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Disclaimer - I'm not a statistician. Most of my knowledge on the subject is self-taught.

I'm trying to grasp how continuous random variables can be used in conjunction with a discrete Poisson process.

Say you measure rainfall data in your area for one year. You can determine the probability if it will rain tomorrow by applying a discrete Poisson distribution. However, let's say you want to take it a step further and predict whether it will rain tomorrow and how much it will rain based on gathered data.

If for every time it rained you measured the amount of rainfall, would determining the amount of rainfall given that it rains be considered a Poisson process? Or is it more along the lines of conditional probability alongside a discrete Poisson distribution?

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The Poisson distribution keeps track of counts of things, and it has support $n = 0,1,2,...$, so you wouldn't use it to measure a binary event (e.g., whether it will rain tomorrow). You would use it for something like counting the number of raindrops that fall in an area during a day.

The Poisson process extends this idea to keep track of a count of things that happen according to a Poisson distribution at random times, where the random times between events are continuous exponential random variables. You would use this for something like modelling the times of rainstorms and the number of raindrops that fall within an area during each one.

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  • $\begingroup$ I might be having a hard time explaining my question: If you were solving a curiosity in which the first part was a discrete poisson and the second part was a continuous poisson - is it possible to relate these two since one is discrete and the other is continuous? Is that what a Skellam distribution is? $\endgroup$ – NRH May 4 '18 at 16:31
  • $\begingroup$ Perhaps it would be easier if you explain the thing you are trying to describe, and we can suggest an appropriate distributional family? (There is no continuous Poisson.) $\endgroup$ – Ben May 5 '18 at 0:00
  • $\begingroup$ I have a set of routers that have 10 possible triggered issues ('Capacity', 'Transport', 'Modulation', etc.). Every time one of these issues occurs I measure the impact towards the network for that day (they're floats like 0.996, 0.00430, 0.162, etc.). To determine whether a 'Capacity' issue will occur I used a Poisson distribution (with say a mean of 0.5 occurring per day for that specific Capacity category). However I don't know how to predict the impact towards the network. Because it's randomly occurring I'm guessing it's another Poisson (cumulative?). $\endgroup$ – NRH May 5 '18 at 20:22

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