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In Statistical Inference by Casella, Berger (version 2), Section 2.6.1, we have the following expression for the Taylor expansion of the mgf:

$$M_X(t)=\sum_{r=0}^\infty \frac{(-1)^r\mu_r't^r}{r!}$$

which is then used for comparison against the uniqueness result by Billingsley mentioned earlier. Where does the alternating term $(-1)^r$ come from, since surely the exponential expansion doesn't contain it (but contains everything else)? It makes sense why we would want this form including the alternating form since Feller (1971), which is where it is from, contains work on Laplace transforms, but I'm not sure why we are able to declare equality here.

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    $\begingroup$ Seems like a typo: the result by Feller in An Introdution to Probability Theory and Its Applications vol II (p 234 of my edition) is about the Laplace transform $\varphi(t) = M_X(-t)$. $\endgroup$
    – Yves
    Jun 30 at 14:04
  • $\begingroup$ Perfect, thanks for the clarification! Didn't know where to find that in Feller (just found a pdf online), but that makes a lot more sense. $\endgroup$ Jun 30 at 17:00

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