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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.