Challenge Task $2$ on Display Advertising Assessment

Consider the ordering of probabilities of campaigns in terms of the probability of having the highest CTRMeanVolume. Will these two orderings always be the same? Assume that the number of draws is gigantic so that the Monte Carlo simulation error is negligible. Hint: This is directly related to the following question. Suppose $x$ and $y$ are two independent random variables with respective densities $f(x)$ and $f(y).$ If $E(x)$ is greater than $E(y),$ then is it always true that $P(x>y)$ is greater than $P(y>x),$ or equivalently that $P(x>y)$ is greater than half? If you say this is always true, then give the rationale or proof. If you say it is not always true, then give an example where it is not true and also the rationale for the failure.

How to Submit: These challenges do not need any code writing, they need only some plain text responses possible accompanied by some equations and graphics. You can submit these in a $1-2$ page PDF file.