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We have previously defined
the abstract conditional

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expectation of one random
variable given

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another random variable.

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And we discussed that it is, by
itself, a random variable.

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In particular, it has
an expectation, or

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mean, of its own.

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What is this mean?

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This is what we want
to find out.

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Let us recall our development.

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We look at the conditional
expectation of a random

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variable given a specific
numerical value of another

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random variable.

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This is a number that
depends on little y.

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And this can be used to define
a function little g.

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The function little g for any
particular little y tells us

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the numerical value of the
conditional expectation.

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Since little g is a well defined
function, we can also

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now define this particular
function, which is now a

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function of a random variable.

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It's a well defined object.

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It's a random variable.

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And then we introduced this
abstract notation.

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We defined this object
to be exactly this

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particular random variable.

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So now we want to calculate
the expected value of this

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object, which is written
this way.

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Now this notation, here, may
look quite formidable, but

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let's see what is happening.

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Inside here, we have
a random variable.

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And we take the expected value
of that random variable.

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Or, more crisply, think of that
as the expected value of

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g of capital Y, where g of
capital Y is defined through

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these correspondences here.

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How do we calculate the expected
value of a function

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of a random variable?

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Here we use the Expected
Value Rule.

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Assuming that Y is a discrete
random variable, the Expected

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Value Rule takes this form.

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And the next step is to
substitute the particular form

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for g of Y that we have.

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g of Y was defined
in this manner.

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So we're dealing with the sum
over all little y's of the

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expected value of X, given that
Y takes the value little

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y, weighted by the
PMF of little y.

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Now if we look at this
expression, then it should

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look familiar.

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It is the expression that
appears in the Total

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Expectation Theorem.

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We take the conditional
expectation under different

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scenarios and weigh those
conditional expectations

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according to the probabilities
of those scenarios.

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And this just gives us the
overall expectation of the

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random variable X.

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So this step, here, was carried
out using the Total

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Expectation Theorem.

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So we have proved this important
fact, that the

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expectation of a conditional
expectation is the same as the

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unconditional expectation.

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This important fact is called
the Law of Iterated

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Expectations.

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The proof was carried out
assuming that Y is discrete.

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So we use this particular
version involving a PMF, but

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the proof is exactly the same
for the continuous case.

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You would be using an integral
and the PDF, instead the PMF.

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As the proof indicates, the Law
of Iterated Expectations

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is nothing but an abstract
version of the Total

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Expectation Theorem.

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It is really the Total
Expectation Theorem written in

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more abstract notation.

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But this turns out to be
powerful and also we avoid

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having to deal separately
with discrete or

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continuous random variables.