WEBVTT
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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.