There are k types of shocks identified that occur, independently, to a system. For 1 ≤ i ≤ k, suppose that shocks of type i occur to the system at a Poisson rate of λ i . Find the probability that the...

There are k types of shocks identified that occur, independently, to a system. For 1 ≤ i ≤ k, suppose that shocks of type i occur to the system at a Poisson rate of λ_i. Find the probability that the nth shock occurring to the system is of type i, 1 ≤ i ≤ n. Hint: For 1 ≤ i ≤ k, let N_i(t) be the number of type i shocks occurring to the system at or prior to t. It can be readily seen that N₁(t) + N₁(t) + · · · + N_k(t) is a Poisson process at a rate of λ = λ₁
+ λ₂
+ · · · + λ_k. Merging these Poisson processes to create a new Poisson process is called superposition of Poisson processes. For 1 ≤ i ≤ k, let Vi be the time of the first shock of type i occurring to the system. Argue that the desired probability is P ! V_i
= min(V₁, V₂, . . . , V_k) " . Then calculate this probability by conditioning on V_i.