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