Let $X$ be an exponentially distributed random variable with mean $\lambda$. What is $E[X^{-1}]$?
The only online resource I've been able to locate that speaks to the issue directly is this Wikipedia stub, which states "Note that the expected value of this random variable does not exist."
I assume this may stem from the fact that $X$ isn't strictly positive, causing a division by zero and a $(-\infty) + \infty$ to appear somewhere in the expectation value integral?
If so, what is the significance of $E[X^{-1}]$ "not existing" from a statistical standpoint? If one generates $n$ exponentially-distributed data and compute the mean of their reciprocals, what does one expect to get as $n \to \infty$? I take it the limit never converges to anything, even $\infty$?