In a probability generating function, what exactly is the parameter of G(z)?

Question

For instance, given $\DeclareMathOperator{\P}{\mathbb{P}} \DeclareMathOperator{\E}{\mathbb{E}} G(z) = \E z^X$, what exactly is $z$? and also what does the generating function actually give you? Because it states that it is returning the expected value of $z^X$ but how exactly is that helpful?

Answer 1

$z$ is nothing of importance, it's simply a variable. Just as the moment generating function $M_X(t)$ is a function in $t$, the probability function $G_X(z)$ is a function in $z$.

The use of generating functions for a discrete probability distribution is that $G_X(z)$ can be used to obtain the probabilities that define the random variable.

You can retrieve the probability $X=k$ by differentiating $G_X(z)$ a total of $k$ times and using the formula:

$$P(X=k) = \frac{G^{(k)}(0)}{k!}$$

So we simply end up evaluating the generating function at $0$ like with the moment generating function.

It follows if we can show $X$ and $Y$, discrete rv, have the same probability generating function, then they have the same distribution.