TITLE: Infinite expected value of a random variable
QUESTION [27 upvotes]: How can a positive random variable $X$ which never takes on the value $+\infty$, have expected value $\mathbb{E}[X] = +\infty$?

REPLY [2 votes]: In your comments, you asked how summing up finite things (or products of finite things with other finite things) can give you something infinite. This shouldn't be too counterintuitive; the idea is that you're summing up infinitely many of those finite things. If anything, it's counterintuitive that you can sometimes sum up infinitely many things and get a finite result. 
I do agree, however, that the idea of a game like the St. Petersburg Game seems counterintuitive. Part of it might be because of the word "expectation." In common usage, when we expect something to happen, we think it's more likely to happen than not. But in probability, that's clearly not the case, because we're taking a weighted average of possible outcomes, and the weighted average itself might be an unlikely, or even impossible outcome. For example, when you roll a die, you "expect" the value of the number shown to be 3.5, even though you know that will never happen. Similarly, we can "expect" the outcome of an experiment to be infinite, even though we know it will be finite. That explanation might not fully satisfy your intuition, but it's a start at least.