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

When the range is given and over something other than [0,1], the above estimation formula apply with D replaced with (D-«lower»)/(«upper»-«lower»). Maximum likelihood estimation of all four parameters when «lower» and «upper» are not known is difficult, but worked out in Johnson, Kotz and Balakrishan (1994), Continuous Univariate Distributions, 2ne ed., Volume II, p. 221-235, John Wiley & sons.

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