Moment Generating Functions

The moment generating function (MGF) is a powerful tool that encodes all the moments of a distribution in a single function. It can be used to find means and variances, prove distributional results, and identify distributions. This topic is part of the AQA Further Mathematics specification.

---

Definition

The moment generating function of a random variable $X$X is defined as:

$M_X(t) = E(e^{tX})$MX​(t)=E(etX)

For a discrete random variable:

$M_X(t) = \sum_x e^{tx} P(X = x)$MX​(t)=∑x​etxP(X=x)

For a continuous random variable:

$M_X(t) = \int_{-\infty}^{\infty} e^{tx} f(x) \, dx$MX​(t)=∫−∞∞​etxf(x)dx

The MGF exists (is finite) for all $t$t in some interval containing 0.

---

Why "Moment Generating"?

Expanding $e^{tX}$etX as a Taylor series:

$e^{tX} = 1 + tX + \frac{(tX)^2}{2!} + \frac{(tX)^3}{3!} + \cdots$etX=1+tX+2!(tX)2​+3!(tX)3​+⋯

Taking expectations:

$M_X(t) = 1 + tE(X) + \frac{t^2}{2!}E(X^2) + \frac{t^3}{3!}E(X^3) + \cdots$MX​(t)=1+tE(X)+2!t2​E(X2)+3!t3​E(X3)+⋯

So the $n$n-th moment is obtained by differentiating $n$n times and evaluating at $t = 0$t=0:

$E(X^n) = M_X^{(n)}(0)$E(Xn)=MX(n)​(0)