Probability of one random variable greater than two times another

2018-02-06 09:24:50

Two random variables $X$ and $Y$ are i.i.d. with probability density function $f$ (unknown). Two variables are nonnegative. What is the probability $P(X>2Y)$ ?

My question is if there is a deterministic answer.

Here is my attempt:

$$Pr(X>2Y) = Pr(Y

$$\int_{0}^{\infty} f(u) \big[\int_{0}^{u/2}f(v)dv \big] du =\int_{0}^{\infty}f(u)F(u/2)du$$

I attempted to use integration by parts but was not able to work out a deterministic answer.

Note that $P(X>Y) = P(Y>X) = 1/2$ by symmetry. I do not know if there is a connection between these two problems.