Conditional Expectation as a random variable of independent rendom variables

Question

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.

Answer 1

You have here

$$E[Y|X_1,X_2]=Y=F(X_1,X_2).$$

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?"