Loss Models: From Data to Decisions, 4th Edition

A.6 Distributions with finite support

For these two distributions, the scale parameter θ is assumed known.

A.6.1.1 Generalized beta—a, b, θ, τ

A.6.1.2 Beta—a, b, θ The case θ = 1 has no special name but is the commonly used version of this distribution.

¹ Some references, such as [2], denote this integral P(α, x) and define Γ(α, x) = ƒ^∞_x t^α-1 e^−t dt. Note that this definition does not normalize by dividing by Γ(α). When using software to evaluate the incomplete gamma function, be sure to note how it is defined.

² There is no inverse transformed beta distribution because the reciprocal has the same distribution, with α and τ interchanged and θ replaced with 1/θ.

³ This is not the same Weibull distribution as in Section A.3.2.3. It is the negative of a Weibull distribution.