the mgf of a sum of independent variables
Adding random variables is one of the most common things we do — total winnings over many bets, total demand from many customers, total error from many noisy measurements. But finding the distribution of a sum directly is genuinely hard: it requires a convolution, an integral that smears one distribution across the other. The mgf offers a shortcut so clean it feels like cheating: under the right condition, adding turns into multiplying.
The rule is M_{X+Y}(t) = M_X(t) times M_Y(t) whenever X and Y are independent. The reason is short. By definition M_{X+Y}(t) = E[e^(t(X+Y))] = E[e^(tX) e^(tY)]. Independence is exactly the property that lets the expectation of a product split into the product of the expectations, so this becomes E[e^(tX)] times E[e^(tY)] = M_X(t) M_Y(t). The hard convolution in the world of distributions has become an ordinary multiplication in the world of mgfs — the same magic by which logarithms once turned multiplication into addition. For a sum of many independent pieces you just keep multiplying.
This single rule, combined with the uniqueness theorem, is the engine behind a whole family of clean results. Multiply the mgfs of independent Poissons and you recognize the mgf of another Poisson, so the sum is Poisson; multiply the mgfs of independent normals and you get the mgf of a normal, so the sum is normal. The crucial fine print is independence: if X and Y are dependent, the expectation of the product does not split, and the product rule fails. The product rule is a statement about independent summands only.
Let X ~ Poisson(a) and Y ~ Poisson(b) be independent. Their mgfs are M_X(t) = e^(a(e^t - 1)) and M_Y(t) = e^(b(e^t - 1)). Multiply: M_{X+Y}(t) = e^((a+b)(e^t - 1)). That is precisely the mgf of a Poisson(a + b), so X + Y ~ Poisson(a + b) — proven by a one-line multiplication.
For independent X and Y, the mgf of the sum is the product of the mgfs.
The product rule needs independence. For dependent variables E[e^(tX) e^(tY)] does not factor, so M_{X+Y}(t) is generally not M_X(t) M_Y(t).