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

Functional relationships for $\;T \sim BETA(\alpha,\beta, c, d),\;\;T \in [c,d + c]\;\; \alpha,\beta, d > 0$

$$ \begin{aligned} f(t|\alpha,\beta,c,d)&=\frac{(t-c)^{\alpha-1}(d+c-t)^{\beta-1}}{B(\alpha,\beta) (d)^{\alpha+\beta-1}}\\\\ F(t|\alpha,\beta,c,d)&=\mathcal{I}{t}(\alpha,\beta,c,d)=\frac{\int{0}^{t}(x-c)^{\alpha-1}(d+c-x)^{\beta-1}dx}{B(\alpha,\beta) (d)^{\alpha+\beta-1}}\\\\ h(t|\alpha,\beta,c,d)&=\frac{f(t)}{1-F(t)}\quad \\\\\ t(p|\alpha,\beta,c,d)&=t(p|\alpha,\beta,0,1) \times d + c \\\\ E[T]&=\frac{\alpha}{\alpha + \beta} \times d + c \\ Var[T]&=\frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)} \times d + c \end{aligned} $$

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