[Math] Calculate probability density function from moment generating function

probabilitystatistics

I have moment generating function

$$ M_z (t) = \dfrac {\lambda^2} {(\lambda-at) (\lambda-t)}, $$ and I'm trying to calculate the PDF from this function. I feel like some kind of inverse transform is required here, but I can't figure it out.

Best Answer

Inverting generating functions is not always easy. It does help to state any constraints on the parameters (which are left unstated here by the OP). Anyway, the approach is: First, convert your mgf into a characteristic function (i.e. replace t -> I t):

enter image description here

Next, invert the characteristic function to yield the pdf (using Mathematica here), using an inverse fourier transform. To get a 'neat' result, it appears we need the parameter constraints {a > 0, $\lambda$ > 0}, so I have added these as assumptions:

enter image description here

HeavisideTheta[x] is a function that is 0 for x<0, and 1 for x > 0, so our pdf is:

enter image description here

Finally, a quick check to make sure all is OK: generate the mgf from the pdf to see it is the same as that we started with (here using the Expect function in mathStatica/Mathematica):

enter image description here

... and all is OK :)

Finally, here is a quick plot of the derived pdf, as parameter $\lambda$ varies:

PlotDensity[f /. {a -> 2, \[Lambda] -> {2,3,4}}]

enter image description here

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