I have the following problem:
A machine has two critically important parts and is subject to three different kinds of shocks. Shocks of type $i$ occur as a Poisson process with rate $\lambda_i$. Shocks of type 1 break part 1, shocks of type 2 break part 2 and shocks of type 3 break both parts. Let $U$ and $V$ be the failure times of the two parts. What is $Pr\{U>s,V>t\}$?
I know that this is looking for the time to the first event, and that $Pr\{T>t\} = e^{-\lambda t}$ where $T$ is the time to the first event and $t$ is an integer. But I'm not sure how to combine the processes as I don't know what parts are independent. Can you offer any advice or tips at all?
For every nonnegative real times $(s,t)$, the event $[U\gt s,V\gt t]$ happens when the first shock of type $i$ happens at time $T_i$ with $T_1\gt s$, $T_2\gt t$ and $T_3\gt\max(s,t)$, hence, by independence, $$P(U\gt s,V\gt t)=\mathrm e^{-\lambda_1 s}\cdot\mathrm e^{-\lambda_2 t}\cdot\mathrm e^{-\lambda_3\max(s,t)}.$$