Moment Generating Functions

A transform that uniquely identifies a distribution and makes sums easy.

The moment generating function (MGF) of $X$ is

M_X(t) = E[e^{tX}]

defined wherever this expectation is finite in a neighborhood of $0$ . When it exists, the MGF uniquely determines the distribution.

Three properties make MGFs powerful:

The $k$ -th moment is the $k$ -th derivative at zero: $E[X^k] = M_X^{(k)}(0)$ .For independent $X$ and $Y$ , the MGF of the sum is the product: $M_{X + Y}(t) = M_X(t)\, M_Y(t)$ .Convergence of MGFs implies convergence in distribution.

Some closed forms worth knowing:

M_{N(\mu, \sigma^2)}(t) = \exp\!\left(\mu t + \frac{\sigma^2 t^2}{2}\right), \qquad M_{\mathrm{Poisson}(\lambda)}(t) = \exp(\lambda(e^t - 1))

The product rule for sums makes MGFs the standard tool for proving the CLT, deriving distributions of sums, and most other classical limit results.