A question in expectation and markov inequality

Question

I've 2 random variables X,Y.

Assume $ X<= Y$.

Prove that for each $\epsilon$ > 0 there exists $\delta$ > 0 so that if:

$|E[Y] - E[X]| < \delta $

Then:

$P(Y>= X + \epsilon) < \epsilon$

Could use some help.

Answer 1

By Markov's Inequality,$$P(Y\geq X+\epsilon)=P(|Y-X|\geq\epsilon)\leq \frac{E[Y-X]}{\epsilon}<\frac{\delta}{\epsilon}.$$ Choose $\delta<\epsilon^2$.

Answer 2

Choose $\delta < \epsilon^2$,

\begin{align} P(Y \geq X+ \epsilon) &=P(Y-X \geq \epsilon) \\ &\leq \frac{E[Y-X]}{\epsilon} \\ &< \frac{\epsilon^2}{\epsilon} \\&=\epsilon \end{align}

In particular, you can choose $\delta = \frac{\epsilon^2}{2}$