## Combined Probability Distribution of two distributions [closed]

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.

-
 Suppose $X$ is zero half the time and one half the time. Suppose $Y_1=X$ and $Y_2=1-X$. Then $Y_1$ and $Y_2$ have the same probability distribution, but the probability that $X$ is less than $Y_1$ is not the same as the probability that $X$ is less than $Y_2$. This question should be closed. – Steven Landsburg Jul 5 at 14:58