PoissonDist (Java Libraries for Stochastic Simulation)

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

Fields
Modifier and Type Field and Description

static double MAXLAMBDA
- Fields inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
  EPSILON

Fields
Modifier and Type	Field and Description
`static double`	`MAXLAMBDA`

Constructor Summary

Constructors
Constructor and Description
`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

Methods
Modifier and Type	Method and Description
`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.
`static double[]`	`getMaximumLikelihoodEstimate(int[] x, int n)` Deprecated.
`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, which should be a real number in the interval [0, 1].
`void`	`setLambda(double lambda)` Sets the λ associated with this object.
`java.lang.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, 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(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)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(int[] x,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - 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 java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class PoissonDist

Field Summary

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

Constructor Summary

Method Summary

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

Methods inherited from class java.lang.Object

Field Detail

MAXLAMBDA

Constructor Detail

PoissonDist

Method Detail

prob

cdf

barF

inverseFInt

getMean

getVariance

getStandardDeviation

prob

cdf

barF

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getLambda

setLambda

getParams

toString