moment generating function
/ MGF /
A moment generating function is a clever single function that packs all the moments of a distribution into one expression, ready to be unpacked on demand. Instead of computing the mean, variance, skewness and so on one painful integral at a time, you build the MGF once and then read each moment off by a simple operation. It is a mathematical zip file for a distribution's moments.
The MGF is defined as M(t) = E[e^(tX)], the expected value of e raised to t times X, treated as a function of the helper variable t. Its magic is twofold. First, taking derivatives at t = 0 produces the moments: the first derivative gives E[X], the second gives E[X^2], and so on. Second, and even more useful, the MGF uniquely identifies a distribution — if two random variables have the same MGF, they have the same distribution. This makes the MGF the go-to tool for proving what the sum of independent random variables looks like, because the MGF of a sum is just the product of the individual MGFs.
For an actuary the MGF is the engine behind aggregate loss models. To find the distribution of total claims — a random number of claims, each a random size — you can often work with MGFs (or the closely related probability and characteristic generating functions) far more easily than with the densities directly. The MGF also underpins the elegant proofs that a sum of independent Poissons is Poisson, or that sums of normals are normal, results that let actuaries combine risks without re-deriving everything from scratch.
If X and Y are independent claim counts with MGFs M_X(t) and M_Y(t), the MGF of their sum is the product M_X(t)·M_Y(t). For two independent Poissons this product is itself a Poisson MGF — proving the combined count is Poisson.
The MGF of a sum of independent variables is the product of their MGFs.
An MGF does not always exist — heavy-tailed distributions like the lognormal and many Paretos have no MGF because E[e^(tX)] is infinite, so actuaries often fall back on the characteristic function, which always exists.