In this lecture we complete our discussion of multiple
continuous random variables.
In the first half, we talk about the conditional
distribution of one random variable, given
the value of another.
We will see that the mechanics are essentially the same as in
the discrete case.
Here, we will actually face some subtle issues, because we
will be conditioning on any event that has 0 probability.
Nevertheless, all formulas will still have the form that
one should expect.
And in particular, we will see natural versions of the total
probability and total expectation theorems.
We will also define independence of continuous
random variables, a concept that has the same intuitive
content as in the discrete case.
That is, when we have independent random variables,
the values of some of them do not cause any revision of our
beliefs about the remaining ones.
Then, in the second half of the lecture, we will focus on
the Bayes rule.
This will be the methodological foundation for
when, later in this course, we dive into
the subject of inference.
The Bayes rule allows us to revise our beliefs about a
random variable.
That is, replace an original probability distribution by a
conditional one, after we observe the value of some
other random variable.
Depending on whether the random variables involved are
discrete or continuous, we will get four different
versions of the Bayes rule.
They all have the same form, with small differences.
And we will see how to apply them through some examples.