I only have rudimentary knowledge of statistics but am currently working on a problem that may (or may not) have an easy solution when expressed as a stochastic problem.
Basically I have two random variables $X[z]$ and $Y[z]$, where the image of $X$ and $Y$ is $[\lambda_1,\lambda_2]$ (a closed compact interval).
I know the probability distribution of $P( X[z] < a)$ and $P( Y[z] > a)$. With this information is there an easy way to find the combined probability distribution (with respect to $a$ of
$$P(X[z] < Y[z])$$ or possibly $$P(X[z] < a< Y[z])$$
or do I need additional information?
or is it a hopeless endeavour?
Any pointers appreciated.