Gamma Distribution

Gamma distribution

Overview

This object defines a gamma distribution with "shape" parameter (kk or α\alpha) and "scale" parameter (θ\theta or 1/β1/\beta). The probability density function, cumulative distribution function, and quantile are defined in Eq. (1), Eq. (2), and Eq. (3), respectively.

f(x;α,β)=βαxα1eβxΓ(α),f(x; \alpha, \beta) = \frac{\beta^\alpha x^{\alpha-1} e^{-\beta x}}{\Gamma(\alpha)},(1)F(x;α,β)=Γ(α,βx),F(x; \alpha, \beta) = \Gamma(\alpha, \beta x),(2)F1(p;α,β)=Γ1(α,p)β,F^{-1}(p; \alpha, \beta) = \frac{\Gamma^{-1}(\alpha, p)}{\beta},(3)

where x,α,β>0x,\alpha,\beta > 0 and 0p10\leq p \leq 1. Γ(a)\Gamma(a) is the gamma function defined by Eq. (4), Γ(a,x)\Gamma(a, x) is the lower incomplete gamma function defined by Eq. (5), and Γ1(a,p)\Gamma^{-1}(a, p) is the inverse of the incomplete gamma function. Γ\Gamma and Γ1\Gamma^{-1} are computed using a custom iterative procedure, which obtains an accuracy of about 101410^{-14}.

Γ(a)=0xa1exdx\Gamma(a) = \int_0^\infty x^{a-1}e^{-x}dx(4)Γ(a,x)=1Γ(a)0xta1etdt\Gamma(a, x) = \frac{1}{\Gamma(a)}\int_0^x t^{a-1}e^{-t}dt(5)

Input Parameters

  • shapeShape parameter (k or alpha).

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Shape parameter (k or alpha).

Required Parameters

  • scale1Scale parameter (theta or 1/beta).

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Scale parameter (theta or 1/beta).

Optional Parameters

  • control_tagsAdds user-defined labels for accessing object parameters via control logic.

    C++ Type:std::vector<std::string>

    Unit:(no unit assumed)

    Controllable:No

    Description:Adds user-defined labels for accessing object parameters via control logic.

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:Set the enabled status of the MooseObject.

Advanced Parameters