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The Time Until the Next Occurrence in a Poisson Process

The Poison process is scalable so that if a random variableis Poisson with occurrences at the rate ofin each time periodand we can writethen events occur in each time periodat the rateand we can also write

Consider a Poisson process, and consider only that there are no occurrences in a timeso thatThis is equal to

The distribution function ofthe time to the first occurrence is then