7

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?

Community
  • 1
Silent
  • 6,668

3 Answers3

5

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.

5xum
  • 126,227
  • 6
  • 135
  • 211
  • 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
4

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.

M.X
  • 723
  • 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
  • This seems to make intuitive sense too, because most of an Exponential distribution is infinite - as x @>>> \infty, so does E(X). – ryanwebjackson Mar 29 '22 at 03:46
2

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!

Community
  • 1
kjetil b halvorsen
  • 7,004