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The generating function of a Poisson distribution is given by

f(z) = Exp[-lamda(1-z)],

wherelambdais the mean and variance of the poisson process.

Now suppose I have an embedded Poisson process, that is,f(f(z)), the new generating function would then be

f(f(z)) = Exp[-lambda(1-Exp[-lambda(1-z)])].

Now the question is how to I get the probability density off(f(z))??

I know I could differentiate and put z=0, but the problem is that I need values for fairly large numbers, that is P(X=100).., hence is not really practical to get the 100th derivative of the generating function.

I have been told I could use a Fast Fourier Transform, but after googling FFT and probability densities I couldn't really find anything comprehensiable for a layperson like me!

So any suggestions as to how I would get the density of f(f(z)), or even some sort of approximation, so I can get an idea of what the distribution looks like? Any help would be great!!

(I actually need to know what the distribution of f(f(f(......))), looks like, but I presume if I can work out f(f(z)), then extrapolating toncases is similar??)

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# Finding the pdf from generating functions?

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