Let $X$ and $Y$ be independent exponential random variables. If $E(X)=1$ and $E(Y)=\dfrac{1}{2}$ then find $P(X>2Y|X>Y).$
Now using memory less property can I claim that $P(X>Y+Y|X>Y)=P(X>Y)?$ I never saw memory less property in two dimensional random variable case.
If it is true then $P(X>Y)=\int_{0}^{\infty}\int_{0}^{x}e^{-x}(2e^{-2y})dydx=e^{-x}(1-e^{-2x})=\int_{0}^{\infty}(e^{-x}-e^{-3x})dx=1-(\frac{1}{3})=\frac{2}{3}$