Suppose that I have a set of known measurable scalar-valued functions $f_{1},\ldots,f_{K}$.

Associated with these functions, I also have a set of known real numbers $a_{1},\ldots,a_{K}$.

Under what conditions on $f_{1},\ldots,f_{K}$ and $a_{1},\ldots,a_{K}$ does there exist a random variable $X$ such that $E[f_{k}(X)] = a_{k}$ for all $k = 1,\ldots,K$?

Can you provide a reference to the literature where such a result is established?

If the general case is too difficult, suppose that $f_{k}(x) = x^{k}$ for all $k$.

As an example, suppose that $K = 2$, $f_{1}(x) = x$, and $f_{2}(x) = x^{2}$.

Then if $a_{1} = 0$ and $a_{2} = 1$, I know that such an $X$ exists---for example take $X \sim N(0,1)$.

On the other hand, if $a_{1} = 2$ and $a_{2} = 1$, then no such $X$ could exist, for if it did we would have $Var(X) = 1 - 2^{2} = -3 < 0$.


This is the Moment problem. The wikipedia article contains brief discussion on results under a variety of assumptions, along with a few references.

| cite | improve this answer | |
  • $\begingroup$ Wonderful! Do you know of any good modern textbook treatments? The latest one on the wikipedia page is 1977. $\endgroup$ – evencoil Jun 12 '16 at 23:37
  • $\begingroup$ @evencoil Sorry, I'm not up to date on the moment problem. You could try a citation search on the referenced articles $\endgroup$ – grand_chat Jun 13 '16 at 3:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.