PoissonDistribution Class 
Namespace: Accord.Statistics.Distributions.Univariate
[SerializableAttribute] public class PoissonDistribution : UnivariateDiscreteDistribution, IFittableDistribution<double, IFittingOptions>, IFittable<double, IFittingOptions>, IFittable<double>, IFittableDistribution<double>, IDistribution<double>, IDistribution, ICloneable, ISampleableDistribution<int>, IDistribution<int>, IRandomNumberGenerator<int>
The PoissonDistribution type exposes the following members.
Name  Description  

PoissonDistribution 
Creates a new Poisson distribution with λ = 1.
 
PoissonDistribution(Double) 
Creates a new Poisson distribution with the given λ (lambda).

Name  Description  

Entropy 
Gets the entropy for this distribution.
(Overrides UnivariateDiscreteDistributionEntropy.)  
Lambda 
Gets the Poisson's parameter λ (lambda).
 
Mean 
Gets the mean for this distribution.
(Overrides UnivariateDiscreteDistributionMean.)  
Median 
Gets the median for this distribution.
(Inherited from UnivariateDiscreteDistribution.)  
Mode 
Gets the mode for this distribution.
(Inherited from UnivariateDiscreteDistribution.)  
Quartiles 
Gets the Quartiles for this distribution.
(Inherited from UnivariateDiscreteDistribution.)  
Standard 
Gets the standard Poisson distribution,
with lambda (rate) equal to 1.
 
StandardDeviation 
Gets the Standard Deviation (the square root of
the variance) for the current distribution.
(Inherited from UnivariateDiscreteDistribution.)  
Support 
Gets the support interval for this distribution.
(Overrides UnivariateDiscreteDistributionSupport.)  
Variance 
Gets the variance for this distribution.
(Overrides UnivariateDiscreteDistributionVariance.) 
Name  Description  

BaseDistributionFunction 
Computes the cumulative distribution function by summing the outputs of the ProbabilityMassFunction(Int32)
for all elements in the distribution domain. Note that this method should not be used in case there is a more
efficient formula for computing the CDF of a distribution.
(Inherited from UnivariateDiscreteDistribution.)  
BaseInverseDistributionFunction 
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p using a numerical
approximation based on binary search.
(Inherited from UnivariateDiscreteDistribution.)  
Clone 
Creates a new object that is a copy of the current instance.
(Overrides DistributionBaseClone.)  
ComplementaryDistributionFunction(Int32) 
Gets P(X > k) the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point k.
This function is also known as the Survival function.
(Inherited from UnivariateDiscreteDistribution.)  
ComplementaryDistributionFunction(Int32, Boolean) 
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point k.
This function is also known as the Survival function.
(Inherited from UnivariateDiscreteDistribution.)  
CumulativeHazardFunction 
Gets the cumulative hazard function for this
distribution evaluated at point x.
(Inherited from UnivariateDiscreteDistribution.)  
DistributionFunction(Int32) 
Gets P(X ≤ k), the cumulative distribution function
(cdf) for this distribution evaluated at point k.
(Inherited from UnivariateDiscreteDistribution.)  
DistributionFunction(Int32, Boolean) 
Gets P(X ≤ k) or P(X < k), the cumulative distribution function
(cdf) for this distribution evaluated at point k, depending
on the value of the inclusive parameter.
(Inherited from UnivariateDiscreteDistribution.)  
DistributionFunction(Int32, Int32) 
Gets the cumulative distribution function (cdf) for this
distribution in the semiclosed interval (a; b] given as
P(a < X ≤ b).
(Inherited from UnivariateDiscreteDistribution.)  
Equals  Determines whether the specified object is equal to the current object. (Inherited from Object.)  
Finalize  Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)  
Fit(Double) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateDiscreteDistribution.)  
Fit(Int32) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateDiscreteDistribution.)  
Fit(Double, IFittingOptions) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateDiscreteDistribution.)  
Fit(Double, Double) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateDiscreteDistribution.)  
Fit(Double, Int32) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateDiscreteDistribution.)  
Fit(Int32, IFittingOptions) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateDiscreteDistribution.)  
Fit(Double, Double, IFittingOptions) 
Fits the underlying distribution to a given set of observations.
(Overrides UnivariateDiscreteDistributionFit(Double, Double, IFittingOptions).)  
Fit(Double, Int32, IFittingOptions) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateDiscreteDistribution.)  
Fit(Int32, Double, IFittingOptions) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateDiscreteDistribution.)  
Fit(Int32, Int32, IFittingOptions) 
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateDiscreteDistribution.)  
Generate 
Generates a random observation from the current distribution.
(Inherited from UnivariateDiscreteDistribution.)  
Generate(Random) 
Generates a random observation from the current distribution.
(Overrides UnivariateDiscreteDistributionGenerate(Random).)  
Generate(Int32) 
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateDiscreteDistribution.)  
Generate(Int32, Double) 
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateDiscreteDistribution.)  
Generate(Int32, Int32) 
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateDiscreteDistribution.)  
Generate(Int32, Random) 
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateDiscreteDistribution.)  
Generate(Int32, Double, Random) 
Generates a random vector of observations from the current distribution.
(Overrides UnivariateDiscreteDistributionGenerate(Int32, Double, Random).)  
Generate(Int32, Int32, Random) 
Generates a random vector of observations from the current distribution.
(Overrides UnivariateDiscreteDistributionGenerate(Int32, Int32, Random).)  
GetHashCode  Serves as the default hash function. (Inherited from Object.)  
GetRange 
Gets the distribution range within a given percentile.
(Inherited from UnivariateDiscreteDistribution.)  
GetType  Gets the Type of the current instance. (Inherited from Object.)  
HazardFunction 
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
(Inherited from UnivariateDiscreteDistribution.)  
InnerComplementaryDistributionFunction 
Gets P(X > k) the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point k.
This function is also known as the Survival function.
(Inherited from UnivariateDiscreteDistribution.)  
InnerDistributionFunction 
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point k.
(Overrides UnivariateDiscreteDistributionInnerDistributionFunction(Int32).)  
InnerInverseDistributionFunction 
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
(Inherited from UnivariateDiscreteDistribution.)  
InnerLogProbabilityMassFunction 
Gets the logprobability mass function (pmf) for
this distribution evaluated at point k.
(Overrides UnivariateDiscreteDistributionInnerLogProbabilityMassFunction(Int32).)  
InnerProbabilityMassFunction 
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
(Overrides UnivariateDiscreteDistributionInnerProbabilityMassFunction(Int32).)  
InverseDistributionFunction 
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
(Inherited from UnivariateDiscreteDistribution.)  
LogCumulativeHazardFunction 
Gets the logcumulative hazard function for this
distribution evaluated at point x.
(Inherited from UnivariateDiscreteDistribution.)  
LogProbabilityMassFunction 
Gets the logprobability mass function (pmf) for
this distribution evaluated at point x.
(Inherited from UnivariateDiscreteDistribution.)  
MemberwiseClone  Creates a shallow copy of the current Object. (Inherited from Object.)  
ProbabilityMassFunction 
Gets the probability mass function (pmf) for
this distribution evaluated at point x.
(Inherited from UnivariateDiscreteDistribution.)  
QuantileDensityFunction 
Gets the first derivative of the
inverse distribution function (icdf) for this distribution evaluated
at probability p.
(Inherited from UnivariateDiscreteDistribution.)  
ToString 
Returns a String that represents this instance.
(Inherited from DistributionBase.)  
ToString(IFormatProvider) 
Returns a String that represents this instance.
(Inherited from DistributionBase.)  
ToString(String) 
Returns a String that represents this instance.
(Inherited from DistributionBase.)  
ToString(String, IFormatProvider) 
Returns a String that represents this instance.
(Overrides DistributionBaseToString(String, IFormatProvider).) 
Name  Description  

