...

where the expectation is with respect to the parameters that are random variables. Thus, the functional form of mixture distributions can quickly become extremely complicated or even intractable. However, Monte Carlo simulation allows us to very simply include mixture distributions in our model, because Crystal Ball generates samples for each iteration in the correct logical sequence. So, for example, a Beta-Binomial(*n*, * a*,

*) distribution is easily generated by constructing a Binomial(Beta(*

*b**,*

*a**), n) model in Excel/Crystal Ball as is shown in the model model BetaBinomial- make sure here that the Binomial distribution in Crystal Ball is "dynamic". This allows Crystal Ball to generate a value first in each iteration from the Beta distribution, then create the appropriate Binomial distribution using this value of*

*b, 1**p*, and finally samples from that Binomial distribution.

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