In the last post we saw that a geometric sum of iid exponential random variables is the interarrival distribution of a split Poisson process. Now, by “magic” geometric random variables will come from considering two Poisson processes.

Let and be independent Poisson processes with intensities and . Let

be the time of first arrival in . What are the distributions of and ?

It is clear that , i.e. has density . By independence, we can compute the distribution of by conditioning on ; for we have

It follows that .

By the independent increment property of Poisson processes, for we have

It follows that .

Now consider the first arrival time of ,

Since , we have , and by symmetry, . What about the distribution of ?

In general, if and are independent, then for

so that .

Therefore

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