0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Perhaps you've made a mistake, since $E(Y/X_1)$ is a number and not a random variable. $\endgroup$ – Marc Apr 3 '15 at 7:45
  • $\begingroup$ Since $X_1$ is a random variable , then $E[Y/X_1]$ is a random variable depending on $X_1$. $\endgroup$ – Nizar Apr 3 '15 at 7:52
  • $\begingroup$ Oh it is supposed to be a conditioning line. Most literature use a straight line $E(Y | X_1)$, to avoid confusion with division;) $\endgroup$ – Marc Apr 3 '15 at 7:53
0
$\begingroup$

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?"

$\endgroup$
  • $\begingroup$ Your discussion is right, thank you, however I edited my post , making the result more general , and at the same time more compatible with my investigation. $\endgroup$ – Nizar Apr 3 '15 at 7:57
  • $\begingroup$ 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?" $\endgroup$ – zoli Apr 3 '15 at 8:00
  • $\begingroup$ I am not sure, but I think $X_1$ and $X_1$ are dependent $\endgroup$ – Nizar Apr 3 '15 at 8:09
  • $\begingroup$ $P(X_1=a,X_1=b)=0 \text{ or } P(X_1=a)$ 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. $\endgroup$ – zoli Apr 3 '15 at 8:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.