Given two independent random variables $X_1$, $X_2$ and $X_3$, then $Y=F(X_1, X_2, X_3)$ is a random variable depending on $X_1$, $X_2$ and $X_3$. Would some one help me to detect whether the two random variables $E[Y|X_1] $ and $E[Y|X_1,X_2]$ are independent or not. Thanks in advance.
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So your question can be reformulated: "$X_1$ and $X_2$ are independent. Are $E[F(X_1,X_2)|X1]$ and $F(X_1,X_2)$ also independent?"
If for instance $F(X_1,X_2)=X_1$ then the question is: "Are $X_1$ and $X_1$ independent?"