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

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