A semi-explicit density function for Kulkarni's bivariate phase-type distribution

Abstract

For a bivariate random variable $(Z_1,Z_2)$ having Kulkarni's bivariate phase-type distribution (see \cite{Ku89}), we derive a simple expression for the semi-explicit density function $E (e^{-sZ_2} 1_{ \{ Z_1 \in dy \} })$. Some immediate consequences are presented as concluding remarks.