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My book as well as Wikipedia gives this definition of expected value:

$\mathbb E(X)=\sum _x xf(x).$ But, $\mathbb E(X)$ is said to exist if and only if that equation is absolute convergent.

But, I see that many places do not follw this def., e.g. here $${\mathbb E} X = \sum_{n=1}^\infty 2^{-n} \cdot 2^n = \sum_{n=1}^\infty 1 = \infty,$$ though we can see that absolute convergence test is failed.

So, can expected value be infinity or negative infinity?

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The expected value of your example does not exist. However, since the series that would define the expected value as a generalized limit of $\infty$, we sometimes (sloppily) say that the expected value is $\infty$. As long as you know what you are talking about, that should not be too big of a problem.

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  • $\begingroup$ Ok, so it is the matter of interpretation, right? And if I stick with bookish definition, I am never going to see $\infty$ or $-\infty$ as expected value, right? $\endgroup$ – Silent May 27 '14 at 10:09
  • $\begingroup$ That is correct. $\endgroup$ – 5xum May 27 '14 at 10:50
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Yes. It can be. Here is an example that I faced in one of my works.

Assume $X$ to be an Exponential distribution ($f_X(x)=e^{-x}$) and $Y=\frac{1}{X}$. For this case, $\mathbb{E}(Y)=\infty$. Indeed, writing the expectation as integral: $$ \mathbb{E}(Y) = \int_0^\infty \frac{1}{x} \mathrm{e}^{-x} \mathrm{d} x $$ you see that the integral diverges at the lower bound. Thus, while it is natural to expect $\mathbb{E}\left(Y=\frac{1}{X}\right) > 0$, the expectation is infinite.

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  • $\begingroup$ In fact, for a non-negative random variable $X$ with a density funftion $f$ which is continuous at zero and with $f(0) > 0$, the expectation of $1/X$ is infinite! Thata is a nice exercise! $\endgroup$ – kjetil b halvorsen May 27 '14 at 12:55
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I just answered a similar question on SE stats, CrossValidated. Rather than copy that answer over here, I give the link: https://stats.stackexchange.com/questions/94402/what-is-the-difference-between-finite-and-infinite-variance

That answer is much longer and more detailed than the ones above here!

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