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How to extract distribution of r.v. $X$ from probability generating function of $X$?

The p.g.f. of $X$ is:

$$G_X(t)=\sum_{k=1}^{\infty}t^kP_X(k).$$

That is, if $G_X(t)$ is known, then (how) can one "figure out" $P_X(k)$?

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    $\begingroup$ Since the values are right in front of you--the probabilities are, by definition, $P_X(k)$ for each $k$--please explain what you mean by "extract." That might depend on how $G_X$ is "known," so perhaps you could say a little more about that, too. $\endgroup$
    – whuber
    Commented Dec 10, 2015 at 20:21
  • $\begingroup$ The values are, but what about the domain? Since the p.g.f. is usually given in e.g. a piece-wise definition. Is the domain the same as the integers that $k$ gets? $\endgroup$
    – mavavilj
    Commented Dec 10, 2015 at 20:23
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    $\begingroup$ That, too, is explicitly in front of you in the summation notation. $\endgroup$
    – whuber
    Commented Dec 10, 2015 at 20:23

1 Answer 1

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You can apply the usual extraction procedure from Taylor series:

$$P_X(k)=\frac{1}{k!}\left.\frac{d^k}{dt^k}G_X(t)\right|_{t=0}$$

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    $\begingroup$ This is being automatically flagged as 'low quality', probably because it is so short. Can you add some text to explicate this? $\endgroup$ Commented Dec 10, 2015 at 21:24
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    $\begingroup$ Maybe there should be a moderator override button to remove 'low quality' designation from high quality short answers which are flagged due to being too short. $\endgroup$ Commented Dec 10, 2015 at 21:31
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    $\begingroup$ @Mark What we (and other active people such as Gung) currently do when automatic flags (or even user-driven flags) look inappropriate is to weigh in with an appropriate comment intended to dissuade people from acting automatically on the flag. That tends to work. $\endgroup$
    – whuber
    Commented Dec 11, 2015 at 14:37

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