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 $\map {\Pi_{X_1} } s, \map {\Pi_{X_2} } s, \ldots, \map {\Pi_{X_n} } s$.

Then:
 * $\displaystyle \map {\Pi_Z} s = \prod_{j \mathop = 1}^n \map {\Pi_{X_j} } s$

Proof
Proof by induction:

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

whenever $m \le n$

$\map P 1$ is true, as this just says $\map {\Pi_{X_1} } s = \map {\Pi_{X_1} } s$.

Basis for the Induction
$\map P 2$ is the case:
 * $\map {\Pi_{X + Y} } s = \map {\Pi_X} s \, \map {\Pi_Y} s$

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 $\map P j$ is true, where $j \ge 2$, then it logically follows that $\map P {k + 1}$ is true.

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

whenever $m \le k$.

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

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 $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

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