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I am considering events. They can be meetings, phone calls, etc. Let's say that a person is calling someone.

Having both the probability distribution of event's start time and probability distribution of an event's duration, can the probability distribution that event is in progress in a given moment be calculated analytically?

In other words: can I calculate the probability that the person of interest is in a call?

Settings of this problem are very similar to the one here, but the measure of interest is different, and I cannot grasp a solution for my problem from that one.

I created a computational solution, but I'm wondering is there any theoretical framework you can point me to.

Oliver
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  • Can you describe your computational solution? Your problem is thorny. The start time distribution would be bimodal with a peak in the early morning and a smaller peak in the early afternoon. Also, the duration distribution depends on the start time (Parkinson's Law). – stretch Nov 22 '22 at 16:16
  • Oh, that was quick, thank you for the comment! – Oliver Nov 22 '22 at 16:42
  • My naïve solution is based on sampling: I've created many samples of the event, then calculated the probability in discrete time intervals. You are right, the observed distribution of the start time exhibits multimodality. But, I found a suitable approximation for PDF, no problem for that. I am wandering if there is a procedure to combine two analytical forms of PDFs for start time and duration into PDF that describe probability of a call in progress. – Oliver Nov 22 '22 at 16:49

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