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Suppose I have two random variables $X$ and $Y$ that are independent. Also suppose that I can sample from $X+Y$ and $Y$. Is it possible to combine those two sampling algorithms to get samples for $X$.

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If you sample from $X+Y$ and $Y$ independently (separately), the answer is no.

For example, suppose $X$ and $Y$ follow the standard normal distribution, then $X+Y$ follows $N(0,2)$. $(X+Y) - Y \sim~ N(0,3)$, instead of expected standard normal distribution. Apart from $(X+Y) - Y$, I cannot find any other way to recreate $X$.

In fact when you sample from $X+Y$ and $Y$ independently, the correlation between $X+Y$ and $Y$ are lost, so it is impossible to recreate $X$ from $X+Y$ and $Y$.

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  • $\begingroup$ The covariance of $Y$ and $X+Y$ is $\text{var}(Y)$, hence they cannot be independent. Ever. $\endgroup$
    – Xi'an
    Oct 9, 2018 at 11:17

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