Finding the pdf from generating functions?

In summary, the conversation discusses using a Poisson distribution and its generating function to find the probability density of f(f(z)), or even f(f(f(...))), with suggestions of using a Fast Fourier Transform or calculating the Fourier transformation of the nth derivative of f(z). However, the process may not be practical for large numbers and further research on Fourier transforms is recommended.
  • #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??)
 
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  • #2
i don't know if it's much help but
if G(w) is Fourier transform of f(z) then
the Fourier transform of the nth derivative of f(z) is equale to (iw)^n multiplied by G(w).
hence u can calculate the Fourier transformation of the nth derivative and continue the calculation from there..
 
  • #3
So for example, if I had the generting function of say, f(f(f(f(z)))), (as defined in the first post), then the 200th derivative of f(f(f(f(z)))), is simply (iw)^200 multiplied by G(w), where G(w) is the Fouirer Transform of f(f(f(f(z))))??

Sorry, I'm just trying to get my head around all this stuff, as I'm kinda new to this stuff and not too sure what I'm doing!
 
  • #4
nope , the Fourier tranformation of nth derivative of f(f(f(f(z)))) is (iw)^n multiplied by g(w).
when u know the Fourier transformation , of a certain function , u can apply the inverse Fourier transformation to obtain the function..
((iw)^200)*g(w) does not equale the nth derivative of f(f(f(f(z)))).
read a bit from this site about the transformation and the inverse transformation
http://en.wikipedia.org/wiki/Fourier_Transform
 

1. What is a generating function?

A generating function is a mathematical tool used in combinatorics and other areas of mathematics to represent a sequence of numbers or other mathematical objects as coefficients of a formal power series. It is often used to simplify calculations and prove identities.

2. How is a generating function related to a probability distribution?

A generating function can be used to derive moments and other properties of a probability distribution. The generating function of a probability distribution is the Laplace transform of the distribution, and the coefficients of the power series expansion of the generating function correspond to the moments of the distribution.

3. How do you find the probability density function (PDF) from a generating function?

To find the PDF from a generating function, you can use the inverse Laplace transform. This involves finding the coefficients of the power series expansion of the generating function and then using them to construct the PDF. The process can be more complex for certain types of generating functions, but the general principle remains the same.

4. Can a generating function be used for any type of distribution?

Yes, a generating function can be used for any type of distribution, as long as the distribution is defined over a discrete set of values. For continuous distributions, a similar tool called the moment-generating function is used.

5. How can generating functions be applied in practical situations?

Generating functions can be applied in a variety of practical situations, such as in probability and statistics, physics, and computer science. They can be used to solve problems involving counting and combinatorics, analyze the behavior of random variables, and model complex systems. They are also useful for deriving closed-form expressions for sums and other mathematical objects.

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