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# BetaBinomialDistribution

 BetaBinomialDistribution represents a beta binomial mixture distribution with beta distribution parameters and , and binomial trials.
• The beta binomial distribution is a binomial distribution with n trials whose probability parameter follows a beta distribution with shape parameters and . »
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 binomial distribution:
Compare its histogram to the PDF:
Distribution parameters estimation:
Assuming known n, 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:
Closed form for symbolic order:
Hazard function:
Quantile function:
 Applications   (5)
CDF of BetaBinomialDistribution is an example of a right continuous function:
The probability of more than 50 successes in 100 trials assuming a beta distribution on :
Define a negative hypergeometric distribution:
Find the probability that black balls were sampled without replacement before a white ball was drawn from an urn initially filled with black and white balls:
Alternatively, compute the probability of drawing a white ball provided that there were black balls in the previous samplings without replacement:
Define the Polya distribution:
Generate random numbers:
Compute probabilities:
Define the Polya-Eggenberg urn distribution:
The distribution models an urn scheme. An urn contains white balls and black balls. When a ball is drawn it is returned to the urn together with additional balls of the same color. The distribution gives the probability of drawing white balls in draws:
Find the number of white balls in 10 draws:
The probability of getting negative integers, integers beyond n, or non-integer numbers is zero:
Relationships to other distributions:
DiscreteUniformDistribution is a special case of a beta binomial distribution:
For and , the beta binomial distribution has a triangular shape but is not a discrete version of TriangularDistribution:
BetaBinomialDistribution is not defined when or is non-positive:
BetaBinomialDistribution is not defined when n is not a positive integer:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
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