Documentation of 'umontreal.iro.lecuyer.probdist.BinomialDist' Java class
BinomialDist
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umontreal.iro.lecuyer.probdist

Class BinomialDist

  • All Implemented Interfaces:
    Distribution

    public class BinomialDist
extends DiscreteDistributionInt
    Extends the class DiscreteDistributionInt for the binomial distribution with parameters n and p, where n is a positive integer and 0 <= p <= 1. Its mass function is given by

    p(x) = nCr(n, x)px(1 - p)n-x = n!/[x!(n - x)!]  px(1 - p)n-x        for x = 0, 1, 2,…n,

    and its distribution function is

    F(x) = ∑j=0xnCr(n, j)  pj(1 - p)n-j        for x = 0, 1, 2,…n,

    where nCr(n, x) is the number of possible combinations of x elements chosen among a set of n elements.

    • Constructor Summary

      Constructors 
      Constructor and Description
      BinomialDist(int n, double p)
      Creates an object that contains the binomial terms, for 0 <= x <= n, and the corresponding cumulative function.

    • Method Summary

      All Methods Static Methods Instance Methods Concrete Methods 
      Modifier and Type Method and Description
      double barF(int x)
      Returns bar(F)(x), the complementary distribution function.
      static double barF(int n, double p, int x)
      Returns bar(F)(x) = P[X >= x], the complementary distribution function.
      double cdf(int x)
      Returns the distribution function F evaluated at x (see).
      static double cdf(int n, double p, int x)
      Computes F(x), the distribution function of a binomial random variable with parameters n and p, evaluated at x.
      static BinomialDist getInstanceFromMLE(int[] x, int m)
      Creates a new instance of a binomial distribution with both parameters n and p estimated using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1.
      static BinomialDist getInstanceFromMLE(int[] x, int m, int n)
      Creates a new instance of a binomial distribution with given (fixed) parameter n, and with parameter p estimated by the maximum likelihood method based on the m observations x[i], i = 0, 1,…, m - 1.
      double getMean()
      Returns the mean of the distribution function.
      static double getMean(int n, double p)
      Computes the mean E[X] = np of the binomial distribution with parameters n and p.
      static double[] getMLE(int[] x, int m)
      Estimates the parameters (n, p) of the binomial distribution using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1.
      static double[] getMLE(int[] x, int m, int n)
      Estimates the parameter p of the binomial distribution with given (fixed) parameter n, by the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1.
      int getN()
      Returns the parameter n of this object.
      double getP()
      Returns the parameter p of this object.
      double[] getParams()
      Returns a table that contains the parameters (n, p) of the current distribution, in regular order: [n, p].
      double getStandardDeviation()
      Returns the standard deviation of the distribution function.
      static double getStandardDeviation(int n, double p)
      Computes the standard deviation of the Binomial distribution with parameters n and p.
      double getVariance()
      Returns the variance of the distribution function.
      static double getVariance(int n, double p)
      Computes the variance Var[X] = np(1 - p) of the binomial distribution with parameters n and p.
      static int inverseF(int n, double p, double u)
      Computes the inverse of the binomial distribution, x = F-1(u), using a linear search starting at the mode if n is small.
      int inverseFInt(double u)
      Returns the inverse distribution function F-1(u), where 0 <= u <= 1.
      double prob(int x)
      Returns p(x), the probability of x, which should be a real number in the interval [0, 1].
      static double prob(int n, double p, double q, int x)
      A generalization of the previous method.
      static double prob(int n, double p, int x)
      Computes and returns the binomial probability p(x) in eq..
      void setParams(int n, double p)
      Resets the parameters to these new values and recomputes everything as in the constructor.
      java.lang.String toString() 

      • Methods inherited from class java.lang.Object

        equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

    • Field Detail

      • MAXN

        public static double MAXN

    • Constructor Detail

      • BinomialDist

        public BinomialDist(int n,
                    double p)
        Creates an object that contains the binomial terms, for 0 <= x <= n, and the corresponding cumulative function. These values are computed and stored in dynamic arrays, unless n exceeds MAXN.

    • Method Detail

      • prob

        public double prob(int x)
        Description copied from class: DiscreteDistributionInt
        Returns p(x), the probability of x, which should be a real number in the interval [0, 1].
        Specified by:
        prob in class DiscreteDistributionInt
        Parameters:
        x - value at which the mass function must be evaluated
        Returns:
        the mass function evaluated at x

      • cdf

        public double cdf(int x)
        Description copied from class: DiscreteDistributionInt
        Returns the distribution function F evaluated at x (see).
        Specified by:
        cdf in class DiscreteDistributionInt
        Parameters:
        x - value at which the distribution function must be evaluated
        Returns:
        the distribution function evaluated at x

      • barF

        public double barF(int x)
        Description copied from class: DiscreteDistributionInt
        Returns bar(F)(x), the complementary distribution function. See the WARNING above.
        Overrides:
        barF in class DiscreteDistributionInt
        Parameters:
        x - value at which the complementary distribution function must be evaluated
        Returns:
        the complementary distribution function evaluated at x

      • inverseFInt

        public int inverseFInt(double u)
        Description copied from class: DiscreteDistributionInt
        Returns the inverse distribution function F-1(u), where 0 <= u <= 1. The default implementation uses binary search.
        Overrides:
        inverseFInt in class DiscreteDistributionInt
        Parameters:
        u - value in the interval (0, 1) for which the inverse distribution function is evaluated
        Returns:
        the inverse distribution function evaluated at u

      • getMean

        public double getMean()
        Description copied from interface: Distribution
        Returns the mean of the distribution function.

