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In this segment, we
pursue two themes.

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Every concept has a conditional
counterpart.

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We know about PDFs, but if
we live in a conditional

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universe, then we deal with
conditional probabilities.

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And we need to use
conditional PDFs.

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The second theme is that
discrete formulas have

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continuous counterparts in which
summations get replaced

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by integrals, and
PMFs by PDFs.

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So let us recall the definition
of a conditional

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PMF, which is just the same as
an ordinary PMF but applied to

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a conditional universe.

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In the same spirit, we can start
with a PDF, which we can

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interpret, for example, in
terms of probabilities of

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small intervals.

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If we move to a conditional
model in which event A is

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known to have occurred,
probabilities of small

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intervals will then be
determined by a conditional

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PDF, which we denote
in this manner.

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Of course, we need to assume
throughout that the

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probability of the conditioning
event is positive

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so that conditional
probabilities are

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well-defined.

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Let us now push the
analogy further.

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We can use a PMF to calculate
probabilities.

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The probability that X takes [a]
value in a certain set is

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the sum of the probabilities
of all the possible

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values in that set.

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And a similar formula is true
if we're dealing with a

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conditional model.

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Now, in the continuous case, we
use a PDF to calculate the

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probability that X takes values
in a certain set.

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And by analogy, we use a
conditional PDF to calculate

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conditional probabilities.

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We can take this relation here
to be the definition of a

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conditional PDF.

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So a conditional PDF is a
function that allows us to

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calculate probabilities by
integrating this function over

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the event or set of interest.

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Of course, probabilities
need to sum to 1.

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This is true in the
discrete setting.

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And by analogy, it should
also be true in

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the continuous setting.

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This is just an ordinary PDF,
except that it applies to a

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model in which event A is
known to have occurred.

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But it still is a
legitimate PDF.

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It has to be non-negative,
of course.

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But also, it needs to
integrate to 1.

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When we condition on an event
and without any further

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assumption, there's not much we
can say about the form of

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the conditional PDF.

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However, if we condition on an
event of a special kind, that

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X takes values in a certain
set, then we can actually

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write down a formula.

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So let us start with a random
variable X that has a given

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PDF, as in this diagram.

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And suppose that A is a subset
of the real line, for example,

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this subset here.

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What is the form of the
conditional PDF?

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We start with the interpretation
of PDFs and

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conditional PDFs in terms of

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probabilities of small intervals.

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The probability that X lies in
a small interval is equal to

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the value of the PDF somewhere
in that interval times the

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length of the interval.

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And if we're dealing with
conditional probabilities,

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then we use the corresponding
conditional PDF.

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To find the form of the
conditional PDF, we will work

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in terms of the left-hand side
in this equation and try to

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rewrite it.

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Let us distinguish two cases.

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Suppose that little X lies
somewhere out here, and we

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want to evaluate the conditional
PDF at that point.

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So trying to evaluate this
expression, we consider a

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small interval from little
x to little x plus delta.

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And now, let us write the
definition of a conditional

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probability.

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A conditional probability, by
definition, is equal to the

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probability that both events
occur divided by the

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probability of the conditioning
event.

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Now, because the set A and
this little interval are

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disjoint, these two events
cannot occur simultaneously.

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So the numerator here
is going to be 0.

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And this will imply that
the conditional PDF is

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also going to be 0.

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This, of course, makes sense.

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Conditioned on the event that
X took values in this set,

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values of X out here
cannot occur.

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And therefore, the conditional
density out here

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should also be 0.

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So the conditional PDF is 0
outside the set A. And this

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takes care of one case.

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Now, the second case to consider
is when little x lies

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somewhere inside here inside the
set A. And in that case,

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our little interval from little
x to little x plus

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delta might have this form.

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In this case, the intersection
of these two events, that X

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lies in the big set and X lies
in the small set, the

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intersection of these two events
is the event that X

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lies in the small set.

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So the numerator simplifies just
to the probability that

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the random variable X takes
values in the interval from

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little x to little
x plus delta.

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And then we rewrite
the denominator.

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Now, the numerator is just an
ordinary probability that the

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random variable takes values
inside a small interval.

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And by our interpretation of
PDFs, this is approximately

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equal to the PDF evaluated
somewhere in that small

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interval times delta.

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At this point, we notice that
we have deltas on both sides

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of this equation.

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By cancelling this delta with
that delta, we finally end up

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with a relation that the
conditional PDF should be

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equal to this expression
that we have here.

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So to summarize, we have
shown a formula for

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the conditional PDF.

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The conditional PDF is 0 for
those values of X that cannot

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occur given the information that
we are given, namely that

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X takes values at
that interval.

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But inside this interval, the
conditional PDF has a form

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which is proportional to
the unconditional PDF.

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But it is scaled by a
certain constant.

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So in terms of a picture,
we might have

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something like this.

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And so this green diagram
is the form of

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the conditional PDF.

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The particular factor that we
have here in the denominator

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is exactly that factor that is
required, the scaling factor

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that is required so that the
total area under the green

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curve, under the conditional
PDF is equal to 1.

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So we see once more the
familiar theme, that

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conditional probabilities
maintain the same relative

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sizes as the unconditional
probabilities.

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And the same is true for
conditional PMFs or PDFs,

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keeping the same shape as the
unconditional ones, except

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that they are re-scaled so that
the total probability

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under a conditional
PDF is equal to 1.

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We can now continue the same
story and revisit everything

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else that we had done for
discrete random variables.

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For example, we have the
expectation of a discrete

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random variable and the
corresponding conditional

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expectation, which is just the
same kind of object, except

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that we now rely on conditional
probabilities.

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Similarly, we can take the
definition of the expectation

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for the continuous case and
define a conditional

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expectation in the same manner,
except that we now

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rely on the conditional PDF.

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So this formula here is the
definition of the conditional

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expectation of a continuous
random variable given a

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particular event.

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We have a similar situation with
the expected value rule,

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which we have already seen for
discrete random variables in

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both of the unconditional and
in the conditional setting.

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We have a similar formula
for the continuous case.

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And at this point, you can
guess the form that the

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formula will take in the

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continuous conditional setting.

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This is the expected value
rule in the conditional

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setting, and it is proved
exactly the same way as for

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the unconditional continuous
setting, except that here in

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the proof, we need to work with
conditional probabilities

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and conditional PDFs, instead
of the unconditional ones.

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So to summarize, there is
nothing really different when

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we condition on an event in the
continuous case compared

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to the discrete case.

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We just replace summations
with integrations.

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And we replace PMFs by PDFs.