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Number of one Poisson type before another

For independent Poisson processes of rates

λ_{1}

and

λ_{2}

, the number

K

of type-two arrivals before the first type-one arrival satisfies

P (K = k) = (\frac{λ _{2}}{λ _{1} + λ _{2}})^{k} \frac{λ _{1}}{λ _{1} + λ _{2}} .

This follows by marking the superposed process by its arrival type.

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