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We have defined the conditional
expectation of a

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random variable given another
as an abstract object, which

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is itself a random variable.

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Let us now do something
analogous with the notion of

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[the]

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conditional variance.

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Let us start with the definition
of the variance,

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which is the following.

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We look at the deviation of the
random variable from its

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mean, square it, then take the
average of that quantity.

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If we live in a conditional
universe where we are told the

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value of some other random
variable, capital Y, then

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inside that conditional universe
the variance becomes

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the following.

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It is defined the same way.

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Well, in the conditional
universe, this is the expected

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value of X.

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So this quantity here is the
deviation of X from its

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expected value in that
conditional universe.

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We square this quantity, we find
the squared deviation,

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and we look at the
expected value of

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that squared deviation.

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But because we live in a
conditional universe, of

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course, this expectation has to
be a conditional one given

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the information that
we have available.

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So this is nothing but the
ordinary variance, but it's

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the variance of the conditional
distribution of

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the random variable, capital
X. This is an

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equality between numbers.

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If I tell you the value of
little y, the conditional

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variance is defined
by this particular

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quantity, which is a number.

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Now, we proceed in the same way
as we proceeded for the

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case where we defined the
conditional expectation as a

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random variable.

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Namely, we think of this
quantity as a function of

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little y, and that function can
be now used to define a

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random variable.

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And that random variable, which
would denote this way,

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this is the random variable
which takes this specific

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value when capital Y happens
to be equal to little y.

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Once we know the value of
capital Y, then this quantity

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takes a specific value.

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But before we know the value
of capital Y, then this

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quantity is not known.

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It's random.

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It's a random variable.

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Let us look at an example to
make this more concrete.

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Suppose that Y is a
random variable.

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We draw that random variable.

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And we're told that conditioned
on the value of

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that random variable, X is going
to be uniform on this

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particular interval
from 0 to Y.

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So if I tell you that capital
Y takes on a specific

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numerical value, then the random
variable X is uniform

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on the interval from
0 to little y.

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A random variable that's uniform
on an interval of

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length little y has a variance
that we know what it is.

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It's y squared over 12.

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So this is an equality
between numbers.

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For any specific value of
little y, this is the

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numerical value of the
conditional variance.

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Let us now change this equality
between numbers into

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an abstract equality between
random variables.

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The random variable, variance
of X given Y, is a random

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variable that takes this
value whenever

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capital Y is little y.

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But that's the same as
this random variable.

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This is a random variable that
takes this value whenever

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capital Y happens to be
equal to little y.

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So we have defined the
abstract concept of a

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conditional variance, similar
to the case of conditional

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expectations.

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For conditional expectations,
we had the law of iterated

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expectations.

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That tells us that the
expected value of the

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conditional expectation is the
unconditional expectation.

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Is it true that the expected
value of the conditional

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variance is going to
be the same as the

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unconditional variance?

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Unfortunately, no.

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Things are a little
more complicated.

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The unconditional variance is
equal to the expected value of

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the conditional variance, but
there is an extra term, that

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is, the variance of the
conditional expectation.

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The entries here in red are
all random variables.

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So the conditional variance has
been defined as a random

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variable, so it has an
expectation of its own.

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The conditional expectation, as
we have already discussed,

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is also a random variable, so it
has a variance of its own.

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And by adding those terms, we
get the total variance of the

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random variable X.

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So what we will do next will
be first to prove this

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equality, and then give a number
of examples that are

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going to give us some intuition
about what these

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terms mean and why this
equality makes sense.