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Question

Let {N₁(t),t ≥ 0} and {N₂(t),t ≥ 0} be two independent Poisson processes, with rates λ₁ and λ₂, respectively. We defineN(t) = N₁(t) - N₂(t).

(a) Explain why the stochastic process {N(t) t ≥ 0} is not a Poisson process.

(b) Give a formula for the probability P[N(t₂) - N(t₁) = n], for t₂ > t₁ ≥ 0 and n ∈ { 0 , ± 1 , ± 2 , . . . }.