Prerequisites & Notation

Before You Begin

This chapter assumes comfort with the linear algebra from Chapter 1 (especially vector spaces, inner products, and matrix decompositions) as well as single-variable calculus (limits, series, integration). The notation table below introduces the probabilistic symbols used throughout.

Linear algebra (Chapter 1)(Review ch01)
Self-check: Can you multiply a matrix by a vector, compute eigenvalues, and state what the SVD is?
Single-variable calculus
Self-check: Can you integrate $\int_0^\infty x e^{-x}\,dx$ and differentiate under the integral sign?
Basic combinatorics
Self-check: Do you know the binomial coefficient $\binom{n}{k}$ and the geometric series formula?
Complex exponentials
Self-check: Can you evaluate $e^{j\theta}$ and relate it to $\cos\theta + j\sin\theta$ ?

Chapter 2 Notation

Probabilistic notation used throughout this chapter. Vectors and matrices follow the conventions of Chapter 1.

Symbol	Meaning	Introduced
$\Omega$	Sample space	s01
$\mathcal{F}$	$\sigma$ -algebra (collection of events)	s01
$P(\cdot)$	Probability measure	s01
$P(A \mid B)$	Conditional probability of $A$ given $B$	s01
$X, Y, Z$	Random variables (scalar, uppercase italic)	s02
$f_X(x)$	Probability density function (PDF) of $X$	s02
$F_X(x)$	Cumulative distribution function (CDF) of $X$	s02
$E[X]$ , $\mu_X$	Expectation (mean) of $X$	s02
$\mathrm{Var}(X)$ , $\sigma_X^2$	Variance of $X$	s02
$M_X(s)$	Moment-generating function $E[e^{sX}]$	s03
$\Phi_X(\omega)$	Characteristic function $E[e^{j\omega X}]$	s03
$\mathbf{x}$	Random vector (boldface lowercase)	s04
$f_{\mathbf{x}}(\mathbf{x})$	Joint PDF of random vector $\mathbf{x}$	s04
$\mathbf{R}_{\mathbf{x}}$ , $\mathbf{C}_{\mathbf{x}}$	Correlation matrix, covariance matrix of $\mathbf{x}$	s04
$\mathcal{CN}(\boldsymbol{\mu}, \mathbf{R})$	Circularly symmetric complex Gaussian distribution	s04
$X_n \xrightarrow{P} X$	Convergence in probability	s05
$X_n \xrightarrow{d} X$	Convergence in distribution	s05
$\{X(t)\}$ , $\{X_n\}$	Continuous-time / discrete-time stochastic process	s06
$R_X(\tau)$	Autocorrelation function of process $X(t)$	s06
$S_X(f)$	Power spectral density of process $X(t)$	s06
$\mathbf{P}$	Transition probability matrix (Markov chain)	s08
$\boldsymbol{\pi}$	Stationary distribution vector	s08
$N(t)$	Counting process (Poisson process)	s09
$\lambda$	Rate parameter (Poisson / exponential)	s09

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