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Model
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Domain
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SDRF
$g(x)$
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CDF or
$F(x)$
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PDF or
$f(x)$
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Quantile
(m2)
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Power
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$x\in(0,\: \infty)$
$\lambda >0,\: k>0$
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Undefined
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$\lambda x^{k}$
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$\lambda kx^{k-1}$
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Undefined
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Power
(shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0,\: k>0$
$z=x-a$
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Undefined
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$\lambda z^{k}$
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$\lambda kz^{k-1}$
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Undefined
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Logarithmic
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$x\in(a,\: \infty)$
$\lambda >0,\: k>0$
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Undefined
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$k\ln x+\lambda$
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$\dfrac{k}{x}$
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Undefined
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Logarithmic
(shifted)
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$x\in(a,\: \infty)$
$\lambda >0,\: k>0$
$z=x-a$
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Undefined
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$k\ln z+\lambda$
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$\dfrac{k}{z}$
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Undefined
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Generalized
exponential
(shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0,\: k>0$
$z=\dfrac{x-a}{\lambda}$
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$\dfrac{k}{\lambda}\dfrac{\left(1-e^{-z}\right)^{k-1}e^{-z}}{1-\left(1-e^{-z}\right)^{k}}$
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$\left(1-e^{-z}\right)^{k}$
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$\dfrac{k}{\lambda}\dfrac{e^{-z}}{\left(1-e^{-z}\right)^{(1-k)}}$
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$a-\lambda\ln\left(1-p^{1/k}\right)$
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Weibull
(Shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0,\: k>0$
$z=\dfrac{x-a}{\lambda}$
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$\dfrac{k}{\lambda}z^{k-1}$
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$1-e^{-z^{k}}$
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$\dfrac{k}{\lambda}z^{k-1}e^{-z^{k}}$
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$a+\lambda\left(\ln\dfrac{1}{1-p}\right)^{1/k}$
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Exponential
(shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0$
$z=\dfrac{x-a}{\lambda}$
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$\dfrac{1}{\lambda}$
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$1-e^{-z}$
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$\dfrac{1}{\lambda}e^{-z}$
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$a-\lambda\ln(1-p)$
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Inverse
Weibull
(shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0,\: k>0$
$z=\dfrac{\lambda}{x-a}$
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$\dfrac{k}{\lambda}z^{k+1}\dfrac{e^{-z^{k}}}{1-e^{-z^{k}}}$
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$e^{-z^{k}}$
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$\dfrac{k}{\lambda}z^{k+1}e^{-z^{k}}$
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$a+\dfrac{\lambda}{(-\ln p)^{1/k}}$
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Inverse
exponential
(shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0$
$z=\dfrac{\lambda}{x-a}$
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$\dfrac{1}{\lambda}z^{2}\dfrac{e^{-z}}{1-e^{-z}}$
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$e^{-z}$
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$\dfrac{1}{\lambda}z^{2}e^{-z}$
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$a-\dfrac{\lambda}{\ln p}$
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Normal
(left-
truncated)
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$x\in(a,\: \infty)$
$a\ge 0,\: \sigma >0$
$z=\dfrac{x-\mu}{\sigma}$
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$\dfrac{\varphi(z)}{\sigma[1-\Phi(z)]}$
$\Phi(z)=\dfrac{1}{2}{erfc}\left(-\dfrac{z}{\sqrt{2}}\right)$
$\varphi(z)=\dfrac{1}{\sqrt{2\pi}}e^{-z^{2}/2}$
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$\dfrac{\Phi(z)-\alpha}{1-\alpha}$, $\alpha =\Phi\left(z_{a}\right)$
$z_{a}=\dfrac{a-\mu}{\sigma}$
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$\dfrac{\varphi(z)}{(1-\alpha)\sigma}$
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$\mu +\sigma\Phi^{-1}[\alpha +(1-\alpha)p]$
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Lognormal
(shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \sigma >0$
$z=x-a$
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$\dfrac{\sqrt{2/\pi}e^{-\dfrac{1}{2}\left(\dfrac{\ln z-\mu}{\sigma}\right)^{2}}}{{erfc}\left(\dfrac{\ln{z}-\mu}{\sqrt{2}\sigma}\right)\sigma{z}}$
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$\Phi(z)=\dfrac{1}{2}{erfc}\left(-\dfrac{\ln{z}-\mu}{\sqrt{2}\sigma}\right)$
