org.apache.commons.math3.distribution

## Class CauchyDistribution

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

- All Implemented Interfaces:
    Serializable, RealDistribution

      public class CauchyDistribution
      extends AbstractRealDistribution

  Implementation of the Cauchy distribution.
  Since:
    1.1 (changed to concrete class in 3.0)
  See Also:
    Cauchy distribution (Wikipedia), Cauchy 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`

  - ### Constructor Summary

    Constructors
    Constructor and Description
    `CauchyDistribution()` Creates a Cauchy distribution with the median equal to zero and scale equal to one.
    `CauchyDistribution(double median, double scale)` Creates a Cauchy distribution using the given median and scale.
    `CauchyDistribution(double median, double scale, double inverseCumAccuracy)` Creates a Cauchy distribution using the given median and scale.
    `CauchyDistribution(RandomGenerator rng, double median, double scale)` Creates a Cauchy distribution.
    `CauchyDistribution(RandomGenerator rng, double median, double scale, double inverseCumAccuracy)` Creates a Cauchy 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` | `density(double x)` Returns the probability density function (PDF) of this distribution evaluated at the specified point `x`.
    `double` | `getMedian()` Access the median.
    `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()` Access the scale parameter.
    `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.
    `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.

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

      `cumulativeProbability, logDensity, probability, probability, reseedRandomGenerator, sample, 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.
      Since:
        2.1
      See Also:
        Constant Field Values

  - ### Constructor Detail

    - #### CauchyDistribution

          public CauchyDistribution()

      Creates a Cauchy distribution with the median equal to zero and scale equal to one.

    - #### CauchyDistribution

          public CauchyDistribution(double median,
                            double scale)

      Creates a Cauchy distribution using the given median and scale.

      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:
        `median` - Median for this distribution.
        `scale` - Scale parameter for this distribution.

    - #### CauchyDistribution

          public CauchyDistribution(double median,
                            double scale,
                            double inverseCumAccuracy)

      Creates a Cauchy distribution using the given median and scale.

      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:
        `median` - Median for this distribution.
        `scale` - Scale parameter for this distribution.
        `inverseCumAccuracy` - Maximum absolute error in inverse cumulative probability estimates (defaults to `DEFAULT_INVERSE_ABSOLUTE_ACCURACY`).
      Throws:
        `NotStrictlyPositiveException` - if `scale <= 0`.
      Since:
        2.1

    - #### CauchyDistribution

          public CauchyDistribution(RandomGenerator rng,
                            double median,
                            double scale)

      Creates a Cauchy distribution.
      Parameters:
        `rng` - Random number generator.
        `median` - Median for this distribution.
        `scale` - Scale parameter for this distribution.
      Throws:
        `NotStrictlyPositiveException` - if `scale <= 0`.
      Since:
        3.3

    - #### CauchyDistribution

          public CauchyDistribution(RandomGenerator rng,
                            double median,
                            double scale,
                            double inverseCumAccuracy)

      Creates a Cauchy distribution.
      Parameters:
        `rng` - Random number generator.
        `median` - Median for this distribution.
        `scale` - Scale parameter for this distribution.
        `inverseCumAccuracy` - Maximum absolute error in inverse cumulative probability estimates (defaults to `DEFAULT_INVERSE_ABSOLUTE_ACCURACY`).
      Throws:
        `NotStrictlyPositiveException` - if `scale <= 0`.
      Since:
        3.1

  - ### Method Detail

    - #### 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.
      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`

    - #### getMedian

          public double getMedian()

      Access the median.
      Returns:
        the median for this distribution.

    - #### getScale

          public double getScale()

      Access the scale parameter.
      Returns:
        the scale parameter for this distribution.

    - #### 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.
      Parameters:
        `x` - the point at which the PDF is evaluated
      Returns:
        the value of the probability density function at point `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`.
      Returns `Double.NEGATIVE_INFINITY` when `p == 0` and `Double.POSITIVE_INFINITY` when `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`

    - #### 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. The mean is always undefined no matter the parameters.
      Returns:
        mean (always Double.NaN)

    - #### getNumericalVariance

          public double getNumericalVariance()

      Use this method to get the numerical value of the variance of this distribution. The variance is always undefined no matter the parameters.
      Returns:
        variance (always Double.NaN)

    - #### 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 always negative infinity no matter the parameters.
      Returns:
        lower bound of the support (always 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}`.

      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`

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