# Conditional Expectation as a random variable of independent rendom variables

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.

• Perhaps you've made a mistake, since $E(Y/X_1)$ is a number and not a random variable. – Marc Apr 3 '15 at 7:45
• Since $X_1$ is a random variable , then $E[Y/X_1]$ is a random variable depending on $X_1$. – Nizar Apr 3 '15 at 7:52
• Oh it is supposed to be a conditioning line. Most literature use a straight line $E(Y | X_1)$, to avoid confusion with division;) – Marc Apr 3 '15 at 7:53

$$E[Y|X_1,X_2]=Y=F(X_1,X_2).$$
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?"
• My answer does not really have to be changed: If $Y=X_1$ then the absurd question is the same: "Are $X_1$ and $X_1$ independent?" – zoli Apr 3 '15 at 8:00
• I am not sure, but I think $X_1$ and $X_1$ are dependent – Nizar Apr 3 '15 at 8:09
• $P(X_1=a,X_1=b)=0 \text{ or } P(X_1=a)$ if $a=b$. But not $P(X_1=a)P(X_1=b)$ except if $P(X_1=a)=P(X_1=b)=1$ or $0$. Or you are teazing me. – zoli Apr 3 '15 at 8:29