# How can expected value be infinite?

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?

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.

• 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? – Silent May 27 '14 at 10:09
• That is correct. – 5xum May 27 '14 at 10:50

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.

• 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! – kjetil b halvorsen May 27 '14 at 12:55

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!