4.5.3—Sums of random variables

If X and Y are independent non-negative integer-valued variables, G_{X+Y}(s)=G_X(s)G_Y(s). Independence makes the generating function factorise because E(s^{X+Y})=E(s^X)E(s^Y).

Multiply the functions, then read coefficients or differentiate the product to obtain the sum’s distribution or moments. Different distributions can be combined when their supports are compatible.

The sum of independent Bin(n₁,p) and Bin(n₂,p) variables has PGF (1−p+ps)^{n₁+n₂}, hence Bin(n₁+n₂,p).

The product rule fails without independence; adding means is always valid under finite expectations, but multiplying PGFs is not.