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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

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    $\begingroup$ 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. $\endgroup$ Commented Jun 23, 2019 at 5:55
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    $\begingroup$ The fourth bullet distinguishes a homogeneous from an inhomogeneous Poisson process. In the latter, the probabilities vary by location. In a post at 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. $\endgroup$
    – whuber
    Commented Jun 23, 2019 at 11:54

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i.e. how can random events give uniform distribution within a time period if themselves are probability-less?

The probability density describing the population should be uniform.

Uniform refers to the uniform probability, not to a uniform distribution of the observed counts/events.

For a specific sample you will not get a uniform distribution.

Imagine the ticks from a Geiger-Müller counter measuring ionizing radiation from a constant source of radiation*. You will hear ticks with variations in intervals. The sample is definitely not uniform. However, for any point in time the probability to observe a count/tick is constant.

The inhomogeous Poisson process, the case when the 4th point is violated, might give some additional intuition about it.


*approximately constant, theoretically the source is not constant since the decay makes that the radiation intensity decreases, and there could be other fluctuations due to environmental circumstances influencing the transmission between the source and the counter.

The Poisson process is an approximation and in reality nothing is like a Poisson distribution (in a similar way a lot of processes are in reality not truly normal distributed): What kind of physical processes are well modeled as poisson processes?

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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.

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