When $ E(1/X) = 1/E(X)$?

Question

1. Given a random variable $X$ that is nonzero a.e. and $E(X) \neq 0$, $E(X) * E(1/X) = 1$ means $X$ and $1/X$ are uncorrelated.

   Let's rule out the case when $X$ is constant a.e.. Then $X$ and $1/X$ are not independent. Then when are $X$ and $1/X$
   uncorrelated? Or some well known cases or examples in probability and stochastic processes?

2. My question comes from that in Little's law in queueing theory, the arrival's rate is defined as $1/E(X_i)$, where $X_i$ is an interarrival time. Can it be defined as $E(1/X_i)$?

Thanks.

Answer 1

If $X>0$ then $E(1/X)\ge 1/E(X)$ because $1/x$ is concave up. For example if $X$ takes three values, equally likely, $$E(X)E(1/X)=\frac{a+b+c}3\frac{1/a+1/b+1/c}3=\frac{3+\frac ab+\frac ba+\frac ac+\frac ca + \frac bc+\frac cb}9\geq1$$ with equality when $a=b=c$.