Properties of Probability Generating Function

Theorem
Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$.

Then $\Pi_X \left({s}\right)$ has the following properties:

PGF defines PMF
The coefficient of $s^x$ in $\Pi_X \left({s}\right)$ is $p_X \left({x}\right)$, where $p_X$ is the probability mass function of $X$.

PGF determines Distribution
The probability generating function uniquely determines a discrete probability distribution, and vice versa.

That is:
 * $\Pi_X \left({s}\right) = \Pi_Y \left({s}\right) \iff \forall k \in \N: p_X \left({k}\right) = p_Y \left({k}\right)$

PGF of 0

 * $\Pi_X \left({0}\right) = p_X \left({0}\right)$

PGF of 1

 * $\Pi_X \left({1}\right) = 1$

Proof that PGF defines PMF
It is clear that:
 * $\forall k \in \N: p_X \left({k}\right) = p_Y \left({k}\right) \implies \Pi_X \left({s}\right) = \Pi_Y \left({s}\right)$

from the method of construction of the probability generating function.

So, suppose that $\Pi_X \left({s}\right) = \Pi_Y \left({s}\right)$.

By definition of probability generating function:

If $\Pi_X \left({s}\right) = \Pi_Y \left({s}\right)$ then it is clear that these two power series have identical coefficients.

Hence the result.

Proof that PGF determines Distribution
We note that the coefficient of $s^x$ in the PGF is $p_X \left({x}\right)$ for each $x$.

So the probability mass function determines the PGF uniquely, and vice versa.