# Table 1 T-X{Y} families based on different quantile functions

Random variable Y and support of r( t) The quantile function Q Y ( λ) Family of probability density function g( x) defined in (2.4)
Exponential (0, ∞) - b log(1 - λ), b > 0 $bf x 1 - F x r - b log 1 - F x$
Weibull (0, ∞) γ{-log(1 - λ)}1/c,  γ, c > 0 $γ f x r γ - log 1 - F x 1 / c c 1 - F x - log 1 - F x c - 1 / c$
Rayleigh (0, ∞) {-2b2 log(1 - λ)}1/2,  b > 0 $bf x r - 2 b 2 log 1 - F x 1 / 2 1 - F x - 2 log 1 - F x 1 / 2$
Dagum (0, ∞) $β λ 1 / p 1 - λ 1 / p 1 / α , α , β , p > 0$ $β f x r β F 1 / α p x / 1 - F 1 / p x 1 / α α p F 1 - 1 / α p x 1 - F 1 / p x 1 + 1 / α$
Lomax (0, ∞) $1 α 1 - 1 - λ 1 / k 1 - λ 1 / k , α , k > 0$ $f x k α 1 - F x 1 / k + 1 r 1 - 1 - F x 1 / k α 1 - F x 1 / k$
Log-logistic (0, ∞) $α λ 1 - λ 1 / β , α , β > 0$ $α f x r α F x / 1 - F x 1 / β β F β - 1 / β x 1 - F x β + 1 / β$
Exponentiated Exponential (0, ∞) $- 1 θ log 1 - λ 1 / α , θ , α > 0$ $f x r - 1 / θ log 1 - F 1 / α x α θ F α - 1 / α x 1 - F 1 / α x$
Cauchy (-∞, ∞) a + b {tan(π(λ - 0.5))}, b > 0 $π b f x r a + b tan π F x - 1 / 2 cos 2 π F x - 1 / 2$
Extreme value(Gumbel) (-∞, ∞) a - b log(-log λ),  b > 0 $bf x r a - b log - log F x - F x log F x$
Laplace (-∞, ∞) $a + b log 2 λ , λ < 0.5 a - b log 2 1 - λ , λ ≥ 0.5 a , b > 0$ $bf x r a + b log 2 F x F x , F x < .5 bf x r a - b log 2 1 - F x 1 - F x , F x ≥ . 5$
Logistic (-∞, ∞) $a + b log λ 1 - λ , b > 0$ $bf x r a + b log F x / 1 - F x F x 1 - F x$
Generalized logistic (II) (-∞, ∞) $log 1 - 1 - λ 1 / α 1 - λ 1 / α , α > 0$ $f x r log 1 - 1 - F x 1 / α / 1 - F x 1 / α α 1 - F x 1 - 1 - F x 1 / α$