Section 14.4

Moments and Generating Functions

Moment generating functions and their uses: lognormal moments, skewness, moments of compound variables, variables with infinite moments.

Toolkit

The moment generating function $M_X(\theta)=\mathbb{E}[e^{\theta X}]$ encodes all moments via $\mathbb{E}[X^n]=M_X^{(n)}(0)$ , and $\log M_X(\theta)$ generates cumulants, with the third cumulant controlling skewness. Key facts:

\mathcal{N}(\mu,\sigma^2):\ M(\theta)=e^{\mu\theta+\sigma^2\theta^2/2},\qquad \mathbb{E}|S|=\sigma\sqrt{2/\pi}\ (S\sim\mathcal{N}(0,\sigma^2)).

\text{Poisson}(\lambda):\ \mathbb{E}\left[\binom{N}{k}\right]=\frac{\lambda^k}{k!}\quad\text{(factorial moments via shifting the sum).}

Symmetry of a density about its mean forces all odd central moments to vanish, hence zero skewness. For compound/lognormal variables, moments follow by substituting into the normal MGF at integer $\theta$ . When $\mathbb{E}|X|=\infty$ (e.g. heavy-tailed or Zeta-type laws), expectations, covariances and higher moments may fail to exist even under independence; check absolute convergence before manipulating sums or differences of expectations. Divergent series (harmonic-type) signal infinite expectation in waiting-time problems.

Problems