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MultivariatePoissonDistribution
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MultivariatePoissonDistribution

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MultivariatePoissonDistribution[μ0,{μ1,μ2,…}]

represents a multivariate Poisson distribution with mean vector {μ0+μ1,μ0+μ2,…}.

Details

  • The multivariate Poisson distribution corresponds to the distribution of {x0+x1,x0+x2,…}, where xi is Poisson distributed with mean μi.
  • The parameters μi can be any positive numbers.
  • MultivariatePoissonDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

  • MultivariatePoissonDistribution[μ0,{μ1,μ2,…,μn}] represents a discrete multivariate statistical distribution supported over the subset of , consisting of all tuples of integers satisfying and characterized by the property that each of the ^(th) (univariate) marginal distributions is a PoissonDistribution for . In other words, each of the variables satisfies xkPoissonDistribution[μ0+μk] for . The multivariate Poisson distribution is parametrized by a positive real number μ0 and by a vector {μ1,μ2,…,μn} of real numbers, which together define the associated mean, variance, and covariance of the distribution. The multivariate Poisson distribution has a probability density function (PDF) that is discrete and unimodal.
  • Care must be exhibited to distinguish the multivariate Poisson distribution from the similarly named multiple Poisson distribution. The latter distribution became the focus of study in the late 1950s and is characterized by being a joint distribution of its univariate marginals. (In contrast, MultivariatePoissonDistribution is not a product of its univariate marginals.)
  • The study of the multivariate Poisson distribution began in the 1930s in the special case when (the so-called bivariate Poisson distribution), while analysis of the general multivariate case began in the late 1950s. Unlike other multivariate distributions such as the MultinormalDistribution, the multivariate Poisson distribution has been redefined several times, with many of its incarnations garnering mixed receptions and criticism from experts. In its most standard incarnation, the multivariate Poisson distribution (as implemented here) is considered to be the most natural multivariate extension of the univariate Poisson distribution and has found use as an alternative to hidden Markov models in the analysis and classification of neuronal spike patterns.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a multivariate Poisson distribution. Distributed[x,MultivariatePoissonDistribution[μ0,{μ1,μ2,…,μn}]], written more concisely as xMultivariatePoissonDistribution[μ0,{μ1,μ2,…,μn}], can be used to assert that a random variable x is distributed according to a multivariate Poisson distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions for multivariate Poisson distributions may be given using PDF[MultivariatePoissonDistribution[μ0,{μ1,μ2,…,μn}],{x1,x2,…,xn}] and CDF[MultivariatePoissonDistribution[μ0,{μ1,μ2,…,μn}],{x1,x2,…,xn}]. The mean, median, variance, covariance, raw moments, and central moments may be computed using Mean, Median, Variance, Covariance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with a multivariate Poisson distribution, EstimatedDistribution to estimate a multivariate Poisson parametric distribution from given data, and FindDistributionParameters to fit data to a multivariate Poisson distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic multivariate Poisson distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic multivariate Poisson distribution.
  • TransformedDistribution can be used to represent a transformed multivariate Poisson distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a multivariate Poisson distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving multivariate Poisson distributions.
  • MultivariatePoissonDistribution is related to a number of other distributions. MultivariatePoissonDistribution is connected to PoissonDistribution, as discussed above, and while the one-dimensional marginal PDFs of MultivariatePoissonDistribution each satisfy a PoissonDistribution, each of the multivariate marginals is again an instance of MultivariatePoissonDistribution. MultivariatePoissonDistribution is a limiting case of MultinomialDistribution in a complicated but precise way, and because of its relation to the univariate PoissonDistribution, MultivariatePoissonDistribution is also related to BinomialDistribution, PolyaAeppliDistribution, and PoissonConsulDistribution.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Probability mass function:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-z0gx1v
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-8bh4mo
Out[2]=2

Cumulative distribution function:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-s78bcd
Out[1]=1

Mean and variance:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-dt6jww
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-zj6lsa
Out[2]=2

Covariance matrix:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-r16sy2

Scope  (8)Survey of the scope of standard use cases

Generate a sample of pseudorandom vectors from a multivariate Poisson distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-ztp9aa
In[2]:=2
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-fjrvns
Out[2]=2

Distribution parameters estimation:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-z3xm5i

Estimate the distribution parameters from sample data:

