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Reading http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter10.pdf pages 368-370. it states "if we delete the hypothesis that have finite range in the above theorem, then the conclusion is no longer necessarily true".

So my question is

Do you know a (simple?) example of two discrete random variables with finite moments equal for each order but different distributions?

user340605
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    See 11.7 and 11.8 in Counterexamples in Probability: Third Edition, by Jordan M. Stoyanov. (This is called the moment problem and you should check keywords such as "Carleman's condition".) – Did May 18 '16 at 20:30
  • Relevant: http://math.stackexchange.com/questions/1057663/does-mgf-determine-cdf – Clement C. May 18 '16 at 20:31
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    For example, for every integer $q\geqslant2$ and real number $|u|\leqslant1$, the formulas $$P(X_u=q^n)=e^{-q}q^n\left(\frac1{n!}+\frac{(-1)^nu}{(q-1)(q^2-1)\cdots(q^n-1)}\right)$$ define a family of random variables $(X_u)$ on the set ${q^n\mid n\geqslant0}$ with the same moments for every $u$. The distribution of $X_0$ is log-Poisson, meaning that $X_0=q^Y$ with $Y$ some Poisson random variable. – Did May 18 '16 at 20:38

1 Answers1

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Not quite what you're asking for, but a standard example of continuous random variables with the same moments yet distinct distributions is

$f_1(x) = e^{- (\log{x})^2/ 2} \, / \,(x\sqrt{2\pi})$

$f_2(x) = f_1(x)(1 + \sin(2 \pi \log{x})/2)$

where $ x > 0$.

Joe C.
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