TITLE: Expectation of inverse of sum of random variables
QUESTION [13 upvotes]: Let $X_i$'s ($i=1,..,n$) be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$.
Is there a method that can be used to compute $\mathbb{E}[1/(X_1+...+X_n)]$?

REPLY [6 votes]: As others have commented, you need more information.  However, you do have an estimate
(using convexity): assuming the random variables are positive,
$$E\left[ \frac{1}{X_1 + \ldots + X_n} \right] \ge \frac{1}{E[X_1 + \ldots + X_n]} = \frac{1}{n\mu}$$