The Pooled Poisson Process

If events of typeoccur at an average rate with each type of event independent of any other and each can be modelled by a Poisson process with parameter respectively, then the occurrence of any type of event will occur at a rateand it seems plausible to model the occurrences of these events by a Poisson process with parameter

More concisely letbe the waiting time for the occurrence of any event. Ifis the time for the event I to occur thenso

Since each process is Poisson,hence


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