PolyaAeppliDistribution

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`PolyaAeppliDistribution`

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PolyaAeppliDistribution[θ,p]

represents a Pólya–Aeppli distribution with shape parameters θ and p.

Details

The Pólya–Aeppli distribution is a compound geometric Poisson distribution, i.e. the distribution of a sum of independent identically distributed geometric random variates where the number of variates follows Poisson distribution.
The probability for positive integer value in a Pólya–Aeppli distribution is proportional to $p^(x-1) TemplateBox[{{1, -, x}, 2, {-, {{(, {{(, {1, -, p}, )}, , theta}, )}, /, p}}}, Hypergeometric1F1]$ .
PolyaAeppliDistribution allows θ to be any positive real number, and p is a number between 0 and 1.
PolyaAeppliDistribution allows θ and p to be a dimensionless quantity. »
PolyaAeppliDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

PolyaAeppliDistribution[θ,p] represents a discrete statistical distribution defined for integer values and determined by the positive real parameters θ and p (called "shape parameters"), where . The Pólya–Aeppli distribution has a probability density function (PDF) that is discrete and unimodal and whose overall shape (its height, its spread, and the horizontal location of its maximum) is determined by the values of θ and p. The Pólya–Aeppli distribution is sometimes referred to as the geometric Poisson distribution, though it should not be confused with either the geometric (GeometricDistribution) or Poisson (PoissonDistribution) distributions.
The Pólya–Aeppli distribution dates back to the dissertation work of Swiss mathematician Alfred Aeppli and the subsequent investigations of Aeppli's adviser George Pólya throughout the 1920s and 1930s. Classically, the Pólya–Aeppli distribution is the distribution of a sum of independent identically distributed geometric (GeometricDistribution) random variates where the number of variates follows a Poisson distribution (PoissonDistribution), and in particular, the distribution can be described as an urn model in which the number of urns is Poisson distributed, while the number of marbles in each urn follows a geometric distribution. Since its inception, the Pólya–Aeppli distribution has been used in biometrics and in the study of Markov models, as well as in the modeling of phenomena in fields like biology, queueing theory, accident statistics, and bioinformatics.
RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Pólya–Aeppli distribution. Distributed[x,PolyaAeppliDistribution[θ,p]], written more concisely as xPolyaAeppliDistribution[θ,p], can be used to assert that a random variable x is distributed according to a Pólya–Aeppli distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
The probability density and cumulative distribution functions may be given using PDF[PolyaAeppliDistribution[θ,p],x] and CDF[PolyaAeppliDistribution[θ,p],x], though one should note that there is no closed-form expression for its PDF. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. These quantities can be visualized using DiscretePlot.
DistributionFitTest can be used to test if a given dataset is consistent with a Pólya–Aeppli distribution, EstimatedDistribution to estimate a Pólya–Aeppli parametric distribution from given data, and FindDistributionParameters to fit data to a Pólya–Aeppli distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Pólya–Aeppli distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Pólya–Aeppli distribution.
TransformedDistribution can be used to represent a transformed Pólya–Aeppli 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 Pólya–Aeppli distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Pólya–Aeppli distributions.
PolyaAeppliDistribution is related to a number of other statistical distributions. It has PoissonDistribution as a limiting case in the sense that the limit of the PDF of PolyaAeppliDistribution[θ,p] as p→0 (for ) is precisely equivalent to the PDF of PoissonDistribution[θ]. PolyaAeppliDistribution is also closely related to GeometricDistribution, PoissonConsulDistribution, SkellamDistribution, and CompoundPoissonDistribution.

Examples

open allclose all

Basic Examples (3)Summary of the most common use cases

Probability mass function:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-c9vzr0

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-nj5ej4

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-bekedt

Out[3]=3

Cumulative distribution function:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-bkh9gk

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-jchef4

Out[2]=2

Mean and variance:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-1xp74b

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-6jelxa

Out[2]=2

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

Generate a sample of pseudorandom numbers from a Pólya–Aeppli distribution:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-qhtk5j

Compare its histogram to the PDF:

In[2]:=2

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-03mwaz

Out[2]=2

Distribution parameters estimation:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-45b7g2

Estimate the distribution parameters from sample data:

In[2]:=2

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-epi747

Out[2]=2

Compare a density histogram of the sample with the PDF of the estimated distribution:

In[3]:=3

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-f8ui5o

Out[3]=3

Skewness:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-mr87mq

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-ys4lg0

Out[2]=2

Limiting values:

In[3]:=3

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-shu7fz

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-2sb5wh

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-x3orfq

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-7pkep0

Out[6]=6

Kurtosis:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-m3lct8

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-r5qshn

Out[2]=2

Limiting values:

In[3]:=3

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-6b77vp

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-pzqmqc

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-hjc4qw

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-ztkg00

Out[6]=6

Different moments with closed forms as functions of parameters:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-js043h

Moment:

In[2]:=2

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-rx074o

Out[2]=2

CentralMoment:

In[3]:=3

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-pknsqa

Out[3]=3

FactorialMoment:

In[4]:=4

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-zg9ct4

Out[4]=4

Closed form for symbolic order:

In[5]:=5

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-b37y8j

Out[5]=5

Cumulant:

In[6]:=6

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-9gzmth

Out[6]=6

Hazard function:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-kijbt5

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-xpjrzc

Out[2]=2

Quantile function:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-f5db4g

In[2]:=2

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-bzwvpb

Out[2]=2

Use dimensionless Quantity to define PolyaAeppliDistribution:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-cdiqpr

Out[1]=1

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

The CDF of PolyaAeppliDistribution is an example of a right-continuous function:

In[1]:=1

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https://wolfram.com/xid/0colxd37mrzbmimb6umy-5w2lx5

Out[1]=1

The number of hotbeds of a contagious disease follows PoissonDistribution with mean 10, while the number of sick people within the hotbed follows GeometricDistribution with mean 7. Find the probability that the total number of sick people is greater than 70:

In[1]:=1