Table 1 Families of T - R { Y } distributions based on different choices for the random variables Y ^∗ and T

(a) Uniform

p

(1) Exponentiated-Exponential

$F_T\left(x\right)={(1-e^{-\mathit{\lambda }x})}^{\mathit{\alpha }}$

(b) Exponential

-log(1-p)

(2) Weibull

$F_T\left(x\right)=1-e^{-{(x/\gamma )}^c}$

(c) Log-logistic

p/(1-p)

(3) Logistic

$F_T\left(x\right)=1-{(1+e^{-\mathit{\lambda }x})}^{-1}$

(d) Logistic

log(p/(1-p))

(4) Log-logistic

$F_T\left(x\right)=1-{[1+{(x/\mathit{\alpha })}^{\mathit{\beta }}]}^{-1}$

(e) Extreme value

log(-log(1-p))

(5) Pareto

$F_T\left(x\right)=1-{(\mathit{\alpha }/x)}^{\mathit{\beta }}$

(6) Cauchy

$f_T\left(x\right)={\{\pi \mathit{\beta }(1+{\left[\right(x-\mathit{\alpha })/\mathit{\beta }]}^2\left)\right\}}^{-1}$

(7) Pascal

$f_T\left(x\right)=0.5\mathit{\lambda }{e}^{-\mathit{\lambda }\left|x\right|}$

(8) Gamma

$f_T\left(x\right)=x^{\mathit{\alpha }-1}{e}^{-x/\mathit{\beta }}/[{\mathit{\beta }}^{\mathit{\alpha }}\Gamma (\mathit{\alpha }\left)\right]$