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Please refers to the below for requirements for poisson distribution.

I'm confused about the 2nd bullet point with the 4th. If an event is random, that means the event cannot be associated with a probability in any means. Then, how can it be possible to get the situation in 4th bullet point? i.e. how can random events give uniform distribution within a time period if themselves are probability-less?

Please explain with example. Better to provide proof as to why Poisson distribution requires the first 4 bullet points as requirements. Thank you!

men

Hing Wong
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    The fact that an event is random does NOT mean that it has no probability. A coin toss has a random outcome, with a probability of 50% for heads and 50% for tails. – Itamar Mushkin Jun 23 '19 at 05:55
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    The fourth bullet distinguishes a *homogeneous* from an *inhomogeneous* Poisson process. In the latter, the probabilities vary by location. In a post at https://stats.stackexchange.com/a/215253/919 I use this characterization of a homogeneous Poisson process to derive formulas for the Poisson distribution: a study of the reasoning there might shed some light on what these requirements mean, how they differ, and why they are important. – whuber Jun 23 '19 at 11:54

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The second bullet point the occurrences must be random just means that the events should be produced by some random process, that is, they should not be deterministic. An odometer or a (Japanese) train schedule is not random in this sense. This in no way is in contradiction with the fourth bullet point, which says that this process must be homogeneous. If it were not, we would get an inhomogeneous Poisson process.

kjetil b halvorsen
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