Assume $f(x)$ is the probability density function (pdf) of a random variable $X\geq0$. Select a number $a$ between $0$ and $X$ uniformly and randomly and split the random variable to make two random variables $a$ and $X-a$. Find the joint pdf of $a$ and $X-a$.
My effort: we have to take the derivative of the following CDF with respect to $r$ and $s$:
$P(a\leq r,X-a\leq s)=P(X-a\leq s|a\leq r) P(a\leq r)$
I can calculate $P(a\leq r)$, but $P(X-a\leq s|a\leq r)$ is an ugly intergral.
Any idea? I thought it might be already in the literature, any good keywords to search for? Ideally, I want to do divide $X$ into $m$ random variables uniformly and find the joint pdf of the outcome (here $m=2$).