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Let $X$ be an integer-valued $rv$ with $\mathrm{pgf}$ $P(s)$ (probability generating functions) and suppose that $\mathrm{mgf}$ $M(s)$ (moment generating functions) exist for $sāˆˆ(-s_0,s_0),s_0>0$. How can we proof that $M(s)$ and $P(s)$ are related??

kjetil b halvorsen
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PsychoMath
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1 Answers1

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(Assuming you mean that $X$'s support is over the nonnegative integers.)

From http://en.wikipedia.org/wiki/Moment_generating_function#Relation_to_other_functions:

The pgf is defined as $G_X(z) = E[z^X]$ whenever the expectation exists.

The mgf is defined as $M_X(t) = E[e^{t X}]$, for at least all reals $t$ with $\lvert t \rvert < s_0$ by assumption.

So then $G_X(e^t) = E[ (e^t) ^ X ] = E[e^{t X}] = M_X(t)$, for any $\lvert t \rvert < s_0$. Because the mgf is assumed to converge, the pgf must as well.

Danica
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  • can you show me how can $mgf$ exist for $sāˆˆ(-s_0,s_0),s_0>0$, this is will be more interested since they are related. thanks ā€“ PsychoMath Aug 04 '13 at 01:41
  • You posed that as an assumption in the original problem. http://en.wikipedia.org/wiki/Moment_generating_function#Examples has some examples; for instance, the Poisson's mgf is $e^{\lambda (e^t - 1)}$. By our result here, the pgf is then $e^{\lambda (t - 1)}$. ā€“ Danica Aug 04 '13 at 02:48