umontreal.iro.lecuyer.probdist
Class PoissonDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
      umontreal.iro.lecuyer.probdist.PoissonDist

All Implemented Interfaces:

Distribution

public class PoissonDist
extends DiscreteDistributionInt

Extends the class DiscreteDistributionInt for the Poisson distribution with mean λ >= 0. The mass function is

p(x) = e^-λλ^x/(x!), for x = 0, 1,...

and the distribution function is

F(x) = e^-λ∑_j=0^x λ^j/(j!), for x = 0, 1,....

If one has to compute p(x) and/or F(x) for several values of x with the same λ, where λ is not too large, then it is more efficient to instantiate an object and use the non-static methods, since the functions will then be computed once and kept in arrays.

For the static methods that compute F(x) and bar(F)(x), we exploit the relationship F(x) = 1 - G_x+1(λ), where G_x+1 is the gamma distribution function with parameters (α, λ) = (x + 1, 1).

Field Summary

static double

MAXLAMBDA

Fields inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt

EPSILON

Constructor Summary

PoissonDist(double lambda)
Creates an object that contains the probability and distribution functions, for the Poisson distribution with parameter lambda, which are computed and stored in dynamic arrays inside that object.

Method Summary

static double	barF(double lambda, int x) Computes and returns the value of the complementary Poisson distribution function, for λ = lambda.
double	barF(int x) Returns bar(F)(x), the complementary distribution function.
static double	cdf(double lambda, int x) Computes and returns the value of the Poisson distribution function F(x) for λ = lambda.
double	cdf(int x) Returns the distribution function F evaluated at x (see).
static PoissonDist	getInstanceFromMLE(int[] x, int n) Creates a new instance of a Poisson distribution with parameter λ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
double	getLambda() Returns the λ associated with this object.
double	getMean() Returns the mean of the distribution function.
static double	getMean(double lambda) Computes and returns the mean E[X] = λ of the Poisson distribution with parameter λ.
static double[]	getMLE(int[] x, int n) Estimates the parameter λ of the Poisson distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
double[]	getParams() Return a table containing the parameter of the current distribution.
double	getStandardDeviation() Returns the standard deviation of the distribution function.
static double	getStandardDeviation(double lambda) Computes and returns the standard deviation of the Poisson distribution with parameter λ.
double	getVariance() Returns the variance of the distribution function.
static double	getVariance(double lambda) Computes and returns the variance = λ of the Poisson distribution with parameter λ.
static int	inverseF(double lambda, double u) Performs a linear search to get the inverse function without precomputed tables.
int	inverseFInt(double u) Returns the inverse distribution function F-1(u), where 0 <= u <= 1.
static double	prob(double lambda, int x) Computes and returns the Poisson probability p(x) for λ = lambda..
double	prob(int x) Returns p(x), the probability of x.
void	setLambda(double lambda) Sets the λ associated with this object.
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

MAXLAMBDA

public static double MAXLAMBDA

Constructor Detail

PoissonDist

public PoissonDist(double lambda)

Creates an object that contains the probability and distribution functions, for the Poisson distribution with parameter lambda, which are computed and stored in dynamic arrays inside that object.

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(double lambda,
                          int x)

Computes and returns the Poisson probability p(x) for λ = lambda..

cdf

public static double cdf(double lambda,
                         int x)

Computes and returns the value of the Poisson distribution function F(x) for λ = lambda.

barF

public static double barF(double lambda,
                          int x)

Computes and returns the value of the complementary Poisson distribution function, for λ = lambda. WARNING: The complementary distribution function is defined as bar(F)(x) = P[X >= x].

inverseF

public static int inverseF(double lambda,
                           double u)

Performs a linear search to get the inverse function without precomputed tables.

getMLE

public static double[] getMLE(int[] x,
                              int n)

Estimates the parameter λ of the Poisson distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The maximum likelihood estimator hat(λ) satisfy the equation hat(λ) = bar(x)_n, where bar(x)_n is the average of x[0],…, x[n - 1] (see).

Parameters:

x - the list of observations used to evaluate parameters

n - the number of observations used to evaluate parameters

Returns:

returns the parameter [ hat(λ)]

getInstanceFromMLE

public static PoissonDist getInstanceFromMLE(int[] x,
                                             int n)

Creates a new instance of a Poisson distribution with parameter λ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.

Parameters:

x - the list of observations to use to evaluate parameters

n - the number of observations to use to evaluate parameters

getMean

public static double getMean(double lambda)

Computes and returns the mean E[X] = λ of the Poisson distribution with parameter λ.

Returns:

the mean of the Poisson distribution E[X] = λ

getVariance

public static double getVariance(double lambda)

Computes and returns the variance = λ of the Poisson distribution with parameter λ.

Returns:

the variance of the Poisson distribution Var[X] = λ

getStandardDeviation

public static double getStandardDeviation(double lambda)

Computes and returns the standard deviation of the Poisson distribution with parameter λ.

Returns:

the standard deviation of the Poisson distribution

getLambda

public double getLambda()

Returns the λ associated with this object.

setLambda

public void setLambda(double lambda)

Sets the λ associated with this object.

getParams

public double[] getParams()

Return a table containing the parameter of the current distribution.

toString

public String toString()

Overrides:

toString in class Object