Extends the class DiscreteDistributionInt for the Poisson distribution [118] (page 325) with mean \(\lambda\ge0\). More...

Inheritance diagram for PoissonDist:

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Collaboration diagram for PoissonDist:

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Public Member Functions
	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.

double	prob (int x)

double	cdf (int x)

double	barF (int x)

int	inverseFInt (double u)

double	getMean ()
	Returns the mean of the distribution function.

double	getVariance ()
	Returns the variance of the distribution function.

double	getStandardDeviation ()
	Returns the standard deviation of the distribution function.

double	getLambda ()
	Returns the \(\lambda\) associated with this object.

void	setLambda (double lambda)
	Sets the \(\lambda\) associated with this object.

double []	getParams ()
	Return a table containing the parameter of the current distribution.

String	toString ()
	Returns a `String` containing information about the current distribution.

Public Member Functions inherited from DiscreteDistributionInt
abstract double	prob (int x)
	Returns \(p(x)\), the probability of \(x\). More...

double	cdf (double x)
	Returns the distribution function \(F\) evaluated at \(x\) (see ( FDistDisc )). More...

abstract double	cdf (int x)
	Returns the distribution function \(F\) evaluated at \(x\) (see ( FDistDisc )). More...

double	barF (double x)
	Returns \(\bar{F}(x)\), the complementary distribution function. More...

double	barF (int x)
	Returns \(\bar{F}(x)\), the complementary distribution function. More...

int	getXinf ()
	Returns the lower limit \(x_a\) of the support of the probability mass function. More...

int	getXsup ()
	Returns the upper limit \(x_b\) of the support of the probability mass function. More...

double	inverseF (double u)
	Returns the inverse distribution function \(F^{-1}(u)\), where. More...

int	inverseFInt (double u)
	Returns the inverse distribution function \(F^{-1}(u)\), where. More...

Static Public Member Functions
static double	prob (double lambda, int x)
	Computes and returns the Poisson probability \(p(x)\) for \(\lambda= \) `lambda`, as defined in ( fmass-Poisson ). More...

static double	cdf (double lambda, int x)
	Computes and returns the value of the Poisson distribution function \(F(x)\) for \(\lambda= \) `lambda`, as defined in ( FPoisson ). More...

static double	barF (double lambda, int x)
	Computes and returns the value of the complementary Poisson distribution function, for \(\lambda= \) `lambda`. More...

static int	inverseF (double lambda, double u)
	Performs a linear search to get the inverse function without precomputed tables.

static double []	getMLE (int[] x, int n)
	Estimates the parameter \(\lambda\) of the Poisson distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...

static PoissonDist	getInstanceFromMLE (int[] x, int n)
	Creates a new instance of a Poisson distribution with parameter \(\lambda\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...

static double	getMean (double lambda)
	Computes and returns the mean \(E[X] = \lambda\) of the Poisson distribution with parameter \(\lambda\). More...

static double	getVariance (double lambda)
	Computes and returns the variance \(= \lambda\) of the Poisson distribution with parameter \(\lambda\). More...

static double	getStandardDeviation (double lambda)
	Computes and returns the standard deviation of the Poisson distribution with parameter \(\lambda\). More...

Static Public Attributes
Constant
static double	MAXLAMBDA = 100000
	The value of the parameter \(\lambda\) above which the tables are not precomputed by the constructor.

Static Public Attributes inherited from DiscreteDistributionInt
static double	EPSILON = 1.0e-16
	Environment variable that determines what probability terms can be considered as negligible when building precomputed tables for distribution and mass functions. More...

Additional Inherited Members
Protected Attributes inherited from DiscreteDistributionInt
double	cdf [] = null

double	pdf [] = null

int	xmin = 0

int	xmax = 0

int	xmed = 0

int	supportA = Integer.MIN_VALUE

int	supportB = Integer.MAX_VALUE

Static Protected Attributes inherited from DiscreteDistributionInt
static final double	EPS_EXTRA = 1.0e-6

Detailed Description

Extends the class DiscreteDistributionInt for the Poisson distribution [118] (page 325) with mean \(\lambda\ge0\).

The mass function is

\begin{align} p(x) & = \frac{e^{-\lambda} \lambda^x}{x!}, \qquad\mbox{for } x=0,1,…\tag{fmass-Poisson} \end{align}

and the distribution function is

\begin{align} F(x) & = e^{-\lambda} \sum_{j=0}^x\; \frac{\lambda^j}{j!}, \qquad\mbox{for } x=0,1,…. \tag{FPoisson} \end{align}

If one has to compute \(p(x)\) and/or \(F(x)\) for several values of \(x\) with the same \(\lambda\), where \(\lambda\) 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 \(F(x) = 1 - G_{x+1}(\lambda)\), where \(G_{x+1}\) is the gamma distribution function with parameters \((\alpha,\lambda) = (x+1, 1)\).

Member Function Documentation

◆ barF()

static double barF	(	double	lambda,
		int	x
	)

static

Computes and returns the value of the complementary Poisson distribution function, for \(\lambda= \) lambda.

WARNING: The complementary distribution function is defined as \(\bar{F}(x) = P[X \ge x]\).

◆ cdf()

static double cdf	(	double	lambda,
		int	x
	)

static

Computes and returns the value of the Poisson distribution function \(F(x)\) for \(\lambda= \) lambda, as defined in ( FPoisson ).

To compute \(F(x)\), all non-negligible terms of the sum are added if \(\lambda\le200\); otherwise, the relationship \(F_{\lambda}(x) = 1 - G_{x + 1}(\lambda)\) is used, where \(G_{x+1}\) is the gamma distribution function with parameter \(\alpha= x+1\) (see GammaDist ).

◆ getInstanceFromMLE()

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

static

Creates a new instance of a Poisson distribution with parameter \(\lambda\) 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()

static double getMean ( double lambda )

static

Computes and returns the mean \(E[X] = \lambda\) of the Poisson distribution with parameter \(\lambda\).

Returns: the mean of the Poisson distribution \(E[X] = \lambda\)

◆ getMLE()

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

static

Estimates the parameter \(\lambda\) of the Poisson distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).

The maximum likelihood estimator \(\hat{\lambda}\) satisfy the equation \(\hat{\lambda} = \bar{x}_n\), where \(\bar{x}_n\) is the average of \(x[0], …, x[n-1]\) (see [118] (page 326)).

Parameters

x	the list of observations used to evaluate parameters
n	the number of observations used to evaluate parameters

Returns: returns the parameter [ \(\hat{\lambda}\)]

◆ getStandardDeviation()

static double getStandardDeviation ( double lambda )

static

Computes and returns the standard deviation of the Poisson distribution with parameter \(\lambda\).

Returns: the standard deviation of the Poisson distribution

◆ getVariance()

static double getVariance ( double lambda )

static

Computes and returns the variance \(= \lambda\) of the Poisson distribution with parameter \(\lambda\).

Returns: the variance of the Poisson distribution \(\mbox{Var}[X] = \lambda\)

◆ prob()

static double prob	(	double	lambda,
		int	x
	)

static

Computes and returns the Poisson probability \(p(x)\) for \(\lambda= \) lambda, as defined in ( fmass-Poisson ).

If \(\lambda\ge20\), this (static) method uses the logarithm of the gamma function, defined in ( Gamma ), to estimate the density.

The documentation for this class was generated from the following file:

PoissonDist.java

Public Member Functions

Static Public Member Functions

Static Public Attributes

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf()

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getStandardDeviation()

◆ getVariance()

◆ prob()