Chapter Overview¶
Where We Are¶
Chapters 2 - 7 have primarily focused on probability problems involving a single random variable. Most practical problems involve multiple, related, random variables. So, we will need to go beyond one-dimensional problems.
Where We’re Going¶
In this chapter, we will step up to two dimensions. Thankfully, most of the work required to go from one to two dimensions carries over to three, or four, or ten, or a hundred dimensions. So, we’ll focus on the two-dimensional case now, and where it is easy to generalize our results, will suggest the generalization.
We will:
Define a random vector and discuss different coordinate systems that could be used to represent the random vector (see Section 8.1). We will:
Introduce basic vector operations.
Learn how to measures lengths and angles using inner products
Contrast Cartesian coordinates with polar coordinates.
Show how to define functions of multiple variables (see Section 8.2).
In particular, we will focus on different conventions for visualizing surfaces.
Combine these ideas to define joint densities (see Section 8.3).