Beta distribution

Beta(X,Y,lower,upper)

The Beta distribution.

Creates a continuous distribution of numbers between 0 and 1 with X / (X+Y) representing the mean, if the optional parameters lower and upper are omitted. For bounds other than 0 and 1, specify the optional lower and upper bounds to offset and expand the distribution.

X and Y must be positive.

When to use

Use a beta distribution if the uncertain quantity is bounded by 0 and 1 (or 100%), is continuous, and has a single mode. This distribution is particularly useful for modeling an opinion about the fraction of a population that has some characteristic. For example, if you have observed n members of the population, of which r display the characteristic c, you can represent the uncertainty about the true fraction with c using a beta distribution with parameters X = r and Y = n - r.

If the uncertain quantity has lower and upper bounds other than 0 and 1, include the lower and upper bounds parameters to obtain a transformed beta distribution. The transformed beta is a very flexible distribution for representing a wide variety of bounded quantities.

Library

Distributions

Parameter Estimation

Suppose D contains sampled historical data indexed by I, and you want to estimate the «X» and «Y» parameters of the beta distribution from this historical data. With your data in D normalized to be between the known bounds of 0 and 1, the parameters can be obtained from the following estimation formulas:

«X» := Var m := Mean(D,I);
       Var v := Variance(D,I);
       (m^2 - m^3 - v*m) / v
«Y» := Var m := Mean(D,I);
       Var s := Variance(D,I);
       (m*(1-m)^2 - v * (1-m)) / v

See Also

• BetaFn -- the complete beta function
• BetaI -- the incomplete beta function, gives the cumulative density analytically.
• BetaIInv -- the inverse of BetaI.
• Dens_Beta -- the probability density of Beta(a,b) at x
• Pert - A parametric variation on the beta distribution