org.apache.commons.math3.distribution

## Class FDistribution

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

- All Implemented Interfaces:
    Serializable, RealDistribution

      public class FDistribution
      extends AbstractRealDistribution

  Implementation of the F-distribution.
  See Also:
    F-distribution (Wikipedia), F-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
    `FDistribution(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom)` Creates an F distribution using the given degrees of freedom.
    `FDistribution(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom, double inverseCumAccuracy)` Creates an F distribution using the given degrees of freedom and inverse cumulative probability accuracy.
    `FDistribution(RandomGenerator rng, double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom)` Creates an F distribution.
    `FDistribution(RandomGenerator rng, double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom, double inverseCumAccuracy)` Creates an F distribution.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `protected double` | `calculateNumericalVariance()` used by `getNumericalVariance()`
    `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` | `getDenominatorDegreesOfFreedom()` Access the denominator degrees of freedom.
    `double` | `getNumeratorDegreesOfFreedom()` Access the numerator degrees of freedom.
    `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.
    `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`.

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

      `cumulativeProbability, inverseCumulativeProbability, 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

    - #### FDistribution

          public FDistribution(double numeratorDegreesOfFreedom,
                       double denominatorDegreesOfFreedom)
                        throws NotStrictlyPositiveException

      Creates an F distribution using the given degrees of freedom.

      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:
        `numeratorDegreesOfFreedom` - Numerator degrees of freedom.
        `denominatorDegreesOfFreedom` - Denominator degrees of freedom.
      Throws:
        `NotStrictlyPositiveException` - if `numeratorDegreesOfFreedom <= 0` or `denominatorDegreesOfFreedom <= 0`.

    - #### FDistribution

          public FDistribution(double numeratorDegreesOfFreedom,
                       double denominatorDegreesOfFreedom,
                       double inverseCumAccuracy)
                        throws NotStrictlyPositiveException

      Creates an F distribution using the given degrees of freedom and inverse cumulative probability accuracy.

      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:
        `numeratorDegreesOfFreedom` - Numerator degrees of freedom.
        `denominatorDegreesOfFreedom` - Denominator degrees of freedom.
        `inverseCumAccuracy` - the maximum absolute error in inverse cumulative probability estimates.
      Throws:
        `NotStrictlyPositiveException` - if `numeratorDegreesOfFreedom <= 0` or `denominatorDegreesOfFreedom <= 0`.
      Since:
        2.1

    - #### FDistribution

          public FDistribution(RandomGenerator rng,
                       double numeratorDegreesOfFreedom,
                       double denominatorDegreesOfFreedom)
                        throws NotStrictlyPositiveException

      Creates an F distribution.
      Parameters:
        `rng` - Random number generator.
        `numeratorDegreesOfFreedom` - Numerator degrees of freedom.
        `denominatorDegreesOfFreedom` - Denominator degrees of freedom.
      Throws:
        `NotStrictlyPositiveException` - if `numeratorDegreesOfFreedom <= 0` or `denominatorDegreesOfFreedom <= 0`.
      Since:
        3.3

    - #### FDistribution

          public FDistribution(RandomGenerator rng,
                       double numeratorDegreesOfFreedom,
                       double denominatorDegreesOfFreedom,
                       double inverseCumAccuracy)
                        throws NotStrictlyPositiveException

      Creates an F distribution.
      Parameters:
        `rng` - Random number generator.
        `numeratorDegreesOfFreedom` - Numerator degrees of freedom.
        `denominatorDegreesOfFreedom` - Denominator degrees of freedom.
        `inverseCumAccuracy` - the maximum absolute error in inverse cumulative probability estimates.
      Throws:
        `NotStrictlyPositiveException` - if `numeratorDegreesOfFreedom <= 0` or `denominatorDegreesOfFreedom <= 0`.
      Since:
        3.1

  - ### 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.
      Parameters:
        `x` - the point at which the PDF is evaluated
      Returns:
        the value of the probability density function at point `x`
      Since:
        2.1

    - #### 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)`.
      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. The implementation of this method is based on
      - F-Distribution, equation (4).
      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`

    - #### getNumeratorDegreesOfFreedom

          public double getNumeratorDegreesOfFreedom()

      Access the numerator degrees of freedom.
      Returns:
        the numerator degrees of freedom.

    - #### getDenominatorDegreesOfFreedom

          public double getDenominatorDegreesOfFreedom()

      Access the denominator degrees of freedom.
      Returns:
        the denominator degrees of freedom.

    - #### 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 denominator degrees of freedom parameter `b`, the mean is
      - if `b > 2` then `b / (b - 2)`,
      - else undefined (`Double.NaN`).
      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 numerator degrees of freedom parameter `a` and denominator degrees of freedom parameter `b`, the variance is
      - if `b > 4` then `[2 * b^2 * (a + b - 2)] / [a * (b - 2)^2 * (b - 4)]`,
      - else undefined (`Double.NaN`).
      Returns:
        the variance (possibly `Double.POSITIVE_INFINITY` as for certain cases in `TDistribution`) or `Double.NaN` if it is not defined

    - #### calculateNumericalVariance

          protected double calculateNumericalVariance()

      used by `getNumericalVariance()`
      Returns:
        the variance of this distribution

    - #### 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 0 no matter the parameters.
      Returns:
        lower bound of the support (always 0)

    - #### 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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