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In this lecture we complete
our discussion of multiple

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continuous random variables.

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In the first half, we talk
about the conditional

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distribution of one random
variable, given

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the value of another.

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We will see that the mechanics
are essentially the same as in

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the discrete case.

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Here, we will actually face some
subtle issues, because we

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will be conditioning on any
event that has 0 probability.

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Nevertheless, all formulas will
still have the form that

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one should expect.

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And in particular, we will see
natural versions of the total

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probability and total
expectation theorems.

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We will also define independence
of continuous

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random variables, a concept that
has the same intuitive

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content as in the
discrete case.

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That is, when we have
independent random variables,

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the values of some of them do
not cause any revision of our

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beliefs about the
remaining ones.

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Then, in the second half of the
lecture, we will focus on

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the Bayes rule.

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This will be the methodological
foundation for

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when, later in this course,
we dive into

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the subject of inference.

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The Bayes rule allows us to
revise our beliefs about a

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random variable.

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That is, replace an original
probability distribution by a

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conditional one, after we
observe the value of some

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other random variable.

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Depending on whether the random
variables involved are

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discrete or continuous, we
will get four different

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versions of the Bayes rule.

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They all have the same form,
with small differences.

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And we will see how to apply
them through some examples.