In[2]:=2
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https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-sbri0q
Out[2]=2

Goodness-of-fit test:

In[3]:=3
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https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-drlyn5
Out[3]=3

Skewness for each component depends on μ and mu_(i):

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-2nilic
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-1fzeew
Out[2]=2

Kurtosis for each component depends on μ and mu_(i):

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-499ss7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-uw5kys
Out[2]=2

Correlation:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-8btepn

Different mixed moments for a bivariate Poisson distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-slatx7
In[2]:=2
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-2dtdhm
Out[2]=2

Mixed central moments:

In[3]:=3
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-s6xlj5
Out[3]=3

Mixed factorial moments:

In[4]:=4
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https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-igch4a
Out[4]=4

Mixed cumulants:

In[5]:=5
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-oedqsj
Out[5]=5

Closed form for a symbolic order:

In[6]:=6
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https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-b0qml0
Out[6]=6

Hazard function:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-mmwmdw
Out[1]=1

Marginal distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-d9gpup
In[2]:=2
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https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-ercdpx
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-fvtm8s
Out[3]=3

Applications  (2)Sample problems that can be solved with this function

In clinical studies, medicine A on average caused an adverse reaction in 12 people per 100000 and medicine B in 9 people per 100000. It has also been determined that while some people will show no adverse reaction to medicine A or B alone, the combination of both caused an adverse reaction on average in 1 person per 500000. Assuming a Poisson model, find the adverse reaction distribution in the population of 10000:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-b3sthr
In[2]:=2
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https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-hsvx2i
Out[2]=2

Find the probability that there are at most 3 adverse reactions to medicine A and at most 4 adverse reactions to medicine B:

In[3]:=3
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-z24c8h
Out[3]=3

A university campus lies completely within twin cities A and B. On a given day there are, on average, 10 car accidents on campus; outside of campus there are 5 more in city A and 10 more in city B. Find the joint distribution of the number of accidents in the twin cities:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-rp1dgd

Probability density function:

In[2]:=2
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-9hgkk6
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-642bps
Out[3]=3

Find the average number of accidents in each city:

In[4]:=4
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-dwhm7d
Out[4]=4

Find the average total number of accidents in the twin cities:

In[5]:=5
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-6xpq53
Out[5]=5

Find the probability that on a given day there are more accidents in city A than in city B:

In[6]:=6
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-6kr7iu
Out[6]=6

Use a random sample to find the probability that there are at least 12 accidents per day in the twin cities:

In[7]:=7
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-ckpy66
Out[7]=7

Properties & Relations  (4)Properties of the function, and connections to other functions

Multivariate Poisson distribution is closed under addition:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-6w74ru
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-lm10tl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-0n5j29
Out[3]=3

One-dimensional multivariate Poisson distribution is a PoissonDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-05dup9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-x9s5mz
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-qcvc5u
Out[3]=3

The components are correlated for all allowed values of parameters:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-02xn3h

Multivariate Poisson cannot be represented as a product of its marginal distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-3wedmo

Find marginal distributions:

In[2]:=2
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-k06qr1
Out[2]=2

Find ProductDistribution of marginal distributions:

In[3]:=3
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-ql5dfu

Compare covariance matrices:

In[4]:=4
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-70or29
Out[4]=4

Compare PDFs:

In[5]:=5
✖
https://wolfram.com/xid/0bywcrcvd8hxwtkanecb7fe-4am0i4
Out[5]=5
Wolfram Research (2010), MultivariatePoissonDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MultivariatePoissonDistribution.html.

Text

Wolfram Research (2010), MultivariatePoissonDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MultivariatePoissonDistribution.html.

CMS

Wolfram Language. 2010. "MultivariatePoissonDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MultivariatePoissonDistribution.html.

APA

Wolfram Language. (2010). MultivariatePoissonDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MultivariatePoissonDistribution.html

BibTeX

@misc{reference.wolfram_2025_multivariatepoissondistribution, author="Wolfram Research", title="{MultivariatePoissonDistribution}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MultivariatePoissonDistribution.html}", note=[Accessed: 25-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_multivariatepoissondistribution, organization={Wolfram Research}, title={MultivariatePoissonDistribution}, year={2010}, url={https://reference.wolfram.com/language/ref/MultivariatePoissonDistribution.html}, note=[Accessed: 25-April-2025 ]}