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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.