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

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?

$$z$$ 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.