Probability mass function:

Cumulative distribution function:

Mean and variance:

Generate a sample of pseudorandom numbers from a Borel–Tanner distribution:

Compare sample histogram to the PDF of the Borel-Tanner distribution:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

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

Skewness:

Skewness attains its minimum independently of n:

Limiting values:

For large n, distribution becomes more symmetric:

Kurtosis:

Limiting values:

For large values of n, kurtosis converges to the kurtosis of the standard NormalDistribution:

Different moments with closed forms as functions of parameters:

Moment:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Use dimensionless Quantity to define BorelTannerDistribution:

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

Customers arrive at a service desk at unit rate per unit time. Serving each customer takes constant time . At the time of the opening of the service desk, the line contained people. The total number of customers served before there is no one in line follows a BorelTannerDistribution:

Note that the time must be less than 1 for the line to eventually disappear:

Show the distribution mass function for specific parameters:

Expected number of customers served before there is no one in line:

Compute for the specific parameters used above:

Sum of random variates from BorelTannerDistribution with common α parameter also follows BorelTannerDistribution:

PoissonConsulDistribution is a parameter mixture of BorelTannerDistribution and PoissonDistribution:

Estimating via method of moments may yield invalid parameters:

Use the definition of the second parameter to specify its value: