WEBVTT
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By this point in this class, you
must have realized that a
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lot of revolves around the
concept of conditioning.
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Conditional expectations
play a central role.
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For this reason, it is useful
to revisit this concept and
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view it in a more
abstract manner.
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The basic idea is that the
value of a conditional
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expectation is affected by a
random quantity by the value
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of the random variable Y on
which we are conditioning.
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It is a function of Y and,
therefore, a random variable.
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Based on this observation, we
will redefine the conditional
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expectation as a random variable
and then try to
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understand its properties.
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In particular, we will develop
a formula for the expected
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value of the conditional
expectation.
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This will be what as known
as the law of iterated
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expectations.
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After doing all this, we will
follow a similar program for
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the conditional variance.
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Once more, we will see
that it can be
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viewed as a random variable.
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And then we will relate its
expected value with the
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unconditional variance.
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This will be the so-called
law of total variance.
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As an illustration of the tools
we are introducing in
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this lecture, we will consider
various examples that will
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hopefully clarify the
concepts involved.
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Our final and most important
example will involve the sum
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of a random number of
independent random variables.
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The setting here is more
challenging than the case
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where we add a fixed number
of random variables.
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But by using conditioning,
we will be able to derive
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formulas for the mean
and the variance.