I have trouble understanding what exact relationship a probability generating function describes. On wikipedia it says that:
"In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable"
but in the formulas i dont understand what the parameter z in G(z) represents or why we take the expected value of raising it to a random variable X like so: G(z) = E(z^X) = [Sum]
how does z depend on the distribution of X and how does one actually calculate probabilities of X taking a specific value once the PGF has been constructed with parameter z.
Im studying PGF:s in relation to stochastic processes and how they are used to express the rows of infinite transition matrices. I am so confused. Sorry I couldn't figure out how to represent the formula in latex here