Another important application in probability that can involve improper double integrals is the calculation of expected values. First we define this concept and then show an example of a calculation.
Definition
In probability theory, we denote the
expected values
and
respectively, as the most likely outcomes of the events. The expected values
and
are given by
where
is the sample space of the random variables
and
Finding expected value
Find the expected time for the events ‘waiting for a table’ and ‘completing the meal’ in
[link] .
Using the first quadrant of the rectangular coordinate plane as the sample space, we have improper integrals for
and
The expected time for a table is
A similar calculation shows that
This means that the expected values of the two random events are the average waiting time and the average dining time, respectively.
A general bounded region
on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.
To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.
We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.
We can use Fubini’s theorem for improper integrals to evaluate some types of improper integrals.
Key equations
Iterated integral over a Type I region
Iterated integral over a Type II region
In the following exercises, specify whether the region is of Type I or Type II.
The region
bounded by
and
as given in the following figure.