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$\dfrac{1}{\sqrt{2\pi}\sigma z}e^{-\dfrac{1}{2}\left(\dfrac{\ln z-\mu}{\sigma}\right)^{2}}$
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$a+\Phi^{-1}(p)$
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Logistic
(left-
truncated)
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$x\in(a,\: \infty)$
$a\ge 0,\: s>0$
$z=\dfrac{x-\mu}{s}$
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$\dfrac{1}{s\left(1+e^{-z}\right)}$
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$\dfrac{\Phi(z)-\alpha}{1-\alpha}$, $\alpha =\Phi\left(z_{a}\right)$
$\Phi(z)=\dfrac{1}{1+e^{-z}}$, $z_{a}=\dfrac{a-\mu}{s}$
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$\dfrac{\varphi(z)}{(1-\alpha)s}$
$\varphi(z)=\dfrac{e^{-z}}{\left(1+e^{-z}\right)^{2}}$
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$\mu +s\ln\dfrac{\alpha /(1-\alpha)+p}{1-p}$
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Generalized
logistic
power
(shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0,\: k>0,\: \beta >0$
$z=x-a$
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$\dfrac{\beta\lambda kz^{\beta k-1}}{\left(\lambda +z^{k}\right)\left[\left(\lambda
+z^{k}\right)^{\beta}-z^{\beta k}\right]}$
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$\left(1-\dfrac{\lambda}{\lambda +z^{k}}\right)^{\beta}$
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$\dfrac{\beta\lambda kz^{\beta k-1}}{\left(\lambda +z^{k}\right)^{\beta +1}}$
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$a+\left(\dfrac{\lambda p^{1/\beta}}{1-p^{1/\beta}}\right)^{1/k}$
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Logistic
power
(shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0,\: k>0$
$z=x-a$
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$k\dfrac{z^{k-1}}{\lambda +z^{k}}$
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$1-\dfrac{\lambda}{\lambda +z^{k}}$
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$\dfrac{\lambda kz^{k-1}}{\left(\lambda +z^{k}\right)^{2}}$
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$a+\left(\dfrac{\lambda p}{1-p}\right)^{1/k}$
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Monad
(shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0$
$z=x-a$
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$\dfrac{1}{\lambda +z}$
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$\dfrac{z}{\lambda +z}$
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$\dfrac{\lambda}{(\lambda +z)^{2}}$
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$a+\dfrac{\lambda p}{1-p}$
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Gamma
(Shifted)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0,\: k>0$
$z=\dfrac{x-a}{\lambda}$
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$g(z)=\dfrac{z^{k-1}e^{-z}}{\lambda[\gamma(k)-\gamma(k,\: z)]}$
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$\dfrac{\gamma(k,\: z)}{\gamma(k)}$
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$\dfrac{1}{\lambda\gamma(k)}z^{k-1}e^{-z}$
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$a+\lambda\gamma^{-1}[p\gamma(k)]$
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Gumbel
(left-
truncated)
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$x\in(a,\: \infty)$
$a\ge 0,\: \lambda >0$
$z=\dfrac{x-\mu}{\lambda}$
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$\dfrac{e^{-z}}{\lambda\left[e^{e^{-z}}-1\right]}$
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$\dfrac{\Phi(z)-\alpha}{1-\alpha}$, $\alpha =\Phi\left(z_{a}\right)$
$\Phi(z)=e^{-e^{-z}}$, $z_{a}=\dfrac{a-\mu}{\lambda}$
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$\dfrac{\varphi(z)}{(1-\alpha)\lambda}$
$\varphi(z)=e^{-(z+e^{-z})}$
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$\mu -\lambda\delta$
$\delta =\ln\left(\ln\dfrac{1}{\alpha +(1-\alpha)p}\right)$
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Beta
(Shifted)
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$x\in(a,\: b)$
$a\ge 0,\: \lambda >0,\: k>0$
$z=\dfrac{x-a}{b-a}$
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$\dfrac{z^{\lambda -1}(1-z)^{k-1}}{(b-a)[B(\lambda ,\: k)-B(z;\lambda ,\: k)]}$
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$\dfrac{B(z;\lambda ,\: k)}{B(\lambda ,\: k)}$
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$\dfrac{z^{\lambda -1}(1-z)^{k-1}}{(b-a)B(\lambda ,\: k)}$
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$a+(b-a)I^{-1}(p)$
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Sine power
(Shifted)
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$x\in(a,\: \lambda)$
$a\ge 0,\: \lambda >0,\: k>0$
$z=\dfrac{x-a}{\lambda}$
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$\dfrac{\pi k}{\lambda}z^{k-1}\tan\left(\dfrac{\pi}{2}z^{k}\right)$
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$\dfrac{1}{2}\left[1-\cos\left(\pi z^{k}\right)\right]$
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$\dfrac{\pi k}{2\lambda}z^{k-1}\sin\left(\pi z^{k}\right)$
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$a+\lambda\left[\dfrac{\cos^{-1}(1-2p)}{\pi}\right]^{1/k}$
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Sine
(Shifted)
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$x\in(a,\: \lambda)$
$a\ge 0,\: \lambda >0$
$z=\dfrac{x-a}{\lambda}$
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$\dfrac{\pi}{\lambda}\tan\left(\dfrac{\pi}{2}z\right)$
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$\dfrac{1}{2}[1-\cos(\pi z)]$
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$\dfrac{\pi}{2\lambda}\sin(\pi z)$
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$a+\lambda\left[\dfrac{\cos^{-1}(1-2p)}{\pi}\right]$
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