Class ParetoDistribution

Method Details

• 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 invalid input: '<'= 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 invalid input: '<'= 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.