SSJ
V. 2.6.

umontreal.iro.lecuyer.probdist
Class BinomialDist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
      extended by umontreal.iro.lecuyer.probdist.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.


Field Summary
static double MAXN
           
 
Fields inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
EPSILON
 
Constructor Summary
BinomialDist(int n, double p)
          Creates an object that contains the binomial terms, for 0 <= x <= n, and the corresponding cumulative function.
 
Method Summary
 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 x = F-1(u), the inverse of the binomial distribution.
 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.
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.
 String toString()
           
 
Methods inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
barF, cdf, getXinf, getXsup, inverseF
 
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.

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 x = F-1(u), the inverse of the binomial distribution.


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 String toString()
Overrides:
toString in class Object

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.