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In this segment we revisit the
concept of conditional

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expectation and view it
as an abstract object

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of a special kind.

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To get going, let us start with
something simple, the

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concept of a function.

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Let's say a function
h that maps real

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numbers to real numbers.

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As a concrete instance,
consider the quadratic

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function that maps a number
x to its square.

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Consider now a random variable,
capital X. What do

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we mean when we write h of X?

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For h defined--

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for example in this particular
way as a quadratic function--

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h of X is defined to be
a random variable.

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Which random variable?

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It is the random variable that
takes the value little x

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squared whenever capital X, the
random variable, happens

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to take the value little x.

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And this is the random variable
that we usually

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denote as the random
variable X squared.

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Now let this come to conditional
expectations.

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The conditional expectation of
a discrete random variable is

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defined by this formula.

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It is like the ordinary
expectation except that we now

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live in a conditional universe
in which the random variable

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capital Y is known to have
taken a value little y.

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And therefore, instead of using
the ordinary formula for

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expectations that involve the
PMF of X, we now use that

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formula but with the conditional
PMF of X, which is

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the appropriate PMF
that applies to

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this conditional universe.

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And if it happens that the
random variable capital X is

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continuous, we would have an
alternative formula but of the

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same kind, where the summation
is replaced by an integral and

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the PMF is replaced by a PDF.

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Now let us look at this
quantity here.

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We have fixed some particular
little y.

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Calculate this quantity.

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And what we get is a number.

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It is a number, but the value of
that number depends on the

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choice of little y.

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If I give you a different little
y then you will get

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another number for this
conditional expectation.

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This means that this quantity
here is really a

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function of little y.

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And let us give a name
to this function.

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Let us call this function g.

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Now that we have defined g we
can ask, what is this object?

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It's a function of
capital Y. It's a

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function of a random variable.

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So it should be a random
variable by itself.

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By analogy, with the earlier
concrete example, it is the

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random variable that takes the
numerical value g of little y

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whenever capital Y happens to
take the value little y.

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But g of little y has been
defined to be the same as this

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conditional expectation.

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So it's the random variable
whose value is this

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conditional expectation, which
is a particular number, if

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capital y happens to take
the value little y.

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This particular random variable
that we have defined

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here, g of capital Y, we call
it the abstract conditional

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expectation of the random
variable X, given the random

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variable Y.

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To summarize, this notation
here stands

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for a random variable.

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It is the random variable whose
numerical value turns

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out to be this one if the value
of the random variable

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capital Y happens
to be little y.

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It is a function of capital Y.
Once we know the value of

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capital Y, then the value of the
conditional expectation is

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well defined.

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It is known.

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And it's equal to this
particular number.

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It is of course a
random variable.

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And as a random variable, it
has all the attributes that

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random variables have.

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For example, it has a
distribution, that

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is, a PMF or a PDF.

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It has a mean of its own.

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And it has a variance
of its own.

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So what will be next in our
agenda is to talk about these

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attributes of this special
random variable, and also to

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use it in several examples.