Skip to main content

Table 2 Special EMO distributions

From: Exponentiated Marshall-Olkin family of distributions

Distribution

q(·)

G(x)

g(x)

E M O N(p,α,λ,μ,σ 2)

\(\frac {\alpha \lambda (1-p)}{\sigma }\)

\(\phi \left (\frac {x-\mu }{\sigma }\right)\)

\(\phi \left (\frac {x-\mu }{\sigma }\right)\)

E M O F r(p,α,λ,β,σ)

α λ(1−p)β

\(\exp \left \{-\left (\frac {\sigma }{x}\right)^{\beta }\right \}\)

\(\sigma ^{\beta } x^{-\beta -1}\exp \left \{-\left (\frac {\sigma }{x}\right)^{\beta }\right \}\)

E M O G a(p,α,λ,a,b)

\(\frac {\alpha \lambda (1-p)b^{a}}{\Gamma (a)}\)

\(\frac {\gamma (a,bx)}{\Gamma (a)}\)

\(\frac {b^{a}}{\Gamma (a)}x^{a-1}e^{-bx}\)

E M O B(p,α,λ,a,b)

\(\frac {\alpha \lambda (1-p)}{B(a,b)}\)

\(\frac {{\int \limits _{0}^{x}}w^{a-1}(1-w)^{b-1}dw}{B(a,b)}\)

\(\frac {1}{B(a,b)}x^{a-1}(1-x)^{b-1}\)

E M O G u(p,α,λ,μ,σ)

\(\frac {\alpha \lambda (1-p)}{\sigma }\)

\(\text {exp}\left \{{-\text {exp}\left [-\left (\frac {x-\mu }{\sigma }\right)\right ]}\right \}\)

\(\text {exp}\left \{-\text {exp}\left [-\frac {(x-\mu)}{\sigma }\right ]-\frac {(x-\mu)}{\sigma }\right \}\)