the uniqueness theorem for the mgf

The mgf is only worth trusting if a distribution cannot disguise itself. Imagine two distributions that happen to share the same mgf — then everything you proved by manipulating mgfs would be ambiguous. The uniqueness theorem rules this out: when an mgf exists, it is a fingerprint. No two different distributions can produce the same mgf, so identifying the mgf identifies the distribution.

Precisely: if X and Y have moment generating functions that are equal and finite for all t in some open interval around 0, then X and Y have exactly the same distribution. This is what makes the standard proof strategy legitimate. You want to know what distribution some complicated thing follows — say a sum of independent variables. You compute its mgf, you do some algebra, and you find the answer matches the mgf of, say, a normal. By uniqueness you are now allowed to conclude the thing IS normal, not merely that it shares a few moments with one. The mgf became a name tag you could read off.

Two honest cautions sit beside this clean statement. First, it needs the mgf to exist near 0; for heavy-tailed distributions the mgf is not defined, so this theorem says nothing and you must reach for the characteristic function, which has its own uniqueness theorem with no such restriction. Second, do not confuse this with matching moments. Two distributions can share every single moment E[X^k] and still differ — the moment sequence alone is not always a fingerprint. It is the existence of the mgf in a neighbourhood of 0, a stronger condition than mere finite moments, that guarantees uniqueness.