BetaBinomialDistribution

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:

Moment:

CentralMoment:

FactorialMoment:

Closed form for symbolic order:

Cumulant:

Hazard function:

Quantile function:

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 a mixture of BinomialDistribution and BetaDistribution:

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: