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

TL;DR

This paper derives a semi-explicit density function for Kulkarni's bivariate phase-type distribution, facilitating easier analysis of the joint distribution of the variables.

Contribution

It provides a new semi-explicit expression for the density function of Kulkarni's bivariate phase-type distribution, simplifying its analysis.

Findings

Derived a semi-explicit density function expression

Presented immediate consequences of the new formula

Enhanced understanding of Kulkarni's bivariate 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.