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What is the most general distribution for which the expected value of the multiplicative inverse equals the multiplicative inverse of the expected value?

Motivation: I'm into modelling dynamics on graphs and I found a problem which is easily solvable in cases where the degree distribution of the vertices is a distribution where $E[1/k] = 1/E[k]$. ($k_i$ is the degree of the $i$th vertex) From this solution I may gain an insight into how to unify multiple models.

So particularly I'm looking for a distribution which consists of non-negative, finite integers. But I'm also interested in continuous solutions. Distributions where $E[1/k^n]=1/E[k^n]$ may also help unifying the models.

What I do know so far, that $k_i=1$ is a particular solution. In the continuous case every function where $f(x)=f(1/x)$ and $E[x]=1$ is a solution. I know what momentum generating functions are and they seem like a good direction to try in, but I failed so far.

What is the most general form of this distribution? Does it have a name? It sounds like something trivial, like a "famous" distribution, but I can't find it.

bacsi bela
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  • similarly $k_i=c$ for any constant $c$ is also a solution. So any regular graph. And that is it. – Henry Jul 12 '21 at 22:01

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By Jensen's inequality applied to the convex function $f(x) = x^{-1}$ on $(0,\infty)$,

$$ \frac{1}{\mathbb{E}[X]} < \mathbb{E} \left[ \frac{1}{X} \right] $$

for any non-constant nonnegative random variable $X$. (See also this for a discussion of the equality cases of Jensen's inequality.) Thus, the constant random variable is the only such random variable.

eepperly16
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