Let $(Z\_t)\_{t \geq 0}$ be a stochastic proces defined by $Z\_t = \max\{X\_t , Y\_t \}$ with $(X\_t)\_{t \geq 0}$ and $(Y\_t)\_{t \geq 0}$ independent Poisson processes with parameters $\lambda$ an $\mu$. How do I show that the first inter-arrival time of $(Z\_t)\_{t \geq 0}$ has an exponential distribution?