This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# WaringYuleDistribution

 WaringYuleDistribution[] represents the Yule distribution with shape parameter . WaringYuleDistributionrepresents the Waring distribution with shape parameters and .
• The probability for integer value in a Waring-Yule distribution is proportional to:
Probability density function of Yule distribution:
Cumulative distribution function of Yule distribution:
Mean and variance of Yule distribution:
Probability density function of Waring distribution:
Cumulative distribution function of Waring distribution:
Mean and variance of Waring distribution:
Probability density function of Yule distribution:
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Cumulative distribution function of Yule distribution:
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Mean and variance of Yule distribution:
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Probability density function of Waring distribution:
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Cumulative distribution function of Waring distribution:
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Mean and variance of Waring distribution:
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 Scope   (7)
Generate a set of pseudorandom numbers that have the Waring distribution:
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 for Yule distribution:
Skewness attains its minimum:
Skewness for Waring distribution:
Kurtosis for Yule distribution:
Kurtosis attains its minimum:
Kurtosis for Waring distribution:
Different moments with closed forms as functions of parameters:
Moment for Yule distribution:
CentralMoment for Yule distribution:
FactorialMoment for Yule distribution:
Cumulant for Yule distribution:
Moment for Waring distribution:
CentralMoment for Waring distribution:
FactorialMoment for Waring distribution:
Cumulant for Waring distribution:
Hazard function for Yule distribution:
Waring distribution:
Quantile function:
 Applications   (3)
The CDF of WaringYuleDistribution is an example of a right-continuous function:
Creation of new species within genera occurs at rate , while birth of new genera occurs at a slower rate . Limiting frequency distribution of sizes of genera of all ages is given by WaringYuleDistribution. Assuming :
Plot the logarithm of the PDF:
Find the probability that a genus will have no more than 5000 species:
Generate a collection of words by randomly selecting characters and white space. The resulting word sizes can be modeled using a WaringYuleDistribution:
Relationships to other distributions:
Yule distribution is a special case of BetaNegativeBinomialDistribution:
Waring distribution is a special case of BetaNegativeBinomialDistribution:
Waring distribution simplifies to Yule distribution for :
Yule distribution can be obtained as a parameter mixture of GeometricDistribution and UniformDistribution:
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