Suppose $X,Y$ are independent nonnegative random variables and $\delta>0$ is a constant. I am interested in a general upper bound on $$ E\left(\left.P\left(X>\frac{Y}{Y+\delta}\right| Y\right)\right). $$ Using the conditional Markov inequality one would get $$ E(X)E\left(\frac{Y+\delta}{Y}\right), $$ which is problematic if the expectation of the reciprocal of $Y$ is not well defined. Does anyone have a better idea?

Thank you in advance!

  • $\begingroup$ I cannot figure out how to get from $$ E\left(\left.P\left(X>\frac{Y}{Y+\delta}\right| Y\right)\right) $$ to $$ P\left(X>\frac{Y+\delta}{Y}\right). $$ The fraction seems to be inverted? $\endgroup$ – Marc Jun 20 '17 at 13:36
  • $\begingroup$ Right, what I meant is that the conditional aspect is misleading, you are simply after an upper bound of $$P\left(X>\frac{Y}{Y+\delta}\right)=P\left(X\cdot(Y+\delta)>Y\right)=P\left((X-1)\cdot Y+\delta\cdot X>0\right)$$ $\endgroup$ – Did Jun 20 '17 at 16:57

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