convolution as the distribution of a sum

Suppose you add two independent random variables — two independent dice, two independent waiting times — and ask how the SUM is distributed. There is a single mechanical operation that produces the answer from the two ingredient distributions: it is called convolution.

The idea is to enumerate every way the sum can hit a target and add up the probabilities. For independent discrete X and Y, the pmf of S = X + Y is p_S(s) = sum over k of p_X(k) p_Y(s - k): for each way to split the total s into a value k for X and the rest s - k for Y, multiply the two probabilities (they multiply because of independence) and add over all splits. For continuous independent variables the sum becomes an integral: f_S(s) = integral of f_X(t) f_Y(s - t) dt — slide one density across the other, multiply, and integrate the overlap. That sliding-overlap operation is the convolution of the two densities.

Convolution is why sums are central to probability and why certain families are 'closed under addition': independent Poissons sum to a Poisson, independent normals to a normal, independent gammas with the same rate to a gamma, independent binomials with the same p to a binomial. It can be tedious to compute directly, which is exactly why transforms exist — the moment generating function and characteristic function turn convolution (a messy integral) into ordinary multiplication, the slick route to the distribution of a sum and the engine behind proofs of the central limit theorem.