HasMethod 
Checks whether an object implements a method with the given name.
(Defined by ExtensionMethods.)  
IsEqual 
Compares two objects for equality, performing an elementwise
comparison if the elements are vectors or matrices.
(Defined by Matrix.)  
To(Type)  Overloaded.
Converts an object into another type, irrespective of whether
the conversion can be done at compile time or not. This can be
used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.)  
ToT  Overloaded.
Converts an object into another type, irrespective of whether
the conversion can be done at compile time or not. This can be
used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.) 
The Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event.
References:
The following example shows how to instantiate a new Poisson distribution with a given rate λ and how to compute its measures and associated functions.
// Create a new Poisson distribution with var dist = new PoissonDistribution(lambda: 4.2); // Common measures double mean = dist.Mean; // 4.2 double median = dist.Median; // 4.0 double var = dist.Variance; // 4.2 // Cumulative distribution functions double cdf1 = dist.DistributionFunction(k: 2); // 0.21023798702309743 double cdf2 = dist.DistributionFunction(k: 4); // 0.58982702131057763 double cdf3 = dist.DistributionFunction(k: 7); // 0.93605666027257894 double ccdf = dist.ComplementaryDistributionFunction(k: 2); // 0.78976201297690252 // Probability mass functions double pmf1 = dist.ProbabilityMassFunction(k: 4); // 0.19442365170822165 double pmf2 = dist.ProbabilityMassFunction(k: 5); // 0.1633158674349062 double pmf3 = dist.ProbabilityMassFunction(k: 6); // 0.11432110720443435 double lpmf = dist.LogProbabilityMassFunction(k: 2); // 2.0229781299813 // Quantile function int icdf1 = dist.InverseDistributionFunction(p: cdf1); // 2 int icdf2 = dist.InverseDistributionFunction(p: cdf2); // 4 int icdf3 = dist.InverseDistributionFunction(p: cdf3); // 7 // Hazard (failure rate) functions double hf = dist.HazardFunction(x: 4); // 0.47400404660843515 double chf = dist.CumulativeHazardFunction(x: 4); // 0.89117630901575073 // String representation string str = dist.ToString(CultureInfo.InvariantCulture); // "Poisson(x; λ = 4.2)"
This example shows hows to call the distribution function to compute different types of probabilities.
// Create a new Poisson distribution var dist = new PoissonDistribution(lambda: 4.2); // P(X = 1) = 0.0629814226460064 double equal = dist.ProbabilityMassFunction(k: 1); // P(X < 1) = 0.0149955768204777 double less = dist.DistributionFunction(k: 1, inclusive: false); // P(X ≤ 1) = 0.0779769994664841 double lessThanOrEqual = dist.DistributionFunction(k: 1, inclusive: true); // P(X > 1) = 0.922023000533516 double greater = dist.ComplementaryDistributionFunction(k: 1); // P(X ≥ 1) = 0.985004423179522 double greaterThanOrEqual = dist.ComplementaryDistributionFunction(k: 1, inclusive: true);