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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 $Var(X) = 1 - 2^{2} = -3 < 0$.

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

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  • $\begingroup$ 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

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