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# BorelTannerDistribution

 BorelTannerDistribution represents a Borel-Tanner distribution with shape parameters and n.
• The probability for integer value in a Borel-Tanner distribution is proportional to for , and is zero for .
Probability density function:
Cumulative distribution function:
Mean and variance:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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 Scope   (7)
Generate a set of pseudorandom numbers that are Borel-Tanner distributed:
Compare its histogram to the PDF:
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 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:
Hazard function:
Quantile function:
 Applications   (2)
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 parameter also follows BorelTannerDistribution:
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