﻿ Beta-Binomial distribution

# Beta-Binomial distribution

Format: VoseBetaBinomial(n, a,b, U)

BetaBinomial equations

A Beta-Binomial distribution returns a discrete value between 0 and n. An example of a Beta-Binomial(30,10,7) distribution, together with its best matching Binomial distribution is given below:

##### Uses

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. Thus:

VoseBetaBinomial(n,a,b) = VoseBinomial(n,VoseBeta(a,b))

The extreme flexibility of the shape of the Beta distribution means that it is often a very fair representation of the randomness of p.

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 distribution always has more spread (variance) than its best fitting Binomial distribution, because the Beta distribution adds extra randomness. Thus, when a Binomial distribution does not match observations, because the observations exhibit too much spread, a Beta-Binomial distribution is often used instead.

##### Zero-modified version

When modeling or analyzing counting data, it is often desirable to modify probability of zero of the discrete distribution we use, to more accurately model the probability of "no event occurring". We can make two types of modifications to our distributionfor this:

• Zero-inflated model - we increase the probability of zero. (equations)

• Zero-truncated model - we entirely remove the probability of zero events occurring. (equations)

##### VoseFunctions for this distribution

VoseBetaBinomialObject constructs a distribution object for this distribution

VoseBetaBinomialProb returns the probability mass or cumulative distribution function for this distribution

VoseBetaBinomialProb10 returns the log10 of the probability mass or cumulative distribution function

VoseBetaBinomialFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution

VoseBetaBinomialFitObject constructs a distribution object of this distribution fitted to data

VoseBetaBinomialFitP returns the parameters of this distribution fitted to data