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Let's consider a Poisson process on the line with rate parameter $\lambda$. There are two ways to think about this:

  1. In any interval $[a,b)$ the expected number of events is distributed as a Poisson random variable with mean $\lambda(b-a)$. Conditional on the number of events, the times of the events in $[a,b)$ are uniformly random.

  2. The inter-arrival times of events is exponential with mean $\lambda^{-1}$.

This is all well-explained on Wikipedia for example. (Well, point (2) is rather given short shrift).

Question 1: I would like to treat this equivalence in a (very) elementary manner. What would be a good textbook, or other resource, which covers this? Perhaps via simulation?

(I am teaching this as a "research / modelling" project, and so I cannot just write my own notes and/or set my own simulation tasks. Rather I really want/need to point the students to some existing resources.)

A perhaps harder question to answer is:

Question 2: I'd love a research article, example in a book, etc. which uses this equivalence in some way to model some real data.

Let me comment on this slightly: there are loads of examples (e.g. "phone calls to a helpline in a given hour") which are Poisson distributed. But I don't know of an example which treats this as a process (e.g. the initial data is presented as times of the calls, not already just as counts).

Matthew Daws
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  • I posted an elementary treatment of the equivalence at https://stats.stackexchange.com/questions/214421. As far as question 2 goes, you can find many examples on our site: search for [Poisson exponential](https://stats.stackexchange.com/search?q=+poisson+exponential+is%3Aanswer+score%3A2). In light of this cornucopia of examples, could you highlight any part of your question that still needs an answer? – whuber Oct 01 '19 at 16:32
  • @whuber Your post is actually very nice: thank you for the link. I would still be interested in e.g. a textbook treatment. I'm afraid I don't understand your link for the Q2 however: I don't see that this search returns examples of real data. – Matthew Daws Oct 01 '19 at 16:42
  • By including "data" as a search keyword I found a few. Typically the data are summarized in a question, because we don't have a standard mechanism to provide large datasets. I have found that many questions provide enough detail that (a) you can ascertain these are real data and (b) you can readily simulate a dataset with all the essential characteristics of the real data. Part of the problem is that we don't field questions that only ask for datasets. However, the threads here can point you towards subjects that use Poisson process data, such as accident records, bird sightings, etc. – whuber Oct 01 '19 at 17:25
  • [Here](https://stats.stackexchange.com/questions/424532/passenger-inter-arrival-times/424544#424544) is another Q&A with an elementary discussion of connections between Poisson and exponential distributions. – BruceET Oct 01 '19 at 18:27

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