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If $(X_1,Y_1), (X_2,Y_2)$ are independent random vectors having the same joint distribution function $F$, then is it correct to say:

  1. $E(X_1)=E(X_2)$ and $E(Y_1)=E(Y_2)$ (the same for variance);
  2. Both, $X_1$ and $Y_1$, are independent of $X_2$ and $Y_2$ .

Please argue the answer.


This question arose when I read the equality in my book:

$2Cov(X_1,Y_1)=E[(X_1-X_2)(Y_1-Y_2)]$.

Thanks in advance.

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marked as duplicate by whuber self-study May 16 at 18:17

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Part (1) appears to be trivial, because you ask whether a distribution function determines its moments: it does by the very definitions of distribution and moment. Therefore I have identified duplicates of your question (2). $\endgroup$ – whuber May 16 at 18:19