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Sheldon Ross describes the following as examples of random variables that generally obey the Poisson probability law:

$1.$ The number of customers entering a post office on a given day.

$2.$ The number of vacancies occurring during a year in the federal judicial system.


I am aware that the Poisson random variable may be used as an approximation for a binomial random variable with parameters $(n,p)$ when $n$ is large and $p$ is small enough so that $np$ is of moderate size, but how do I think about the two situations above using this piece of information? I am not too sure of what I'd consider independent trials and what I'd consider a success in these case.

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In (1), you could imagine each member of the population, independently of the others, has a small probability of entering that post office on that day.

In (2), you could imagine that each federal judge, independently of the others, has a small probability of retiring or dying in a given year.

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