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Is it possible that two real random variables $X_1,X_2$ are defined in the same probability space $(Ω,F,P)$ and have the same distribution and be independent ?

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closed as unclear what you're asking by Did, erfink, Shailesh, Daniel W. Farlow, Leucippus Jul 23 '17 at 1:30

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  • $\begingroup$ please provide reasoning for downvoting . $\endgroup$ – liaguridio Jul 22 '17 at 16:31
  • $\begingroup$ Why are you duplicating existing questions/answers? $\endgroup$ – Did Jul 22 '17 at 16:32
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I throw a penny and a dime. $X_1$ is the event that the penny is $H$, $X_2$ is the event that the dime is $H$.

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  • $\begingroup$ Those are discrete random variables. $\endgroup$ – Furrane Jul 22 '17 at 16:15
  • $\begingroup$ Note: events are not random variables $\endgroup$ – Yujie Zha Jul 22 '17 at 16:17
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    $\begingroup$ @liaguridio "If the random variables are independent their sigma algebras are independent" Indeed, actually this is the definition of two random variables being independent. And? $\endgroup$ – Did Jul 22 '17 at 16:32
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    $\begingroup$ I suggest: work with my example (which, of course, is exactly the same example as the one provided by @Did in your link). Compute the sigma algebras generated by my variables. Convince yourself that all is consistent here. $\endgroup$ – lulu Jul 22 '17 at 16:34
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    $\begingroup$ At this point, I think you are asking questions more relevant to your link than to this post. In my discrete example notions of "surely" and "almost surely" are not relevant. I suggest you post your questions on that post rather than here. $\endgroup$ – lulu Jul 22 '17 at 16:37

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