Let $X$, $Y$ be two independent continuous random variables with support on $\mathbb{R}_+$. More precisely, assume that $X$ and $Y$ are distributed Fréchet with location parameter $0$, shape parameter $\alpha$ and scale parameters $s_X$ and $s_Y$ respectively (hence the only difference between these distributions is the scale parameter). Let $a$ denote a positive scalar. How to compute the conditional expectation $$ E\left[X|X\geq aY\right]\ ? $$
My first approach was to assume a fixed $Y$ and then integrate with respect to $Y$, i.e.
$$ E\left[X|X\geq aY\right] =\int_{0}^{\infty}E\left[X|X\geq ay\right]f_{y}\left(y\right)dy =\int_{0}^{\infty}\left[\int_{ay}^{\infty}xf_{x}\left(x\right)dx\right]f_{y}\left(y\right)dy $$
But I got stuck with those huge integrals after substituting $f_x$ and $f_y$ for the Fréchet distribution.
I also tried finding the conditional distribution of $X$ for $X\geq aY$, i.e. $f_{X|X\geq aY}(x,y)$, so I can write
$$ E\left[X|X\geq aY\right] = \int_{0}^\infty x f_{X|X\geq aY}\left(x|y\right)dx $$
But I don't know how to obtain the joint distribution given that the condition depends on both $X$ and $Y$.