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Say we have a type of event that lasts 30 seconds and occurs on average once per second.

I know that you can use an exponential distribution to describe time between events, or a Poisson distribution to describe how many events start in a fixed period. What distribution describes the variable "number of events simultaneously occurring"? I would like to answer questions like:

  • "What is the mean number of events simultaneously occurring?"
  • "During what proportion of time are more than two events co-occurring?" (i.e., $P(x > 2)$)

Is there also an answer if the duration is a distribution instead of fixed?

Edit: This seems to answer my question, but only for durations with an exponential distribution.

Kenneth Allen
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    For the first question: If the length of each event is deterministically 30 seconds, then the number of events simultaneously occurring at a given time $t$ is equivalent to the number of events that started in the time interval $[t-30, t]$, which is Poisson with mean $30$. – angryavian May 04 '21 at 04:56
  • Wow, that's really simple. Thank you. – Kenneth Allen May 05 '21 at 01:38

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