Examples of the case when expectation is infinite or doesn't exist.

Question

In Jacod&Protter's book, Probability Essentials. In page 52, one statement is we can have $E\{X\}=\infty$, even when $X$ is never equal to $+\infty$. Can anybody give an example here?

Also, let $X,Y\in L^1$, $XY$ need not be in $L^1$ in general. I fail to find an example to show this.

For the first, consider the discrete RV $X$ supported on $\mathbb{N}$ which takes the value $n$ with probability $6/(\pi n )^2.$ For the second, try to think of an $L^1$ function which is not $L^2.$ — stochasticboy321, Mar 27 '18 at 19:29
Consider a game where you flip a coin repeatedly until it comes up heads for the first time. Count the number of tails you flipped and call it $k$. You win $2^k$ dollars. Try calculating the expected amount of money that you'll win in this game each time you play. — JMoravitz, Mar 27 '18 at 19:31
When it says X is never equal to infinity. Is it X is almost surely finite? That would help you understand and find examples I think — Frostic, Mar 27 '18 at 19:38
For the last, do you really mean $X,Y$ as random variables, or do you mean functions $f,g$? — Clement C., Mar 27 '18 at 21:03
@YHH https://math.stackexchange.com/questions/116237/the-product-of-integrable-random-variables-need-not-be-integrable — Clement C., Mar 27 '18 at 21:07
Note that you will never find a counterexample if $X,Y$ are independent. (Can you see why?) — Clement C., Mar 27 '18 at 21:08
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Clement C. · Accepted Answer · 2018-03-27T21:12:57.917

Continuous: consider the "one-sided" Cauchy distribution with density function $$ f(x) = \frac{2}{\pi}\frac{1}{1+x^2}\mathbb{1}_{[0,\infty)} $$ (i.e., supported on $[0,\infty)$). Then we have $$ \int_{[0,\infty)} xf(x)dx = \infty\,. $$

Discrete: Consider the probability distribution over $\mathbb{N}$ with $p(n) = \frac{6}{\pi^2}\frac{1}{(n+1)^2}$ for $n\geq 0$. (This is the discrete equivalent of the above). Note that $$\sum_{n=0}^\infty np(n) = \infty\,.$$

As for the last question, note that you will never find a counterexample if you choose $X,Y$ to be independent. (Can you see why?) Now, using either Cauchy-Schwarz or the AM-GM inequality, one can also see that at least one of $X,Y$ needs to not be in $L^2$ (otherwise $XY$ will be integrable).

Based on these two observations, choose the simplest : $X=Y$ with $\mathbb{E}[|X|] < \infty$ but $\mathbb{E}[X^2]= \infty$. Examples abund, but take e.g. a discrete random variable $X$ on $\mathbb{N}$ with $\mathbb{P}\{ X = n\} = \frac{C}{(n+1)^3}$, for $n\geq 0$ (and the right normalizing constant $C>0$).

Examples of the case when expectation is infinite or doesn't exist.

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