How to show that inter-arrival time of stochastic process has an exponential distribution?

Question

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?

Answer 1

Hint: if $T_x,T_y,T_z$ are the first inter-arrival times of $X,Y,Z$ respectively, then $T_z=T_x\wedge T_y$, and using this you can compute $\mathbb{P}(T_z>t)$ for $t>0$.

Answer 2

I will assume that you meant $Z_t = \min\{X_t,Y_t\}$ because otherwise the result is not true. In this case, denoting by $T_n^X$, $T_n^Y$, and $T_n^Z$ the $n^{\mathrm{th}}$ arrival time of $X_t$, $Y_t$, and $Z_t$, respectively, we have for each $t>0$ \begin{align} \mathbb P(T_1^Z>t) &= \mathbb P(\min\{T_1^X,T_1^Y\}>t)\\ &= \mathbb P(T_1^X>t, T_1^Y>t)\\ &= \mathbb P(T_1^X>t)\mathbb P(T_1^Y>t)\\ &= e^{-\lambda t}e^{-\mu t}\\ &= e^{-(\lambda+\mu)t}, \end{align} so that $T_1^Z$ has exponential distribution with parameter $\lambda+\mu$.