= Beta(Min, Max, a,*b*)

Crystal Ball parameter restrictions

The Beta distribution is the only continuous parametric distribution available with Crystal Ball that is bounded at both ends. Four examples of the Beta distribution are shown below. In its standard form (Scale =1) the distribution ranges from zero to one (see the first and second figure), and takes a wide range of shapes. The third figure (left bottom) shows examples in which the Scale > 1 and the fourth graph (right bottom) shows example of the Beta distribution with the notation as used in Crystal Ball.

#### Uses

The Beta distribution has two main uses:

As the description of uncertainty or random variation of a probability, fraction or prevalence;

As a useful distribution one can re-scale and shift to create distributions with a wide range of shapes and over any finite range. As such, it is sometimes used to model expert opinion, for example in the form of the PERT distribution.

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.

#### Comments

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.

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.

Beta(*a*, *b, 1*) = 1 - Beta(*b*, *a, 1*): a property that is readily apparent in the context of modeling uncertainty about a binomial probability. This identity is sometimes useful to keep equations neater.

The Excel function BETADIST(x,a,*b*) returns the Beta cumulative distribution function.

A Beta(z,1) distribution is sometimes called a Power-function distribution with parameter z.