Moment Generating Functions: Theory and Applications

Definition and Properties

The moment generating function (MGF) of X is M(t) = E[e^(tX)], defined for t in some neighborhood of 0. The n-th moment is E[X^n] = M^(n)(0), the n-th derivative evaluated at t = 0. If the MGF exists in a neighborhood of 0, it uniquely determines the distribution. This uniqueness theorem is the key reason MGFs matter for Exam P.

For discrete X: M(t) = sum of e^(tx) * p(x). For continuous X: M(t) = integral of e^(tx) * f(x) dx.

MGFs of Common Distributions

Memorize these for Exam P. Bernoulli(p): M(t) = 1 - p + p*e^t. Binomial(n,p): M(t) = (1 - p + p*e^t)^n. Poisson(lambda): M(t) = exp(lambda*(e^t - 1)). Exponential(lambda): M(t) = lambda/(lambda - t). Normal(mu, sigma^2): M(t) = exp(mu*t + sigma^2*t^2/2). Gamma(alpha, lambda): M(t) = (lambda/(lambda - t))^alpha.

To identify a distribution from its MGF, match the algebraic form to one of these standard forms and read off the parameters.

Applications to Sums

The MGF of a sum of independent random variables is the product of their individual MGFs. If X and Y are independent, M_{X+Y}(t) = M_X(t) * M_Y(t). This property makes it easy to prove that the sum of independent Poissons is Poisson, the sum of independent normals is normal, and the sum of independent gammas (with the same rate) is gamma. Exam P frequently uses this technique to identify the distribution of a sum.