bbinopdf

Beta-binomial probability distribution function.

This m-file returns the beta-binomial probability density function with parameters N, A and B at the values in X. Note: The density function is zero unless N, A and B are integers.

The Beta-binomial distribution is used to model the number of successes in n binomial trials when the probability of success p is a Beta(a,b) random variable. The extreme flexibility of the shape of the Beta distribution means that it is often a very fair representation of the randomness of p.

A variable with a Beta-binomial distribution is distributed as a binomial distribution with parameter p, where p is distribution with a beta distribution with parameters a (alpha) and b (beta). For n trials, it has probability density function:

p(x) = n_C_x [B(x+a, n-x+b) / B(a,b)]

where B(a,b) is a beta function and n_C_x is a binomial coefficient. (http://mathworld.wolfram.com/BetaBinomialDistribution.html) (http://en.wikipedia.org/wiki/Beta-binomial_model)

The probability of success varies randomly, but in any one scenario that probability applies to all trials. For example, you might consider using the Beta-binomial distribution to model:

--The number of life insurance policy holders who will die in any one year, where some external variable (e.g. highly contagious disease, extreme weather) moderates the probability of death of all individual to some degree.
--The number of cars that crash in a race of n cars, where the predominant factor is not the skill of the individual driver, but the weather on the day.
--The number of bottles of wine from a producer that are bad where the predominant factor is not how each bottle is treated, but something to do with the batch as a whole.

The Beta-binomial is a two-dimensional multivariate Polya distribution, as the binomial and beta distributions are special cases of the multinomial and Dirichlet distributions, respectively.

Syntax: function y = bbinopdf(x,n,a,b)

Inputs:
x - number of success
n - number of trials
a – Beta’s alpha parameter
b - Beta‘s beta parameter

Output:
y - beta-binomial probability value