BUILT-IN MATHEMATICA SYMBOL

MultivariatePoissonDistribution

represents a multivariate Poisson distribution with mean vector

.

MORE INFORMATION

The multivariate Poisson distribution corresponds to the distribution of where is Poisson distributed with mean .

The parameters can be any positive numbers.

MultivariatePoissonDistribution can be used with such functions as Mean, CDF, and RandomVariate.

EXAMPLES

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Basic Examples (4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Covariance matrix:

Probability density function:

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Cumulative distribution function:

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Mean and variance:

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Covariance matrix:

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Scope (8)

Generate a set of pseudorandom vectors that are Poisson distributed:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Goodness-of-fit test:

Skewness for each component depends on

and

:

Kurtosis for each component depends on

and

:

Correlation:

Different mixed moments for a bivariate Poisson distribution:

Mixed central moments:

Mixed factorial moments:

Mixed cumulants:

Closed form for a symbolic order:

Hazard function:

Marginal distributions:

Applications (2)

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:

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

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:

Probability density function:

Find the average number of accidents in each city:

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

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

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

Properties & Relations (4)

Multivariate Poisson distribution is closed under addition:

One-dimensional multivariate Poisson distribution is a PoissonDistribution:

The components are correlated for all allowed values of parameters:

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

Find marginal distributions:

Find ProductDistribution of marginal distributions:

Compare covariance matrices:

Compare PDFs: