What is the right distribution to model the duration of (short-term) sick-leave in days? I would like to model the distribution of the duration of (short-term) sick-leave: if some one calls in sick, how long can you expect the sickness to last? Short-term would typically mean between 1 and 14 days.
Distributions that came to mind are the binomial distribution or the Poisson distribution. The binomial distribution is less good of a fit I think, the Poisson is a bit more lenient in the tail.
How could I best reason about this?

We can mathematically prove that the distribution for $n$ independent Bernoulli trials (like a coin flip), each with a success probability of $p$, has a particular distribution that we call "binomial".
There are similar mathematical proofs for other named distributions.
I see no way to mathematically prove how much leave a person will take. Consequently, this is an empirical problem, noting the issues discussed in comments by Stephan Kolassa and Tim.

This will depend very much. People who call in sick on Friday will probably only be sick on Friday (1 day), except if they call in again on Monday after. Otherwise, it may depend on whether policy is that you can excuse yourself for up to (say) three days and need a doctor's note for anything longer. Or whether people need to get a doctor's note in any case, and doctors then "typically" write a note for three days. Best to look at separate histograms per day of week of the initial notification.
-Stephan Kolassa


Something like binomial won't work. Binomial assumes i.i.d. trials, say that you have flu, it's not that you randomly have it one day and another day it just ends, probably at the beginning the probability of it continuing for the next day increases, then it decreases. Also, it might work differently for different cases, e.g. if you had a minor accident, you may recover fast, so different dynamics than flu. What I'm trying to say is that it might be not that easy to find a single, simple distribution to describe it. -Tim

The issue about a Friday (or whatever day comes before typical days off) is insightful. If I am very sick and need four days to recover, I might only take two days of leave if I fall ill on a Friday and take off that Friday and the following Monday. If I am ill when I wake up Tuesday morning, I will be out all four days remaining in the week.
