ParetoDistribution (Apache Commons Math 3.6.1 API)

java.lang.Object
- org.apache.commons.math3.distribution.AbstractRealDistribution
- - org.apache.commons.math3.distribution.ParetoDistribution

All Implemented Interfaces:

Serializable, RealDistribution
```
public class ParetoDistribution
extends AbstractRealDistribution
```
Implementation of the Pareto distribution.
Parameters: The probability distribution function of X is given by (for x >= k):
```
  α * k^α / x^(α + 1)
 
```
- k is the scale parameter: this is the minimum possible value of X,
- α is the shape parameter: this is the Pareto index
Since:

3.3

See Also:
Pareto distribution (Wikipedia), Pareto distribution (MathWorld), Serialized Form

Field Summary

Fields
Modifier and Type Field and Description

static double DEFAULT_INVERSE_ABSOLUTE_ACCURACY
Default inverse cumulative probability accuracy.
- Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution
  random, randomData, SOLVER_DEFAULT_ABSOLUTE_ACCURACY

Fields
Modifier and Type	Field and Description
`static double`	`DEFAULT_INVERSE_ABSOLUTE_ACCURACY` Default inverse cumulative probability accuracy.

Constructor Summary

Constructors
Constructor and Description
`ParetoDistribution()` Create a Pareto distribution with a scale of `1` and a shape of `1`.
`ParetoDistribution(double scale, double shape)` Create a Pareto distribution using the specified scale and shape.
`ParetoDistribution(double scale, double shape, double inverseCumAccuracy)` Create a Pareto distribution using the specified scale, shape and inverse cumulative distribution accuracy.
`ParetoDistribution(RandomGenerator rng, double scale, double shape)` Creates a Pareto distribution.
`ParetoDistribution(RandomGenerator rng, double scale, double shape, double inverseCumAccuracy)` Creates a Pareto distribution.

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`	`cumulativeProbability(double x0, double x1)` Deprecated. See `RealDistribution.cumulativeProbability(double,double)`
`double`	`density(double x)` Returns the probability density function (PDF) of this distribution evaluated at the specified point `x`.
`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()` Returns the scale parameter of this distribution.
`double`	`getShape()` Returns the shape parameter of this distribution.
`protected double`	`getSolverAbsoluteAccuracy()` Returns the solver absolute accuracy for inverse cumulative computation.
`double`	`getSupportLowerBound()` Access the lower bound of the support.
`double`	`getSupportUpperBound()` Access the upper bound of the support.
`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`.
`double`	`sample()` Generate a random value sampled from this distribution.

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

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

- Field Detail
  - DEFAULT_INVERSE_ABSOLUTE_ACCURACY
```
public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY
```
    Default inverse cumulative probability accuracy.
    
    See Also:
    Constant Field Values
- Constructor Detail
  - ParetoDistribution
```
public ParetoDistribution()
```
    Create a Pareto distribution with a scale of 1 and a shape of 1.
  - ParetoDistribution
```
public ParetoDistribution(double scale,
                  double shape)
                   throws NotStrictlyPositiveException
```
    Create a Pareto distribution using the specified scale and shape.
    Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see 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:
    scale - the scale parameter of this distribution
    shape - the shape parameter of this distribution
    
    Throws:
    
    NotStrictlyPositiveException - if scale <= 0 or shape <= 0.
  - ParetoDistribution
```
public ParetoDistribution(double scale,
                  double shape,
                  double inverseCumAccuracy)
                   throws NotStrictlyPositiveException
```
    Create a Pareto distribution using the specified scale, shape and inverse cumulative distribution accuracy.
    Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see 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:
    scale - the scale parameter of this distribution
    shape - the shape parameter of this distribution
    inverseCumAccuracy - Inverse cumulative probability accuracy.
    
    Throws:
    
    NotStrictlyPositiveException - if scale <= 0 or shape <= 0.
  - ParetoDistribution
```
public ParetoDistribution(RandomGenerator rng,
                  double scale,
                  double shape)
                   throws NotStrictlyPositiveException
```
    Creates a Pareto distribution.
    
