A doubt about expectation of a random variable

Question

Let $X$ be a random variable. Then is it true that, $E(X)< \infty \iff E(|X|)< \infty$.

Answer 1

By definition, $E(X)$ does not exist unless $E(|X|) < \infty$. Thus the integral defining expected value (in the continuous case) is a Lebesgue integral, not an improper Riemann integral.

Answer 2

Yes

Necessarily, we are not allow to say $E(X)<\infty$, or anything else about $E(X)$ unless $X$ is integrable, that is unless $E(|X|) <\infty$.

Conversely, the triangle inequality states $|E(X)| \leq E(|X|) <\infty$.

Answer 3

One can construct a random variable $X$ with $E(X)=-\infty<\infty$, but, of course, $X$ would not be integrable. So the correct version is:

$|E(X)|<\infty\Leftrightarrow E(|X|)<\infty $.

It is easy to show if you decompose $X$ in positive $X^+$ and negative $X^-$ parts (both are positive) and use identities $X=X^+-X^-$ and $|X|=X^++X^-$.

It is possible to define expectation for a random variable, which is not integrable, but in that case the support of it should be bounded either from above or from below. For instance, for any positive random variable expectation is defined.