Phase-type distribution

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CDF

1 − α e x S 1 {\displaystyle 1-{\boldsymbol {\alpha }}e^{xS}{\boldsymbol {1}}}

CF

− α ( i t I + S ) − 1 S 0 + α 0 {\displaystyle -{\boldsymbol {\alpha }}(itI+S)^{-1}{\boldsymbol {S}}^{0}+\alpha _{0}}

Mean

− α S − 1 1 {\displaystyle -{\boldsymbol {\alpha }}{S}^{-1}\mathbf {1} }

Median

no simple closed form

MGF

− α ( t I + S ) − 1 S 0 + α 0 {\displaystyle -{\boldsymbol {\alpha }}(tI+S)^{-1}{\boldsymbol {S}}^{0}+\alpha _{0}}

Mode

no simple closed form