Adding independent non-negative integer-valued random variables corresponds to multiplying their probability generating functions. This is the PGF translation of the [Discrete Convolution Formula](/theorems/1123): convolution in the probability domain becomes multiplication in the generating-function domain, just as convolution of signals becomes multiplication of Fourier transforms. This multiplicativity is what makes PGFs a powerful computational tool. To find the distribution of a sum of independent variables, one multiplies their PGFs and reads off the coefficients — far easier than performing the convolution sum directly, especially for sums of more than two variables. For instance, the sum of $n$ independent $\operatorname{Bernoulli}(p)$ variables has PGF $(1 - p + pz)^n$, which is immediately recognisable as the PGF of $\operatorname{Bin}(n, p)$. The sum of independent Poisson variables has PGF $e^{(\lambda_1 + \lambda_2)(z-1)}$, confirming the Poisson addition rule. This theorem also serves as the base case for the more general [PGF of a Random Sum](/theorems/1131), where the number of summands is itself random. The analogous result for moment generating functions is the [MGF of a Sum](/theorems/1144).