# Question: Let X t be a Poisson counting process with arrival

Let X (t) be a Poisson counting process with arrival rate, λ. We form two related counting processes, Y1 (t) and Y2 (t), by randomly splitting the Poisson process, X (t). In random splitting, the i th arrival associated with X (t) will become an arrival in process Y1 (t) with probability p and will become an arrival in process Y2 (t) with probability 1 – p. That is, let S i be the i th arrival time of X (t) and define to be a sequence of IID Bernoulli random variables with Pr (Wi = 1) = p and Pr (Wi = 0) = 1 – p. Then the split processes are formed according to

Find the PMFs of the two split processes, PY1 (k; t) = Pr (Y1 (t) = k) and PY2 (k; t) = Pr (Y2 (t) = k). Are the split processes also Poisson processes?

Find the PMFs of the two split processes, PY1 (k; t) = Pr (Y1 (t) = k) and PY2 (k; t) = Pr (Y2 (t) = k). Are the split processes also Poisson processes?

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