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$x_1, x_2, \dots, x_n, \dots$ - independent random variables.

Is it true that $$ \sum_{i = 1}^{\infty}Ex_i = E(\sum_{i = 1}^{\infty} x_i) $$ ?

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Here it says:

It can be shown that linearity of expectation also holds for countably infinite summations in certain cases. For example, it holds that:

$E\left[\sum\limits_{i=1}^{\infty}X_i\right]=\sum\limits_{i=1}^{\infty}E[X_i]$

if $\sum\limits_{i=1}^{\infty}E[|X_i|]$ converges.

Also This Book says the same thing.

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  • 1
    $\begingroup$ Thanks, but is this condition important? $\endgroup$ – user1761982 Nov 21 '13 at 17:09
  • $\begingroup$ There are cases where your desired equality does not hold. E.g. flip a fair coin repeatedly. Let $T$ be the number of flips until the first head. Let $X_i$ be $-2^i$ if $i<T$, let $X_i=2^i$ if $i=T$, let $X_i=0$ if $i>T$. Then $E[X_i] = 0$, but $\sum_i X_i = 1$ with probability 1. For two more examples, see algnotes.info/on/background/stopping-times/walds. $\endgroup$ – Neal Young Mar 21 '18 at 14:40

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