PGF of Sum of Independent Discrete Random Variables/General Result

Theorem
Let:
 * $Z = X_1 + X_2 + \cdots + X_n$

where each of $X_1, X_2, \ldots, X_n$ are independent discrete random variables with PGFs $\Pi_{X_1} \left({s}\right), \Pi_{X_2} \left({s}\right), \ldots, \Pi_{X_n} \left({s}\right)$.

Then:
 * $\displaystyle \Pi_Z \left({s}\right) = \prod_{j \mathop = 1}^n \Pi_{X_j} \left({s}\right)$

Proof
Proof by induction:

For all $n \in \N_{> 0}$, let $P \left({n}\right)$ be the proposition:
 * $\displaystyle \Pi_Z \left({s}\right) = \prod_{j \mathop = 1}^m \Pi_{X_j} \left({s}\right)$

whenever $m \le n$

$P \left({1}\right)$ is true, as this just says $\Pi_{X_1} \left({s}\right) = \Pi_{X_1} \left({s}\right)$.

Basis for the Induction
$P \left({2}\right)$ is the case:
 * $\Pi_{X + Y} \left({s}\right) = \Pi_X \left({s}\right) \Pi_Y \left({s}\right)$

which is proved in PGF of Sum of Independent Discrete Random Variables.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({j}\right)$ is true, where $j \ge 2$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:
 * $\displaystyle \Pi_Z \left({s}\right) = \prod_{j \mathop = 1}^m \Pi_{X_j} \left({s}\right)$

whenever $m \le k$.

Then we need to show:
 * $\displaystyle \Pi_Z \left({s}\right) = \prod_{j \mathop = 1}^m \Pi_{X_j} \left({s}\right)$

whenever $m \le {k + 1}$.

Induction Step
This is our induction step:

Let $Z = X_1 + X_2 + \cdots + X_k + X_{k + 1}$

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \N: \displaystyle \Pi_Z \left({s}\right) = \prod_{j \mathop = 1}^n \Pi_{X_j} \left({s}\right)$