Probability mass function:

Cumulative distribution function:

Mean and variance:

Generate a sample of pseudorandom numbers from a Poisson–Consul distribution:

Compare sample histogram to the PDF of the Poisson-Consul distribution:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

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

Skewness:

Limiting values:

For large μ, the distribution becomes symmetric:

Kurtosis:

Limiting values:

For large μ, the kurtosis becomes close to the kurtosis of NormalDistribution:

Different moments with closed forms as functions of parameters:

Moment:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Use dimensionless Quantity to define PoissonConsulDistribution:

CDF of PoissonConsulDistribution is an example of a right-continuous function:

The number of customers arriving at a service desk follows PoissonDistribution with mean 0.6, and the number of customers already in line before the service desk opens follows PoissonDistribution with mean 5. The number of customers served until there is no one in line follows PoissonConsulDistribution:

Plot the probability mass function:

Expected number of customers served:

Standard deviation:

Find the probability that at least 15 customers will be served at a busy period:

Simulate numbers of customers served at 30 busy periods:

The initial size of a population has PoissonDistribution with mean μ. The size of each offspring generation is also Poisson distributed with mean proportional to the generation size, with constant λ. Simulate the total progeny:

The total progeny follows a PoissonConsulDistribution:

Simulate the population size for 30 generations:

The sum of random variates from PoissonConsulDistribution with the same λ parameter follows PoissonConsulDistribution with the same λ parameter:

Proof using factorial moment-generating functions:

Relationships to other distributions:

Poisson–Consul distribution simplifies to PoissonDistribution:

The limit of large μ in PoissonConsulDistribution such that is fixed gives InverseGaussianDistribution:

Force to be constant:

PoissonConsulDistribution is a parameter mixture of BorelTannerDistribution and PoissonDistribution: