Beta Distribution

Beta distribution

Overview

This object defines a beta distribution with two shape parameters: "alpha" (α\alpha) and "beta" (β\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)=xα1(1x)β1B(α,β),f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{\Beta(\alpha,\beta)},(1)F(x;α,β)=Ix(α,β),F(x;\alpha,\beta) = I_x(\alpha,\beta),(2)F1(p;α,β)=Ix1(α,β,p),F^{-1}(p; \alpha,\beta) = I_x^{-1}(\alpha, \beta, p),(3)

where α,β>0\alpha,\beta > 0 and 0x,p10\leq x,p \leq 1. B(a,b)\Beta(a,b) is the beta function defined in Eq. (4), Ix(a,b)I_x(a,b) is the regularized incomplete beta function defined in Eq. (5), and Ix1(a,b,p)I_x^{-1}(a, b, p) is the inverse of the incomplete beta function. Each of these quantities are computed using a custom iterative procedure (Press et al., 1992), which obtains an accuracy of about 101410^{-14}.

B(a,b)=Γ(a)Γ(b)Γ(a+b)=01xa1(1x)b1dx\Beta(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} = \int_0^1 x^{a-1}(1-x)^{b-1}dx(4)Ix(a,b)=1B(a,b)0xta1(1t)b1dtI_x(a,b) = \frac{1}{\Beta(a,b)}\int_0^x t^{a-1}(1-t)^{b-1}dt(5)

Input Parameters

  • alphaShape parameter 1.

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Shape parameter 1.

  • betaShape parameter 2.

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Shape parameter 2.

Required 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

References

  1. William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical Recipes in C. Cambridge University Press, USA, 1992. ISBN 0521431085.[BibTeX]