Prerequisites & Notation

Before You Begin

This chapter develops the transform toolkit — moment generating functions, characteristic functions, and probability generating functions — that converts questions about sums and convolutions into questions about products and composition. We assume familiarity with random variables, expectation, and joint distributions from earlier chapters.

Random variables, PMFs, PDFs, and CDFs(Review ch05, ch06)
Self-check: Can you compute $\mathbb{E}[g(X)]$ for both discrete and continuous $X$ ?
Expectation, variance, and higher moments(Review ch05, ch06)
Self-check: Can you derive $\text{Var}(X+Y)$ when $X$ and $Y$ are independent?
Joint distributions and independence(Review ch07)
Self-check: Can you state what $X \perp Y$ implies about $\mathbb{E}[g(X)h(Y)]$ ?
Common distributions: Gaussian, Poisson, Exponential, Gamma, Binomial(Review ch05, ch06)
Self-check: Can you write the PDF of $\mathcal{N}(\mu, \sigma^2)$ and the PMF of $\text{Poi}(\lambda)$ ?
Taylor series and complex exponentials
Self-check: Can you expand $e^{jux}$ as a power series in $u$ ?
Riemann-Stieltjes integration (basics)(Review ch08)
Self-check: Can you express $\mathbb{E}[g(X)]$ as $\int g(x)\,dF(x)$ ?

Notation for This Chapter

Symbols introduced in this chapter. The transforms encode the entire distribution into a single function, converting convolution into multiplication.

Symbol	Meaning	Introduced
$M_X(t)$	Moment generating function: $\mathbb{E}[e^{tX}]$	s01
$\phi_X(u)$	Characteristic function: $\mathbb{E}[e^{juX}]$	s02
$G_X(s)$	Probability generating function: $\mathbb{E}[s^X]$ for nonneg. integer-valued $X$	s05
$m_X(t)$	Cumulant generating function: $\log M_X(t)$	s04
$\kappa_n$	$n$ -th cumulant of $X$	s04
$m_X^*(a)$	Fenchel-Legendre transform (rate function): $\sup_t \{at - m_X(t)\}$	s08
$S_n$	Partial sum: $S_n = \sum_{i=1}^n X_i$	s06
$\xrightarrow{D}$	Convergence in distribution	s07
$\eta$	Probability of ultimate extinction (branching process)	s09
$j$	Imaginary unit: $j = \sqrt{-1}$	s02

← Ch 8 The Moment Generating Function