I want to embark on a project about inverting a Moment-generating function of a probabilitiy distribution. That is given by \begin{equation} M_X(t) = \text{E} \exp(tX) \end{equation} Since I have never done anything like this before, I am searching for some good references for this, especially references with worked examples.
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$\begingroup$ Have you looked at en.wikipedia.org/wiki/Moment_problem yet? $\endgroup$– Tom CopelandCommented Mar 24, 2012 at 23:47
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$\begingroup$ --- Thanks, done so now! But that is more about existence. As I have a moment-generating function calculated from some random variable, existence is not a problem. What I want is the density function of that random variable (which I also do know exists). $\endgroup$– kjetil b halvorsenCommented Mar 25, 2012 at 15:01
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$\begingroup$ Maybe this can be helpful: en.wikipedia.org/wiki/Inverse_Laplace_transform $\endgroup$– Felipe OlmosCommented Mar 25, 2012 at 21:53
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$\begingroup$ The moment problem is the inversion problem from the MGF to the measure or PDF. Try googling the specific moment problem--Hamburger, Stieltjes, Hausdorff--discussed in the wiki. Maybe pitt.edu/~super7/19011-20001/19461.pdf is a good simple overview. $\endgroup$– Tom CopelandCommented Mar 31, 2012 at 2:33
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2 Answers
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The inverse Laplace transform takes place in the complex domain and includes the residue theorem. You can probably do that analytically (as the above forwarded you to Wiki). But usually you can perhaps see how your usual transforms look like and try to invert by converting the result to a commonly used form which is easily transformed back.
For example, if your result is a polynomial, you can separate it to partial fractions on the laplace domain and then the conversion back doesn't need to go through the complex domain and the definition but rather via known formulas.
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Replace $t$ with $it$ in the moment generating function. Then use an inverse Fourier transform to get the distribution.
