probability generating function

[elmarsur]

1. The problem statement, all variables and given/known data

A random variable X has the generating function
f(z) = 1 / (2-z)^2
Find E(X) and Var(X).


2. Relevant equations



3. The attempt at a solution

Would anyone explain in simpler terms the notion of the generating function, such that I may be able to solve problems? All I have found were proofs, but nothing of practical use.

Thank you very much!

[elmarsur]

Thank you very much, LC!

I imagine that there isn't any significant difference between MGF and PGF (probability), the latter being a special application of the first?!
If so, I still have a couple of questions:
1) Do the formulae apply to all sorts of probability distributions/densities?
2) How do I calculate the variance (formula-wise) for these generating functions (which I understand are not really functions but series of terms)?

Thank you very much in advance.

[LCKurtz]

Thank you very much, LC!

I imagine that there isn't any significant difference between MGF and PGF (probability), the latter being a special application of the first?!
If so, I still have a couple of questions:
1) Do the formulae apply to all sorts of probability distributions/densities?
2) How do I calculate the variance (formula-wise) for these generating functions (which I understand are not really functions but series of terms)?

Thank you very much in advance.

I posted that reply quickly as I was about to leave for a movie and hadn't noticed you were asking about a probability generating function instead of moment generating function. I deleted the post but apparently you saw it before I deleted it. PGF's are defined for non-negative integer valued random variables. For a PGF PX(z),

E(X) = P'X(1) and
Var(X) = P''X(1) + P'X(1) - (P'X(1))2

[elmarsur]

Thank you very much, LC!
I hope I find you around again when I cry for help.