Let N 1 (t) and N 2 (t) be two independent Poisson processes with rate 1

Let N₁(t) and N₂(t) be two independent Poisson processes with rate λ₁ and λ₂ respectively. Let N(t) = N₁(t) +N₂(t) be the merged process. Show that given

We can interpret this result as follows: Any arrival in the merged process belongs to N₁(t) with probability

and belongs to N₂(t) with probability

independent of other arrivals.

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N(t) = n, N₁(t) ~ Binomial n₂ λ₁ λ1+12,