In Auburngrads/teachingApps: Apps for Teaching Statistics, R Programming, and Shiny App Development

Functional relationships for $\;X \sim GAM(\kappa,\theta),\;\;X \in [0,\infty),\;\;\; \kappa,\theta\; > 0$

$$ \begin{aligned} f(x|\kappa,\theta)&=\frac{1}{\Gamma\left(\kappa\right)\theta^{\kappa}}x^{\kappa-1}e^{-x/\theta}\\\\ F(x|\kappa,\theta)&=\frac{\Gamma_{I}\left(\kappa,x/\theta\right)}{\Gamma\left(\kappa\right)}\\\\ h(x|\kappa,\theta)&=\frac{x^{\kappa-1}e^{-x/\theta}}{\left(\Gamma\left(\kappa\right)-\Gamma_{I}\left(\kappa,x/\theta\right)\right)\theta^{\kappa}\Gamma\left(\kappa\right)}\\\\\ E[X]&=\kappa\theta\\\\ Var[X]&=\kappa\theta^{2} \end{aligned} $$

• $\kappa \in \mathbb{R}^{+}$ is a shape parameter

• $\theta \in \mathbb{R}^{+}$ is a scale parameter

• $\Gamma(z)$ is the gamma function defined as

$$ \Gamma(z) = \begin{cases} \int_0^{\infty} x^{z-1}e^{-x}dx \hspace{12pt}\text{ if } z \in \mathbb{R}\\ (z - 1)! \hspace{40pt} \mbox{ if } z \in \mathbb{I} \end{cases} $$

• $\Gamma_{I}(a,b)$ is the (lower) incomplete gamma function defined as

$$ \Gamma_{I}(a,b) = \int_{0}^{b} t^{a-1}e^{-t}dt. $$

• $\Gamma(z)$ values can be computed in R using the base function gamma(x)

• $\Gamma_{I}(a,b)$ values can be computed using the gamma_inc(a,b) function from the gsl package

Auburngrads/teachingApps documentation built on June 17, 2020, 4:57 a.m.