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I have been trying to solve a master equation and now finally when I solved it using the method of moment generating function. I don't know how to convert it into a corresponding probability function. The moment generating function $F[x,y,t]$ is of the form of,

$$F[x,y,t] = e^{(k_1 x+k_2 y+\frac{k_3}{\sqrt{xy}}+k_4)t} F[0]$$

Where, $$F[x,y,t]=\sum_{r=0}^{\infty} \sum_{s=0}^{\infty} x^r y^s P[r,s,t]$$ and $P[r,s,t]$ is the probability distribution function that I am looking for and $t$ is time.

What probability generating function does it corresponds to?

Nitin

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    $\begingroup$ What exactly are $x,y,t$? Write how $F$ is defined. $\endgroup$ – zhoraster Mar 29 '16 at 19:44
  • $\begingroup$ Now this is impossible unless $k_3=0$. In the latter case, the answer is easy and independent of $t$ (as $F$ is). $\endgroup$ – zhoraster Mar 31 '16 at 19:33
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I would comment on your question, but I haven't enough rep to do so. I am not entirely sure there exists a closed form PDF for this MGF. It is not particularly difficult to calculate the result analytically via an FFT.

Recall that the MGF is the expected value of the exponent of the product of t and your random variable and that the characteristic function is the expected value of the exponent of the product of ti and your random variable. You can thus use the transformation map of t -> ti to move between them.

Recalling from your measure theory, then, that this resulting characteristic function is the inverse Fourier transform of the PDF, you can then quickly recover (to within the Gibbs artifacts) the PDF via a forward FFT.

If you are not satisfied with numerical solutions then you can attempt the various methods of solving your moment problem analytically.

In general, I highly recommend working in the characteristic function space rather than with MGFs since characteristics are guaranteed to exist while MGFs are not--due to the L1 integrability guarantee around the modulus.

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