Let X and Y be discrete random variables defined on the same state space $(\Omega, \mathcal F,\mathcal P)$. Assume that for an arbitrary non-stochastic natural number k, the following relation holds:
$P(X \le a) \le P(Y\le ak)$
My question is if we can safely state that :
$E[\frac{Y}{k}]\le E[X]$
Yes. If $X$ and $Y$ are also non-negative this can be proved easily: You don't even have to assume that the distributions are discrete. We have the formula $EU=\int_0^{\infty} P(U>t)dt$ for any non-negative random variable $U$. Since $P(X>t) \geq P(\frac Y k >t)$ we get the inequality by integrating w.r.t. $t$.
You should read about 'stochastic ordering'. The given condition is equivalent to the fact that $Eg(\frac Y k) \leq Eg(X)$ for any non-decreasing $g$.