Prerequisites & Notation

Before You Begin

This chapter builds directly on the theory of single random variables developed in Chapters 5 and 6. You will need fluency with PMFs, PDFs, CDFs, expectation, and functions of a single random variable.

Probability mass functions and probability density functions(Review ch05)
Self-check: Can you write down the PMF of a Poisson random variable and compute $\mathbb{P}(X = k)$ ?
Cumulative distribution functions and their properties(Review ch06)
Self-check: Can you state the three defining properties of a CDF?
Expectation, variance, and LOTUS(Review ch05)
Self-check: Can you compute $\mathbb{E}[g(X)]$ using LOTUS without finding the distribution of $g(X)$ ?
Functions of a single random variable (CDF method, change of variables)(Review ch06)
Self-check: If $X \sim \mathcal{N}(0,1)$ and $Y = X^2$ , can you derive $f_{Y}(y)$ ?
Conditional probability and Bayes theorem for events(Review ch02)
Self-check: Can you state Bayes theorem and apply the law of total probability?

Chapter 7 Notation

The following notation is used throughout this chapter. Symbols marked with $f$ , $F$ , etc. are customizable via the notation preferences panel.

Symbol	Meaning	Introduced
$f_{X,Y}(x,y)$	Joint probability density function of $(X,Y)$
$P_{X,Y}(x_i, y_j)$	Joint probability mass function (discrete case)
$F_{X,Y}(x,y)$	Joint cumulative distribution function
$f(y\|x)$	Conditional PDF of $Y$ given $X = x$
$\text{Cov}(X,Y)$	Covariance of $X$ and $Y$
$\rho_{X,Y}$	Correlation coefficient of $X$ and $Y$
$f_{X} * f_{Y}$	Convolution of densities (PDF of sum of independent RVs)
$J_{g^{-1}}(\mathbf{y})$	Jacobian determinant of the inverse transformation
$X_{(k)}$	The $k$ -th order statistic ( $k$ -th smallest value)

← Ch 6 Joint PMFs and PDFs