Prerequisites & Notation

Prerequisites for Chapter 6

This chapter extends the discrete random variable framework of Chapter 5 to the continuous setting. The key new idea is that probabilities are computed as integrals of a density function rather than sums of a mass function, but the CDF remains the unifying object.

Cumulative distribution function (CDF) properties(Review ch05)
Self-check: Can you state the three defining properties of a CDF: right-continuity, monotonicity, and boundary limits?
Discrete random variables and PMFs(Review ch05)
Self-check: Can you compute $\mathbb{E}[g(X)]$ for a discrete RV using the PMF?
Basic integration and differentiation (calculus)
Self-check: Can you evaluate $\int_0^\infty x e^{-\lambda x}\,dx$ by integration by parts?
Series and limits
Self-check: Are you comfortable with the limit $(1 - \lambda/n)^n \to e^{-\lambda}$ as $n \to \infty$ ?

Notation for Chapter 6

We collect the principal symbols used in this chapter. Random variables are uppercase italic, realizations are lowercase italic, and densities use lowercase function notation.

Symbol	Meaning	Introduced
$f(x)$	Probability density function of $X$
$F(x)$	Cumulative distribution function: $F(x) = \mathbb{P}(X \leq x)$
$\\mathcal{N}(\\mu, \\sigma^2)$	Gaussian (normal) distribution with mean $\mu$ and variance $\sigma^2$
$\Q(x)$	Q-function: $Q(x) = \frac{1}{\sqrt{2\pi}}\int_x^{\infty} e^{-t^2/2}\,dt$
$\\Phi(x)$	Standard normal CDF: $\Phi(x) = 1 - Q(x)$
$\\text{Var}(X)$	Variance of $X$
$\\mathbb{E}[X]$	Expectation of $X$
$\\Gamma(t)$	Gamma function: $\Gamma(t) = \int_0^{\infty} x^{t-1} e^{-x}\,dx$
$B(a,b)$	Beta function: $B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$
$\\delta(x)$	Dirac delta function

← Ch 5 PDF, CDF, and the Relationship Between Them