Is it worth migrating this question to MO, from math.SE?
It seems elementary but did not get any response on MSE even after 10 days.
I don't think the question would be good at MO. I would vote to close it as off-topic, and believe so would many others. I see no particular research angle to the question, and I would not consider it otherwise interesting to research mathematicians, either.
That said, I gave you something of an answer at MSE. I don't know if my way qualifies as a nice way of writing your distribution, but I don't see how one could do much better. If you (or anyone else, for that matter) want to discuss mathematical details, let us do so where the question is.