Find upper bound for $P(X>Y+15)$

Question

Let $E(X)=E(Y)=75$ and $Var(X)=10$ and $Var(Y)=12$ and $Cov(X,Y)=-3$. Then find upper bound for this values

a) $P(X>Y+15)$

b) $P(Y>X+15)$

I tried to solve this question by calculate $E(X^2) , E(Y^2) , E(XY)$ but i havn't find the upper bound with this datas.

Answer 1

You can find a nontrivial upperbound by noting that $$ P(X>Y+15)\leq P(X-Y>15)\leq P(|X-Y|>15)=P((X-Y)^2>15^2)=\frac{E[(X-Y)^2]}{15^2} $$ You can compute $E[(X-Y)^2]=EX^2+EY^2-2EXY$ from the information given.