- #1
jimmy1
- 61
- 0
finding the pdf from generating functions??
The generating function of a Poisson distribution is given by
f(z) = Exp[-lamda(1-z)],
where lambda is 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 of f(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 to n cases is similar??)
The generating function of a Poisson distribution is given by
f(z) = Exp[-lamda(1-z)],
where lambda is 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 of f(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 to n cases is similar??)