Table 1 Some Parametric Distributions and their Properties

Weibull

\(\exp \left( { - \left( {\lambda t} \right)^{\alpha } } \right)\)

\(\lambda \alpha \exp \left( {\lambda t} \right)^{\alpha - 1}\)

\(\alpha \lambda \left( {\lambda t} \right)^{\alpha - 1} \times \exp \left( { - \left( {\lambda t} \right)^{\alpha } } \right)\)

Extreme Value

\(\exp \left( { - \exp \left( {\frac{t - \mu }{\sigma }} \right)} \right)\)

\(\sigma^{ - 1} \exp \left( {\frac{t - \mu }{\sigma }} \right)\),\(t \ge 0\)

\(\sigma^{ - 1} \exp \left( {\frac{t - \mu }{\sigma } - \exp \left( {\frac{t - \mu }{\sigma }} \right)} \right)\)

Log-normal

\(1 - \Phi \left( {\alpha \log \left( {\lambda t} \right)} \right)\)

\(\begin{gathered} \frac{{\left( {\frac{1}{t\sigma }} \right)\phi \left( {\frac{\ln t}{\sigma }} \right)}}{{\Phi \left( { - \frac{\ln t}{\sigma }} \right)}}, \hfill \\ \,\,\,\,\,\,\,\,\,t> 0;\sigma> 0. \hfill \\ \end{gathered}\)

\(\left( {2\pi } \right)^{{ - \frac{1}{2}}} \alpha t^{ - 1} \exp \left( {\frac{{ - \alpha^{2} \left( {\log \left( {\lambda t} \right)} \right)^{2} }}{2}} \right)\)

Log-logistic

\(\frac{1}{{1 + \left( {\lambda t} \right)^{\alpha } }}\)

\(\frac{\alpha }{{t\left[ {1 + (t/\lambda )^{ - \alpha } } \right]}}\)

; \(t> 0\)

\(\lambda \alpha \left( {\lambda t} \right)^{\alpha - 1} \left( {1 + \left( {\lambda t} \right)^{\alpha } } \right)^{ - 2}\)

Gamma

\(1 - \frac{{\Gamma_{x} \left( \gamma \right)}}{\Gamma \left( \gamma \right)},x \ge 0,\gamma> 0.\)

\(1 - \frac{{x^{\gamma - 1} e^{ - t} }}{{\Gamma \left( \gamma \right) - \Gamma_{x} \left( \gamma \right)}}\), \(t \ge 0\)

\(\frac{{\lambda^{k} t^{k - 1} }}{\Gamma \left( k \right)}\exp \left( { - \lambda t} \right)\)

Gompertz

\(\exp \left( { - \frac{{e^{\ell } }}{\varphi }(e^{\ell } - 1)} \right)\,\)

\(\begin{gathered} \exp (\,\ell )\exp (\,\varphi t)\, \hfill \\ = \exp (\ell + \varphi t) \hfill \\ \end{gathered}\)

; \(t \ge 0\)

\(\exp \left( { - \frac{{e^{\ell } }}{\varphi }(e^{\ell } - 1) + \ell + \varphi t} \right)\)

Generalised Gamma

\(\int\limits_{0}^{x} {t^{s - 1} } e^{ - t} dt\)

\(\frac{{p\lambda \left( {\lambda t} \right)^{\alpha - 1} e^{ - t} dt}}{{\gamma \left( {\alpha /p,\left( {\lambda t} \right)^{p} } \right)}}\);\(t \ge 0\)

\(p\frac{{\lambda \left( {\lambda t} \right)^{\alpha - 1} e^{{ - \left( {\lambda t} \right)p}} }}{{\Gamma \left( {\alpha /p} \right)}}\)