Or a poisson process of rate lambda the bernoulli arrival

Question

or a Poisson process of rate λ, the Bernoulli arrival approximation assumes that in any very small interval of length ?, there is either 0 arrivals with probability 1-λ? or 1 arrival with probability λ?. Use this approximation to prove Theorem 10.7.

Theorem

Let N(t) = N₁(t) + N₂(t) be the sum of two independent Poisson processes with rates λ₁ and λ₂. Given that the N(t) process has an arrival, the conditional probability that the arrival is from N₁(t) is λ₁/(λ₁ + λ₂).