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I find that in many economics model, many events are often called "Poisson process" even though they will only occur once, like the death of an agent or the exit of a firm or transition from one aggregate state to another aggregate state. However the definition of Poisson process (see e.g. wiki) seems to link with Poisson distribution or Poisson random variables, which basically count numbers of occurrence which can be larger than one. And the classical examples of Poisson process in the textbook are always something like during a certain time how many buses arrive the station. So I am confused why those events with only two states can be also called Poisson process?

In particular, I want to see how we can formally (mathematically) define such an event, say e.g. a representative agent with a flow death rate $\mu$ under continuous time, as a Poisson process.

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the question of "until some event happens" can also be answered under the assumption of a Poisson process.

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Each person can die only once, but many people in a city can die in the same week or month. So number of deaths/period is a plausibly Poisson variable. (Moved from comments)

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