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$X_1$ and $X_2$ are independent. $Y_1|X_1\sim\mathrm{Ber}\left(X_1\right)$, $Y_2|X_2\sim\mathrm{Ber}\left(X_2\right)$. Are $Y_1$ and $Y_2$ necessarily independent? (Assume $\mathrm{P}\left(0, $\mathrm{P}\left(0)

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No. Let $X_1,X_2$ be independent uniform (0,1) random variables, and then define $Y_1=1_{(X_1\geq X_2)}\,\mbox{ and }\, Y_2=1_{(X_2> X_1)}.$

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    @did Thanks for the kind words.2012-08-27