Table 7 Some pmfs of the new family based on different T distributions.
Distribution of t | pmf | Sub family |
|---|---|---|
Exponential | \(e^{- \alpha {x^{\theta }F(x)}}- e^{- \alpha {(x+1)^{\theta }F(x+1)}}\) | DE-\(X^{\theta }\) |
Weibull | \(e^{- \left( \frac{x^{\theta }F(x)}{\beta }\right) ^{\lambda }}- e^{- \left( \frac{(x+1)^{\theta }F(x+1)}{\beta }\right) ^{\lambda }}\) | DW-X\(^{\theta }\) |
Rayleigh | \(e^{- \left( \frac{x^{\theta }F(x)}{\beta }\right) ^{2}}- e^{- \left( \frac{(x+1)^{\theta }F(x+1)}{\beta }\right) ^{2}}\) | DR-X\(^{\theta }\) |
Fréchet | \(e^{-\left( \frac{(x+1)^{\theta }F(x+1)}{\delta }\right) ^{-\kappa }}-e^{-\left( \frac{x^{\theta }F(x)}{\delta }\right) ^{-\kappa }}\) | DFr-X\(^{\theta }\) |
Lomax | \((1+\beta x^{\theta }F(x))^{-\alpha }-(1+\beta (x+1)^{\theta }F(x+1))^{-\alpha }\) | DL-X\(^{\theta }\) |
Burr | \((1+ (x^{\theta }F(x))^{\alpha })^{-\beta } -(1+ ((x+1)^{\theta }F(x+1))^{\alpha })^{-\beta }\) | DB-X\(^{\theta }\) |
Log-logistic | \({1}/\left( {1+({(x+1)^{\theta }F(x+1)}/{\alpha })^{-\beta }}\right) -{1}/\left( {1+({x^{\theta }F(x)}/{\alpha })^{-\beta }}\right)\) | DLL-X\(^{\theta }\) |
Gamma | \({\gamma \left( \alpha , \beta (x+1)^{\theta }F(x+1)\right) }/{\Gamma (\alpha )}-{\gamma \left( \alpha , \beta x^{\theta }F(x)\right) }/{\Gamma (\alpha )}\) | DG-X\(^{\theta }\) |
Log-normal | \(\dfrac{1}{2}\left[ \Phi \left( \dfrac{\ln ((x+1)^{\theta }F(x+1))-\mu }{\sigma \sqrt{2}}\right) -\Phi \left( \dfrac{\ln (x^{\theta }F(x))-\mu }{\sigma \sqrt{2}}\right) \right]\) | DLN-X\(^{\theta }\) |
