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Suppose $X$ has a cdf $F$. Let the cdf of $Y:=-X$ be $G$.

Now change these random variables to be $\tilde{X} = F(X^-)$ and $\tilde{Y} = G(Y^-)$. Note $f(x_0^-)$ means $\lim_{x \to x_0^-} f(x)$.

I was wondering what relation is between $\tilde{X}$ and $\tilde{Y}$? I guess $\tilde{X} + \tilde{Y} = 1$? But I am not sure.

More generally what if $Y:= aX+b$ with $a < 0$?

Thanks and regards!

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