When two independent discrete random variables are added, the distribution of their sum is not obtained by simply adding their probability mass functions — one must account for all the ways their individual values can combine to produce a given total. The convolution formula does exactly this: $\mathbb{P}(X + Y = n)$ is the sum over all $k$ of $\mathbb{P}(X = k)\,\mathbb{P}(Y = n - k)$, pairing every possible value of $X$ with the complementary value of $Y$. This is the discrete analogue of the convolution integral for continuous densities. It is the computational method behind deriving the distribution of sums of familiar random variables — for instance, verifying that the sum of two independent Poisson variables is again Poisson, or that the sum of two independent binomials (with the same $p$) is binomial. In the language of generating functions, the convolution formula is encoded by the multiplicativity of the PGF ([PGF of a Sum](/theorems/1130)): the coefficient of $z^n$ in the product $G_X(z) \cdot G_Y(z)$ is precisely the convolution sum above. Working with PGFs is often more efficient than computing convolutions directly, especially for sums of more than two variables.