Package org.apache.commons.statistics.distribution

Class TrapezoidalDistribution.RegularTrapezoidalDistribution

    • Field Detail

      • divisor

        private final double divisor
        Cached value (d + c - a - b).

      • bma

        private final double bma
        Cached value (b - a).

      • dmc

        private final double dmc
        Cached value (d - c).

      • cdfB

        private final double cdfB
        Cumulative probability at b.

      • cdfC

        private final double cdfC
        Cumulative probability at c.

      • sfB

        private final double sfB
        Survival probability at b.

      • sfC

        private final double sfC
        Survival probability at c.

    • Constructor Detail

      • RegularTrapezoidalDistribution

        RegularTrapezoidalDistribution​(double a,
                               double b,
                               double c,
                               double d)
        Parameters:
        a - Lower limit of this distribution (inclusive).
        b - Start of the trapezoid constant density.
        c - End of the trapezoid constant density.
        d - Upper limit of this distribution (inclusive).

    • Method Detail

      • density

        public double density​(double x)
        Description copied from interface: ContinuousDistribution
        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 - Point at which the PDF is evaluated.
        Returns:
        the value of the probability density function at x.

      • cumulativeProbability

        public double cumulativeProbability​(double x)
        Description copied from interface: ContinuousDistribution
        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 - 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.

      • survivalProbability

        public double survivalProbability​(double x)
        Description copied from interface: ContinuousDistribution
        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 complementary cumulative distribution function.

        By default, this is defined as 1 - cumulativeProbability(x), but the specific implementation may be more accurate.

        Parameters:
        x - Point at which the survival function is evaluated.
        Returns:
        the probability that a random variable with this distribution takes a value greater than x.

      • getMean

        public double getMean()
        Description copied from class: TrapezoidalDistribution
        Gets the mean of this distribution.

        For lower limit \( a \), start of the density constant region \( b \), end of the density constant region \( c \) and upper limit \( d \), the mean is:

        \[ \frac{1}{3(d+c-b-a)}\left(\frac{d^3-c^3}{d-c}-\frac{b^3-a^3}{b-a}\right) \]

        Specified by:
        getMean in interface ContinuousDistribution
        Specified by:
        getMean in class TrapezoidalDistribution
        Returns:
        the mean.

      • getVariance

        public double getVariance()
        Description copied from class: TrapezoidalDistribution
        Gets the variance of this distribution.

        For lower limit \( a \), start of the density constant region \( b \), end of the density constant region \( c \) and upper limit \( d \), the variance is:

        \[ \frac{1}{6(d+c-b-a)}\left(\frac{d^4-c^4}{d-c}-\frac{b^4-a^4}{b-a}\right) - \mu^2 \]

        where \( \mu \) is the mean.

        Specified by:
        getVariance in interface ContinuousDistribution
        Specified by:
        getVariance in class TrapezoidalDistribution
        Returns:
        the variance.

      • nonCentralMoment

        private static double nonCentralMoment​(int k,
                                       double b,
                                       double c)
        Compute the k-th non-central moment of the standardized trapezoidal distribution.

        Shifting the distribution by scale (d - a) and location a creates a standardized trapezoidal distribution. This has a simplified non-central moment as a = 0, d = 1, 0 <= b < c <= 1. 

                       2             1       ( 1 - c^(k+2)           )
 E[X^k] = ----------- -------------- ( ----------- - b^(k+1) )
          (1 + c - b) (k + 1)(k + 2) (    1 - c              )

        Simplification eliminates issues computing the moments when a == b or c == d in the original (non-standardized) distribution.

        Parameters:
        k - Moment to compute
        b - Start of the trapezoid constant density (shape parameter in [0, 1]).
        c - End of the trapezoid constant density (shape parameter in [0, 1]).
        Returns:
        the moment