I have a slightly more complex problem, but I believe the technical nature is captured in the following:
For the $s$-indexed sequence, $X_s$, of iid positive-valued random variables with finite density at $0$, the average of random variable $Y$, with
$Y=\sum_{s=1}^{N}\frac{1}{X_s} , $
diverges because vanishing samples of $X_s$ dominate the sum, leading it to diverge, and thus the reciprocal vanishes.
The average of the reciprocal of $Y$, $Z=1/Y$ is not affected by this and so is still finite. I would like to know how to calculate the average of $Z$ for a given distribution of $X_s$, say exponential. Take $N$ large if it helps.
It is useful to know how to compute inverse distributions and quotient distributions.
I am not so familiar with random variable algebra and found the ill-defined distribution of the intermediate $Y$ variable an impediment to the application of the standard step-wise transformation of random variables.
Thank you in advance for even partial progress towards a solution.
EDIT: mathematica stalls trying to do the expectation for $N=3$. The answer it gives for $N=2$ is $\frac{1}{3\lambda}$, where $\lambda$ is the parameter of the exponential distribution of $X_s$.