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In this last lecture of this
unit, we continue with some of

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our earlier themes, and then
introduce one new notion, the

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notion of independence
of random variables.

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We will start by elaborating a
bit more on the subject of

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conditional probability
mass functions.

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We have already discussed the
case where we condition a

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random variable on an event.

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Here we will talk about
conditioning a random variable

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on another random variable,
and we will develop yet

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another version of the total
probability and total

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expectation theorems.

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There are no new concepts
here, just new notation.

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I should say, however, that
notation is important, because

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it guides you on how to think
about problems in the most

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economical way.

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The one new concept that we will
introduce is the notion

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of independence of
random variables.

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It is actually not an entirely
new concept.

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It is defined more or less the
same way as independence of

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events, and has a similar
intuitive interpretation.

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Two random variables are
independent if information

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about the value of one of them
does not change your model or

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beliefs about the other.

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On the mathematical side, we
will see that independence

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leads to some additional
nice properties

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of means and variances.

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We will conclude this lecture
and this unit on discrete

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random variables by considering
a rather difficult

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problem, the hat problem.

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We will see that by being
systematic and using some of

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the tricks that we have learned,
we can calculate the

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mean and variance of a rather
complicated random variable.