Maximum of two Independent Random Variables with Erlang distribution

Question

While I am deriving the maximum of two Erlang Indepenent random variables $$ z = \begin{cases} l, & \text{if $l$>$h$ and $l$>0} \\ h, & \text{if $h$>$l$ and $l$>0} \\ h, & \text{if $l$<0} \end{cases} $$ $$ E[z]= \int_0^\infty \int_{h=l}^\infty h f(h)f(l) dh dl + \int_0^\infty \int_{h=0}^l h f(h)f(l) dh dl + \int_{-\infty}^0 \int_{h=0}^\infty h f(h)f(l) dh dl $$ However, $l$ will not be less than zero since it is representing a time period. for this condition how can I simplify this term $$\int_{-\infty}^0 \int_{h=0}^\infty h f(h)f(l) dh dl$$ to only represent $l=0$ ?

Answer 1

Erlang random variables have a non-negative support.   So because both $H$ and $L$ are Erlang RV, therefore: $$\begin{align} Z & = \begin{cases} H & : 0\leq L \leq H \\ L & : 0\leq H < L\\ 0 & : \textsf{otherwise}\end{cases} \\[2ex] \mathsf E(Z) & = \int_0^\infty \int_h^\infty l f(h)f(l)\operatorname d l\operatorname d h + \int_0^\infty \int_0^h h f(h)f(l)\operatorname d l\operatorname d h \\[2ex] & = 2\int_0^\infty h f(h)\int_0^h f(l)\operatorname d l\operatorname d h & \textsf{by symmetry} \end{align}$$