Probability that X is less than Y directly from joint CDF? Suppose X and Y are arbitrary random variables with joint cdf F. Is it possible to find $P(X&lt;Y)$ or $P(X\le Y)$ directly from F? I think it's easier to think in terms of the equivalent $P(X-Y&lt;0)$, but I'm not sure how to go from there. I know that if X and Y are continuous, we can differentiate the joint cdf to get the joint pdf and we can integrate over that to solve it. But is there a more general way to get it directly from the joint cdf? e.g., what if X or Y is not continuous or discrete?
$\displaystyle \mathbb{P}(X\leq Y)= \int_{-\infty}^{\infty}\int_{-\infty}^{y} f(x,y)dxdy$. Assuming continuous r.v pdf $f(x,y)$
$\displaystyle \mathbb{P}(X\leq Y)= \int_{-\infty}^{\infty}\int_{-\infty}^{y} f(x,y)dxdy$. Assuming continuous r.v pdf $f(x,y)$