BUILT-IN MATHEMATICA SYMBOL

BetaNegativeBinomialDistribution

represents a beta negative binomial mixture distribution with beta distribution parameters

and

, and n successful trials.

MORE INFORMATION

The beta negative binomial distribution is a negative binomial distribution whose probability parameter p follows a beta distribution with shape parameters and . »

BetaNegativeBinomialDistribution allows , , and n to be any positive real numbers.

BetaNegativeBinomialDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

EXAMPLES

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Basic Examples (3)

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 have a beta negative binomial distribution:

Compare its histogram to the PDF:

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:

Kurtosis:

Different moments with closed forms as functions of parameters:

Moment:

CentralMoment:

FactorialMoment:

Closed form for symbolic order:

Cumulant:

Hazard function:

Quantile function:

Applications (2)

CDF of BetaNegativeBinomialDistribution is an example of a right continuous function:

The probability of at least 50 failures before 10 successes, assuming a beta distribution on

:

Properties & Relations (5)

The probability of getting negative integers, integers beyond n, or non-integer numbers is zero:

Relationships to other distributions:

WaringYuleDistribution is a special case of beta negative binomial distribution:

Beta negative binomial distribution is a mixture of NegativeBinomialDistribution and BetaDistribution:

Possible Issues (2)

BetaNegativeBinomialDistribution is not defined when

,

, or n is non-positive:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful: