In a probability generating function, what exactly is the parameter for 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/what does it mean?

Answer 1

The variable $z$ is just a dummy variable, as mentioned in the comments. Some textbooks also use $\mathbb{E}\left [ s^{k} \right ]$.

The probability generating function has many important properties. For a random variable $X$, $$\mathbb{E}\left [ X \right ] = G'\left ( 1 \right )$$ and $$\text{Var}(X)= G''(1)+G'(1)-\left [ G'(1) \right ]^2.$$ Another interesting property is that it is related to moment generating functions from the fact that $$G_{X}(e^{t})=M_{X}(t).$$ Probability generating functions are important in stochastic processes like branching processes and extinction probabilities.