Transforms and Moment Generating Functions

Convolution (Derived Distributions) finds the distribution of a sum of independent random variables via an integral (or a sum, in the discrete case) — correct, but often algebraically heavy. The moment generating function (MGF), $M_X(s)=\mathbb{E}[e^{sX}]$, converts every distribution into an ordinary function of a real parameter $s$, and under this encoding, convolution becomes multiplication: the MGF of a sum of independent random variables is the product of their MGFs. This single fact makes the MGF the vault's most powerful tool for deriving closure results (Poisson-sum-is-Poisson, normal-sum-is-normal) and, thanks to the inversion property, for identifying an unknown distribution purely by pattern-matching its MGF against a table of known forms.

Learning objectives. By the end of this note you should be able to: