How to show that for $U∼Unif(0,1)$,
(1) $max$ $Cov(X,Y)=Cov(F^{-1}(U),G^{-1}(U))$ and
(2) $min$ $Cov(X,Y)=Cov(F^{-1}(U),G^{-1}(1-U))$
where $F(a)=P(X\leq\ a)$ and $G(a)=P(Y \leq\ a)$ and $X$ and $Y$ are positive random variables.
Find the maximum and minimum of Cov(X,Y)
1
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probability
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0I believe you omitted saying that $U \sim Unif(0,1).$ A fundamental result, often used in simulation, is that $F_X^{-1}(U)$ has the same distribution as $X$. Sometimes the inverse CDF is called the 'quantile' function. – 2017-01-12
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0Yes, $ U∼Unif(0,1)$. Can you be more specific on this question? – 2017-01-12
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0Maybe see [this Wikipedia page](https://en.wikipedia.org/wiki/Inverse_transform_sampling). – 2017-01-12
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0Thanks. Could you please give more hints on the posted problem? – 2017-01-12