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I have a simple question about probability generating functions. If $X$ and $Y$ are two discrete random variables (say, taking values in the set of non-negative integers), and if $E(t^X) = E(t^Y)$ for all $t \in (0,1)$, is it true that $X$ and $Y$ have the same distribution? Note that, I am not assuming anything for $t<0$ or $t>1$ (the cases $t=0$ and $t=1$ are obviously trivial).

Any help will be highly appreciated.

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  • $\begingroup$ do you mean E($e^{tx}$)? $\endgroup$
    – Rivaldo
    Commented Apr 27, 2018 at 21:26

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Each of $Et^X$ and $Et^Y$ is a power series with radius of convergence at least 1, and they agree as functions on the interval $[0,1)$. It's an easy result in complex analysis that their coefficients are equal. But the coefficient of $t^k$ in $Et^X$ is $P(X=k)$, and so on...

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