If I have three random variables $X, Y$ and $Z$ where $Y$ and $Z$ are independent. Is it true that $P(X,Y,Z)=P(X,Y)P(X,Z)$? What about $P(X\vert Y,Z)=P(X\vert Y)P(X\vert Z)$?
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No. Just consider $X$ independent from both $Y$ and $Z$, then
$$P(X,Y,Z) = P(X)P(Y)P(Z)$$ $$P(X,Y)P(X,Z) = P(X)^2P(Y)P(Z).$$
These are only equal if $X$ is constant.
For the second question, still considering independent $X$,
$$P(X|Y,Z) = P(X)$$ $$P(X|Y)P(X|Z) = P(X)^2,$$
which, as before, are only equal if $X$ is constant.
The inference that you can make from your assumptions is
$$P(X,Y,Z) = P(Y,Z)P(X|Y,Z) = P(Y)P(Z)P(X|Y,Z).$$
filipos
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Hi Filipos, thanks for your answer. But what is it like if X is not independent to both Y and Z? – Daisy Mar 04 '16 at 15:02
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Consider $X = (Y, Z)$. We get $P(X)\delta_{X, (Y, Z)}$ versus $P(X)^2\delta_{X, (Y, Z)}$ in the first questions and $\delta_{X, (Y, Z)}$ versus $P(X)\delta_{X, (Y, Z)}$ in the second one. – filipos Mar 04 '16 at 16:22