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.