      • getVariance

        public double getVariance()
        Description copied from interface: Distribution
        Returns the variance of the distribution function.

      • getStandardDeviation

        public double getStandardDeviation()
        Description copied from interface: Distribution
        Returns the standard deviation of the distribution function.

      • prob

        public static double prob(int n,
                          double p,
                          int x)
        Computes and returns the binomial probability p(x) in eq..

      • prob

        public static double prob(int n,
                          double p,
                          double q,
                          int x)
        A generalization of the previous method. Computes and returns the binomial term f (x) = (n!/x!(n-x)!)pxqn-x, where p and q are arbitrary real numbers (q is not necessarily equal to 1 - p). In the case where 0 <= p <= 1 and q = 1 - p, the returned value is a probability term for the binomial distribution.

      • cdf

        public static double cdf(int n,
                         double p,
                         int x)
        Computes F(x), the distribution function of a binomial random variable with parameters n and p, evaluated at x.

      • barF

        public static double barF(int n,
                          double p,
                          int x)
        Returns bar(F)(x) = P[X >= x], the complementary distribution function.

      • inverseF

        public static int inverseF(int n,
                           double p,
                           double u)
        Computes the inverse of the binomial distribution, x = F-1(u), using a linear search starting at the mode if n is small. If n is larger than 10000, the linear search starts from 0 and the cdf static method is used to compute F(x) at different values of x, which is much less efficient.

      • getMLE

        public static double[] getMLE(int[] x,
                              int m)
        Estimates the parameters (n, p) of the binomial distribution using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1. The estimates are returned in a two-element array, in regular order: [n, p].
        Parameters:
        x - the list of observations used to evaluate parameters
        m - the number of observations used to evaluate parameters
        Returns:
        returns the parameters [hat(n), hat(p)]

      • getInstanceFromMLE

        public static BinomialDist getInstanceFromMLE(int[] x,
                                              int m)
        Creates a new instance of a binomial distribution with both parameters n and p estimated using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1.
        Parameters:
        x - the list of observations to use to estimate the parameters
        m - the number of observations to use to estimate the parameters

      • getMLE

        public static double[] getMLE(int[] x,
                              int m,
                              int n)
        Estimates the parameter p of the binomial distribution with given (fixed) parameter n, by the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1. Returns the estimator in an array with a single element.
        Parameters:
        x - the list of observations used to evaluate parameters
        m - the number of observations used to evaluate parameters
        n - the number of success
        Returns:
        returns the parameter [hat(p)]

      • getInstanceFromMLE

        public static BinomialDist getInstanceFromMLE(int[] x,
                                              int m,
                                              int n)
        Creates a new instance of a binomial distribution with given (fixed) parameter n, and with parameter p estimated by the maximum likelihood method based on the m observations x[i], i = 0, 1,…, m - 1.
        Parameters:
        x - the list of observations to use to evaluate parameters
        m - the number of observations to use to evaluate parameters
        n - the parameter n of the binomial

      • getMean

        public static double getMean(int n,
                             double p)
        Computes the mean E[X] = np of the binomial distribution with parameters n and p.
        Returns:
        the mean of the Binomial distribution E[X] = np

      • getVariance

        public static double getVariance(int n,
                                 double p)
        Computes the variance Var[X] = np(1 - p) of the binomial distribution with parameters n and p.
        Returns:
        the variance of the binomial distribution Var[X] = np(1 - p)

      • getStandardDeviation

        public static double getStandardDeviation(int n,
                                          double p)
        Computes the standard deviation of the Binomial distribution with parameters n and p.
        Returns:
        the standard deviation of the binomial distribution

      • getN

        public int getN()
        Returns the parameter n of this object.

      • getP

        public double getP()
        Returns the parameter p of this object.

      • getParams

        public double[] getParams()
        Returns a table that contains the parameters (n, p) of the current distribution, in regular order: [n, p].

      • setParams

        public void setParams(int n,
                      double p)
        Resets the parameters to these new values and recomputes everything as in the constructor. From the performance viewpoint, it is essentially the same as constructing a new BinomialDist object.

      • toString

        public java.lang.String toString()
        Overrides:
        toString in class java.lang.Object
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