Prerequisites & Notation

Before You Begin

This chapter introduces random variables — the bridge from abstract probability spaces to numerical computation. We assume familiarity with the probability axioms and basic counting from earlier chapters.

Probability spaces: sample space $\Omega$ , $\sigma$ -algebra $\mathcal{F}$ , measure $\mathbb{P}$ (Review ch01)
Self-check: Can you define a probability space and verify the axioms for a given example?
Conditional probability and Bayes' rule(Review ch02)
Self-check: Can you compute $\mathbb{P}(A | B)$ and apply Bayes' theorem?
Independence of events(Review ch02)
Self-check: Can you verify whether two events are independent from a joint probability table?
Combinatorics: binomial coefficients, counting arguments(Review ch04)
Self-check: Can you compute $\binom{n}{k}$ and apply it to counting problems?
Basic series: geometric series, Taylor expansion of $e^x$
Self-check: Can you sum $\sum_{k=0}^{\infty} x^k$ and recognize $e^x = \sum_{k=0}^{\infty} x^k/k!$ ?

Notation for This Chapter

Symbols introduced in this chapter. Random variables are uppercase italic ( $X$ ), realizations are lowercase italic ( $x$ ), and sets are calligraphic ( $\mathcal{X}$ ).

Symbol	Meaning	Introduced
$X : \Omega \to \mathbb{R}$	Random variable (measurable function)	s01
$P(x)$	Probability mass function of $X$	s01
$F(x)$	Cumulative distribution function of $X$	s01
$\mathcal{X}$	Support of $X$ (set of values with positive probability)	s01
$\mathbb{E}[X]$	Expectation (mean) of $X$	s02
$\text{Var}(X)$	Variance of $X$	s03
$\sigma_X$	Standard deviation of $X$	s03
$M_X(t)$	Moment generating function: $\mathbb{E}[e^{tX}]$	s04
$H(X)$	Shannon entropy of $X$ : $-\sum_x p(x) \log_2 p(x)$	s05
$\mathbf{1}_A$	Indicator random variable of event $A$	s01

← Ch 4 Definition and Probability Mass Function