Let $X,Y,Z$ be independent random variables. If for all $a\in \mathbb{R}$ $$P(X>a) \geq P(Y>a),\,\,\,\,\,(1)$$ can we conclude $$P(X>Z) \geq P(Y>Z)?\,\,\,\,\,\,\,(2)$$
I am a little confused. It is like we are replacing a constant with a random variable, which is weird. On the other hand, every instantiation of $Z$ is a number $b\in \mathbb{R}$ which satisfies (1) replacing $a$ with $b$. Can we readily conclude (2) from (1), or it is not obvious? Maybe we have to find the distributions of $X-Z$ and $Y-Z$ and then decide?
Just integrate (1) with respect to $F_Z$ and you will get (2). [ $P(X>Z)=\int P(X>a)dF_Z(a)$ etc].