The discussion revolves around the concept of probability generating functions (PGFs) in the context of a random variable X, specifically focusing on the generating function f(z) = 1 / (2-z)^2. Participants are exploring how to derive expected value E(X) and variance Var(X) from this function.
The discussion is active, with participants sharing insights and clarifications. Some guidance has been provided regarding the definitions and relationships between PGFs and MGFs, as well as the formulas for expected value and variance. However, there is no explicit consensus on the broader applicability of these concepts across different distributions.
Participants are navigating the complexities of generating functions, with some expressing confusion over the terminology and the nature of these functions as series of terms rather than traditional functions. There is an acknowledgment of the specific context of non-negative integer valued random variables in relation to PGFs.
elmarsur said: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.