Moment Generating Functions are extremely useful in exploring the behavior of sums of independent random variables. These sums often converge to the Normal Distribution, a result known as the Central Limit Theorem. It is this regular behavior of ir-regular and random objects that is at the heart of probability and statistics.
S1: MGF of Sums
S2: Sums of Normals
S3: Sums of Gammas
S5: Sums of Poissons