Chapter Overview¶
Where We Are¶
Chapter 8 introduced functions of multiple variables and showed how they can be used to define distributions for random vectors. Chapter 9 introduced derivatives of multivariate functions and showed how to use them to solve optimization problems.
Where We’re Going¶
In this chapter, we will study how to integrate functions of multiple variables. Integration in multiple variables is essential since chances and expectations are produced by integrating joint densities over multiple coordinates at once. When working in three dimensions or more, marginal densities are also expressed as an integral of a joint density over multiple coordinates. Geometrically, all of these tasks reduce to finding the volume beneath a surface.
In this chapter we will focus on the continuous, two-dimensional case. All results also apply to double sums.
We will:
Define a double integral as the volume under a surface over a region, and show how to compute double integrals using iterated integration (see Section 10.1). We will:
Introduce Fubini’s Theorem to show that the order of integration does not change the value of the integral.
Factor double integrals of functions that can be expressed as sums, or products, of univariate functions.
Integrate over both rectangular and non-rectangular regions.
Define conditional expectation (see Section 10.2). We will:
Derive the chain rule of expectation (iterated expectation) from conditional expectations by splitting integrals.
Apply the chain rule to find the expectation of geometric random variables and of variables drawn from a mixture distribution.
Practice computing integrals over manifolds (see Section 10.3)
Integrate over lines, then apply this skill to find the density of sums of random variables via the convolution formula.
Integrate over circles, then apply these lessons to integrate symmetric density functions in polar coordinates.