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The Beta distribution is the conjugate prior (meaning it has the same functional form, therefore also often called "convenience prior") to the Binomial likelihood function in Bayesian inference and, as such, is often used to describe the uncertainty about a binomial probability, given a number of trials *n* have been made with a number of recorded successes *s*. In this situations, a1 is set to the value (*s* + *x*), *b* is set to (*n* - *s* + *y*), and Scale = 1, where Beta(*x*, *y, 1*) is the prior.

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A Beta(1, 1, 1) = Uniform(0, 1) is usually used as a non-informative prior, though a Beta(½,½,1) and a Beta(0,0,1) are also sometimes used.

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The Beta distribution has also been used for a wide variety of other applications because it can take a very diverse set of shapes, as illustrated in the graphs above.

In Crystal Ball, the Beta can model a variable that runs from a Minimum to Maximum by using the following formula:

A special version of this four-parameter Beta distribution is called a PERT distribution. It makes the assumption that the mean = (minimum + 4*most likely = maximum) / 6. This extra equation allows the four parameters to be determined from three input values: the minimum, most likely and maximum, which makes it ideal for modeling expert opinion of a variable's uncertainty.

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