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I need to use the Poisson Point Process (PPP) model to randomly distribute a set of 'objects'; over a given area: Let's say that we have N objects to distribute over an area that has been split equally into S subsections. How might I use PPP to decide whether or not a subsection r (where r ∈ S) contains an object t (where t ∈ N)?

Ideally if anyone has a pseudo code solution then let me know, but I would be grateful for any form of help.

If you need me to be more specific let me know.

user3352349
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  • What have you tried so far? Where did you get stuck? I assume you know how to calculate a Poisson distribution. – Davislor Oct 09 '15 at 19:45
  • I've looked at various other examples on stackoverflow(e.g. [this](http://stackoverflow.com/questions/9832919/generate-poisson-arrival-in-java)) which have described producing random values around a mean (basically Knuth's algorithm), and I've been trying to get my head round some of the associated wiki articles. So my problem is one of understanding... how do I apply the various equations and algorithms to the above problem? Thanks – user3352349 Oct 09 '15 at 20:43

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Not writing a complete solution, because a request that says you need to use a Poisson Point Process and aren’t sure what that is sounds a lot like homework.

First, if the sections have been divided equally, any given element is equally likely to be in any of them. You would use a PPP to determine how many elements each section is likely to contain. Keep in mind that, if the sections are divided equally, they all have equal measure, 1/S. The links you followed give the probability of finding exactly x elements in r given its measure and N, so the probability of finding at most x is the cumulative PDF of this, and the probability of finding more than x is the complement of the PDF.

One hint to actually calculate this: memoize a vector of the factorials up to N, and think of an easy way to find the lowest common denominator of a/n! + b/(n-1)! + c/(n-2)! + ….

Davislor
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  • Thanks for the response... I will say though that this is not any sort of homework... my request is deliberately vague and is to assist in furthering my understanding in the described area prior to beginning a project. – user3352349 Oct 13 '15 at 08:27
  • For reference: Each section will contain 0 or 1 elements – user3352349 Oct 13 '15 at 08:30
  • Okay. That sounds more like Bernoulli variables than a Poisson point process. If there’s a known number of balls N that fit into a larger number of boxes S, no more than one per box, then the probability that any given box ends up with a ball is N/S, and the probability that a box will get the next ball is 1/(S-N). The probability of a given arrangement of balls is 1 in S choose N, or (S-N)!/S! if order is significant. – Davislor Oct 13 '15 at 21:46