Beta-Binomial distribution


Format: BetaBinomial(n, a,b)

 

 

 

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

Comments

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 versions

When modeling or analyzing counting data, it is often desirable to modify the probability of zero events 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 events occurring.

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

ModelRisk functions added to Microsoft Excel for the Beta-Binomial distribution

VoseBetaBinomialObject generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

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.

 

ModelRisk functions added to Microsoft Excel for the Zero-Inflated Beta-Binomial distribution

VoseZIBetaBinomialObject generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZIBetaBinomialProb returns the probability mass or cumulative distribution function for this distribution.

VoseZIBetaBinomialProb10 returns the log10 of the probability mass or cumulative distribution function.  

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

VoseZIBetaBinomialFitObject constructs a distribution object of this distribution fitted to data.

VoseZIBetaBinomialFitP returns the parameters of this distribution fitted to data.

 

ModelRisk functions added to Microsoft Excel for the Zero-Truncated Beta-Binomial distribution

VoseZTBetaBinomialObject generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZTBetaBinomialProb returns the probability mass or cumulative distribution function for this distribution.

VoseZTBetaBinomialProb10 returns the log10 of the probability mass or cumulative distribution function.  

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

VoseZTBetaBinomialFitObject constructs a distribution object of this distribution fitted to data.

VoseZTBetaBinomialFitP returns the parameters of this distribution fitted to data.

 

Beta-Binomial distribution equations

Zero-Inflated Beta-Binomial distribution equations

Zero-Truncated Beta-Binomial distribution equations

 

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