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$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $?

My feeling is that this is not necessarily true. But cannot come up with an example.

Can someone provide a counterexample or give a proof for this statement?

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  • $\begingroup$ where are the $X_i$ defined? $\endgroup$
    – mathemagician
    Commented Oct 20, 2014 at 2:03
  • $\begingroup$ The statement is asking for any sequence of $X_n$ so they can be defined on any probability space. $\endgroup$
    – user184389
    Commented Oct 20, 2014 at 2:04

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If $X_n$ are defined on the same probability space then the theorem is true and a proof is demonstrated in this question.

If $X_n$ are defined on different probability spaces then clearly $+$ must be defined on them for the question to have any meaning.

If $X$ is defined as the number of Farmer Brown's sheep in the paddock at 10am and $Y$ as the position of an electron in the CERN supercollider at the same time (for some frame of reference) then clearly it is not meaningful to speak of adding them.

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  • $\begingroup$ Dude, it's countably infinite sum, not finite sum $\endgroup$
    – user184389
    Commented Oct 20, 2014 at 4:39
  • $\begingroup$ The link you gave is not relevant to this question. It states the result for finite sum of r.v.s $\endgroup$
    – user184389
    Commented Oct 20, 2014 at 4:39
  • $\begingroup$ So yea your answer is not correct $\endgroup$
    – user184389
    Commented Oct 20, 2014 at 4:59
  • $\begingroup$ Actually it states for 2 random variables, but the sum of 2 random variables is a random variable so it can be applied over and over: that is all you need for any countable set of random variables. The sum may $\to \pm\infty$ but that doesn't matter. $\endgroup$
    – Dale M
    Commented Oct 20, 2014 at 5:05
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    $\begingroup$ "it can be applied over and over: that is all you need for any countable set of random variables." No, one needs something more, which is called Fubini theorem. $\endgroup$
    – Did
    Commented Oct 20, 2014 at 23:03

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