Finding $P(X < cY)$ when $X$ and $Y$ are iid random variables

Question

If I know that $X$ and $Y$ are iid random variables, is there are special formula for $P(X < cY)$ where $c$ is some nonzero scalar?

Answer 1

If, for example, $X$ (and thus also $Y$) has density $f$ and cdf $F$, we have $$ \mathbb P(X < c Y) = \int_{-\infty}^\infty dy \int_{-\infty}^{cy} dx\; f(x) f(y) = \int_{-\infty}^\infty dy\;F(cy) f(y)$$ and not much more can be said, unless you know the form of $f$. Of course if $c=1$ and the distribution is continuous, $\mathbb P(X < Y) = 1/2$.