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
  - logDensity
```
public double logDensity(double x)
```
    Returns the natural logarithm of 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. Note that due to the floating point precision and under/overflow issues, this method will for some distributions be more precise and faster than computing the logarithm of RealDistribution.density(double). The default implementation simply computes the logarithm of density(x). See documentation of density(double) for computation details.
    
    Overrides:
    
    logDensity in class AbstractRealDistribution
    
    Parameters:
    x - the point at which the PDF is evaluated
    
    Returns:
    the logarithm of 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

Class LevyDistribution

Field Summary

Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

Constructor Summary

Method Summary

Methods inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

Methods inherited from class java.lang.Object

Constructor Detail

LevyDistribution

LevyDistribution

Method Detail

density

logDensity

cumulativeProbability

inverseCumulativeProbability

getScale

getLocation

getNumericalMean

getNumericalVariance

getSupportLowerBound

getSupportUpperBound

isSupportLowerBoundInclusive

isSupportUpperBoundInclusive

isSupportConnected