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Conditional PDFs share most
of the properties

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of conditional PMFs.

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All facts for the discrete case
have continuous analogs.

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The intuition is more or less
the same, although it is much

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easier to grasp in the
discrete case.

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For example, we have seen this
version of the total

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probability theorem.

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There is a continuous
analog in which we

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replace sums by integrals.

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And we replace PMFs by PDFs.

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The proof of this fact is
actually pretty simple.

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By the multiplication rule, the
integrand, here, is just

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the joint PDF of X and Y.

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And we know that if we take the
joint PDF and integrate

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with respect to one variable
then we recover the marginal

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PDF of the other random
variable.

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So this is one theorem
that extends to

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the continuous case.

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Moving along, we have defined
the conditional expectation in

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this manner in the
discrete case.

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And we define it similarly
for the continuous case.

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So actually here we now
have a new definition.

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This definition is also
consistent with the definition

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of the expectation of a
continuous random variable.

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The expected value for
continuous random variable is

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the integral of X times [a]

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density.

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Except that here we live in a
conditional universe where

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we're conditioning
on this event.

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And therefore, we
need to use the

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corresponding conditional PDF.

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Finally, we have the total
expectation theorem in the

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discrete case.

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And there is the obvious analog
in the continuous case

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where we are using an integral
and a density.

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The interpretation is that we
consider all possibilities for

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Y. Under each possibility of Y
we find the expected value of

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X. And then we weigh those
different possibilities

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according to the corresponding
values of the density.

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So we're taking a weighted
average of these conditional

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expectations to obtain the
overall expectation of the

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random variable X.

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The derivation of this fact is
maybe a little instructive

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because it uses various facts
that we have in our hands.

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So let's see how to derive it.

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We start from this expression in
the right-hand side and we

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will show that it is equal to
the expected value of X. The

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expression on the right-hand
side is equal to the

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following, it's the integral
of the density of Y.

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And then, inside here, we have
the conditional expectation

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which is defined this way.

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So we just plug-in
the definition.

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And then what we do, is we take
this term and move it

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inside the integral.

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Which we can do because this
integral is with respect to x.

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And therefore, this is
like a constant.

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And we also interchange the
order of integration.

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Now, the inner integration
is with respect to y.

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As far as Y is concerned, this
term, x, is a constant.

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So we can take it and move it
outside this first integral

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and place it out there.

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So this term disappears
and goes out there.

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What do we have here?

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This part, by the previous fact,
the total probability

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theorem, is just the density of
X. So we're left with the

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integral of x times the
density of x dx.

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And this is the expected
value of X.

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Finally, we have various forms
of the expected value rule,

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which barely deserve
writing down.

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Because they're exactly
what you might expect.

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Consider, for example, an
expression such as this one,

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the expected value of a
function of the random

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variable X but conditioned on
the value of the random

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variable Y. How do we calculate
this quantity?

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Well, the expected value rule
tells us that we should

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integrate g of x times the
density of X. But because,

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here we live in a conditional
universe, we should actually

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use the corresponding
conditional PDF of X. And

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there are many other versions
of the expected value rule.

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Any version that we have seen
for the discrete case has,

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also, a continuous analog which
looks about the same

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except that we integrate and
that we use densities.