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If we know that $X_1$ and $X_2$ are i.i.d random variables, and $Z_1$ and $Z_2$ are also i.i.d random variables, can we say $X_1Z_1$ and $X_2Z_2$ are i.i.d random variables too? suppose that $X_1$ ,$X_2$, $Z_1$ and $Z_2$ are all independent random variables.

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    $\begingroup$ For this edited version (I mean assuming that all the random variables are independent), yes they are. $\endgroup$
    – passerby51
    Apr 15, 2013 at 1:04

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Pick $X_1$, $X_2$ iid, and $Z_2 = X_1$, $Z_1 = X_2$, then $Z_1$, $Z_2$ are also iid. On the other hand $X_1Z_1 = X_1X_2$ and $X_2Z_2 = X_1X_2$ are not iid.

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  • $\begingroup$ Yes you are right. I edited the question. $\endgroup$
    – May
    Apr 15, 2013 at 0:53
  • $\begingroup$ Which should not prevent you from accepting this answer. $\endgroup$
    – Did
    Sep 5, 2013 at 14:36

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