Is it ever true that the identity $E(\frac{1}{X^2}) = \frac{1}{E(X^2)}$ holds for a nontrivial random variable X?
I see that it would work if $X$ is a constant $1$ because $\frac{1}{1^2} = \frac{1}{1^2}$, but I assume this is the trivial solution. If I put in something like $N(0,1)$, then it clearly doesn't work.