This is a problem I've been nerdling on for a few days, hoping it has an elegant solution. Alas, my maths degree is now so far back in the past that I've really not got very far.
I'm trying to figure out the distribution of the following variable (y). x0 and x1 are random picks from uniform(0,1). If x1 is bigger than xo, y = xo. If x1 is smaller, pick x2 from U(0,1) and make the same comparison - as long as each pick is smaller than the last, you get another go, then finally y is the smallest value you reached in the sequence.
Here's how far I got: if f(p, q) is the probability that you end up below p having so far got to q > p (and still having another go available), then either you get below p on your next go, or you get p < x < q so you get another go, or you're bust.
$$ f(p, q) = p + \int_p^qf(p,x)dx = p + F(p, q) - F(p,p) $$
Now I suspect F(p,p) to be zero but I'm not confident of that. In any case, when we've got here is a differential equation of the form $ f'(q) = A + f(q) $. I couldn't think of any function satisfying that. Any help gratefully accepted, particularly if it leads to an elegant closed form solution ... and if I'm somewhere near an interesting (easy-ish) problem if I changed my rules a bit, please do point me to that, as this exercise is for no other purpose than staving off senility for a bit longer