    Parameters:
    rng - Random number generator.
    scale - Scale parameter of this distribution.
    shape - Shape parameter of this distribution.
    
    Throws:
    
    NotStrictlyPositiveException - if scale <= 0 or shape <= 0.
  - ParetoDistribution
```
public ParetoDistribution(RandomGenerator rng,
                  double scale,
                  double shape,
                  double inverseCumAccuracy)
                   throws NotStrictlyPositiveException
```
    Creates a Pareto distribution.
    
    Parameters:
    rng - Random number generator.
    scale - Scale parameter of this distribution.
    shape - Shape parameter of this distribution.
    inverseCumAccuracy - Inverse cumulative probability accuracy.
    
    Throws:
    
    NotStrictlyPositiveException - if scale <= 0 or shape <= 0.
- Method Detail
  - getScale
```
public double getScale()
```
    Returns the scale parameter of this distribution.
    
    Returns:
    the scale parameter
  - getShape
```
public double getShape()
```
    Returns the shape parameter of this distribution.
    
    Returns:
    the shape 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.
    For scale k, and shape α of this distribution, the PDF is given by
    - 0 if x < k,
    - α * k^α / x^(α + 1) otherwise.
    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.
    For scale k, and shape α of this distribution, the CDF is given by
    - 0 if x < k,
    - 1 - (k / x)^α otherwise.
    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
  - cumulativeProbability
```
@Deprecated
public double cumulativeProbability(double x0,
                                      double x1)
                             throws NumberIsTooLargeException
```
    Deprecated. See RealDistribution.cumulativeProbability(double,double)
    
    For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1). The default implementation uses the identity
    P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)
    
    Specified by:
    
    cumulativeProbability in interface RealDistribution
    
    Overrides:
    
    cumulativeProbability in class AbstractRealDistribution
    
    Parameters:
    x0 - the exclusive lower bound
    x1 - the inclusive upper bound
    
    Returns:
    the probability that a random variable with this distribution takes a value between x0 and x1, excluding the lower and including the upper endpoint
    
    Throws:
    
    NumberIsTooLargeException - if x0 > x1
  - getSolverAbsoluteAccuracy
```
protected double getSolverAbsoluteAccuracy()
```
    Returns the solver absolute accuracy for inverse cumulative computation. You can override this method in order to use a Brent solver with an absolute accuracy different from the default.
    
    Overrides:
    
    getSolverAbsoluteAccuracy in class AbstractRealDistribution
    
    Returns:
    the maximum absolute error in inverse cumulative probability estimates
  - getNumericalMean
```
public double getNumericalMean()
```
    Use this method to get the numerical value of the mean of this distribution.
    For scale k and shape α, the mean is given by
    - ∞ if α <= 1,
    - α * k / (α - 1) otherwise.
    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.
    For scale k and shape α, the variance is given by
    - ∞ if 1 < α <= 2,
    - k^2 * α / ((α - 1)^2 * (α - 2)) otherwise.
    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}.
    
    The lower bound of the support is equal to the scale parameter k.
    
    Returns:
    lower bound of the support
  - 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}.
    
    The upper bound of the support is always positive infinity no matter the parameters.
    
    Returns:
    upper bound of the support (always 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.
    The support of this distribution is connected.
    
    Returns:
    true
  - sample
```
public double sample()
```
    Generate a random value sampled from this distribution. The default implementation uses the inversion method.
    
    Specified by:
    
    sample in interface RealDistribution
    
    Overrides:
    
    sample in class AbstractRealDistribution
    
    Returns:
    a random value.

Class ParetoDistribution

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

Field Detail

DEFAULT_INVERSE_ABSOLUTE_ACCURACY

Constructor Detail

ParetoDistribution

ParetoDistribution

ParetoDistribution

ParetoDistribution

ParetoDistribution

Method Detail

getScale

getShape

density

logDensity

cumulativeProbability

cumulativeProbability

getSolverAbsoluteAccuracy

getNumericalMean

getNumericalVariance

getSupportLowerBound

getSupportUpperBound

isSupportLowerBoundInclusive

isSupportUpperBoundInclusive

isSupportConnected

sample