Multiply PGFs only when the variables are independent.
Notes and key methods
If X and Y are independent nonnegative integer variables, \(G_{X+Y}(t)=G_X(t)G_Y(t)\). For m independent copies of X, use \([G_X(t)]^m\).
Differentiate the product to recover moments, or add independent expectations and variances. A deterministic shift c gives \(G_{X+c}(t)=t^cG_X(t)\); multiplying the value of X by c instead gives \(G_{cX}(t)=G_X(t^c)\).
Check the conditions
Independence is essential for PGF multiplication.
A shift and a scale of the variable affect the PGF differently.
Three worked examples
Example 1 · PGFs of independent sums
\(\displaystyle X_{1}\),…,\(\displaystyle X_{m}\) are independent \(\displaystyle \operatorname{Bin}(6,\frac{1}{2})\) variables, with \(\displaystyle m=2\). Let S be their sum. Thus \(G_S(t)=(1-p+pt)^{12}\).
\(\displaystyle X_{1}\),…,\(\displaystyle X_{m}\) are independent \(\displaystyle \operatorname{Bin}(3,\frac{3}{5})\) variables, with \(\displaystyle m=3\). Let S be their sum. Thus \(G_S(t)=(1-p+pt)^{9}\).
\(\displaystyle X_{1}\),…,\(\displaystyle X_{m}\) are independent \(\displaystyle \operatorname{Bin}(4,\frac{1}{5})\) variables, with \(\displaystyle m=4\). Let S be their sum. Thus \(G_S(t)=(1-p+pt)^{16}\).