3.12  Moment-Generating Functions

The moment-generating function (MGF) of a random variable X is defined as:

[3.138]

for w ∈ . We call it the moment-generating function because it provides a means of calculating the moments of X. If the MGF is finite on an open interval about w = 0, then all the moments of X exist and the k^th moment of X equals the k^th derivative with respect to w of the MGF evaluated at w = 0. Heuristically, we motivate this result by applying the Taylor series expansion for the exponential function in definition [3.138]:

[3.139]

If M_X(w) is finite on some interval about the point w = 0, it can be shown that the expectation of the sum equals the sum of the expectations

[3.140]

[3.141]

You may confirm that the k^th derivative of [3.141] with respect to w evaluating at w = 0 yields the k^th moment E(X^k).

Let X be a random variable and a, b ∈ . Define a new random variable Y = bX + a. By definition [3.138], the MGF for the new random variable is related to the MGF of X by

[3.142]

More generally, suppose X is an n-dimensional random vector with independent components X_i, b is an n-dimensional row vector (b₁  b₂  …   b_n), and a ∈ . Define the random variable Y = bX + a. The MGF of Y is

[3.143]

A uniform, U(a,b), random variable has MGF

[3.144]

Those of N(μ,σ²) or χ²(ν,δ²) random variables are, respectively,

[3.145]

[3.146]

The MGF for a lognormal random variable is derived by Leipnik (1991). It is complicated, so we do not present it here.