# Poisson process, time and probabilities

An average of n birds fly through an area, in an hour, following a Poisson process. (I think this means the hours don't matter; there's no influence in the number of birds that fly through, at different parts of the day, hypothetically. Correct me if I'm wrong.) P1 is the probability that exactly m birds fly through between 12:00-14:00 (2 continuous hours). P2 is the probability that exactly m birds fly through between 15:00-16:00 and 17:00-18:00 (a total of 2 hours still, but not together). Please notice m is the same for both situations.

Are the probabilities of P1 and P2 the same or I'm assuming something that's wrong?

Thanks!

(Please tell me if you need more information, I can make it up)

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## 1 Answer

Yes, they are the same. Another crucial assumption in the Poisson process is that what happens now is independent of what happened a moment ago or what will happen in the next moment (or at any other moment, for that matter). Therefore the distribution of events during any (measurable) period of time depends only on the length of time, not on how it is broken up.

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Brilliant. Thanks for the clarification. –  Queops Jan 27 '11 at 23:48
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