WEBVTT
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We now continue with the
development of continuous
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analogs of everything we know
for the discrete case.
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We have already seen a few
versions of the total
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probability theorem, one version
for events and one
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version for PMFs.
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Let us now develop a
continuous analog.
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Suppose, as always, that we have
a partition of the sample
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space into a number of
disjoint scenarios.
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Three scenarios in
this picture.
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More generally, n scenarios
in these formulas.
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Let X be a continuous random
variable and let us take B to
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be the event that the random
variable takes a value less
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than or equal to
some little x.
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By the total probability
theorem, this is the
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probability of the first
scenario times the conditional
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probability of this event given
that the first scenario
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has materialized, and then we
have similar terms for the
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other scenarios.
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Let us now turn this equation
into CDF notation.
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The left-hand side is what we
have defined as the CDF of the
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random variable x.
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On the right-hand side, what we
have is the probability of
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the first scenario multiplied,
again, by a CDF of the random
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variable X. But it is a CDF that
applies in a conditional
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model where event
A1 has occurred.
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And so we use this notation to
denote the conditional CDF,
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the CDF that applies to the
conditional universe.
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And then we have similar terms
for the other scenarios.
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Now, we know that the derivative
of a CDF is a PDF.
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We also know that any general
fact, such as this one that
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applies to unconditional models
will also apply without
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change to a conditional model,
because a conditional model is
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just like any other ordinary
probability model.
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So let us now take derivatives
of both
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sides of this equation.
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On the left-hand side, we
have the derivative of a
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CDF, which is a PDF.
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And on the right-hand side, we
have the probability of the
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first scenario, and then the
derivative of the conditional
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CDF, which has to be the same
as the conditional PDF.
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So we use here the fact that
derivatives of CDFs are PDFs,
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and then we have similar terms
under the different scenarios.
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So we now have a relation
between densities.
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To interpret this relation,
we think as follows.
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The probability of falling
inside the little interval
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around x is determined by the
probability of falling inside
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that little interval under
each one of the different
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scenarios and where each
scenario is weighted by the
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corresponding probability.
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Now, we multiply both sides of
this equation by x, and then
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integrate over all x's.
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We do this on the
left-hand side.
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And similarly, on the right-hand
side to obtain a
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term of this form.
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And we have similar terms
corresponding
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to the other scenarios.
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What do we have here?
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On the left-hand side, we have
the expected value of x.
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On the right-hand side, we
have this probability
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multiplied by the conditional
expectation of X given that
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scenario A1 has occurred.
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And so we obtain a version of
the total expectation theorem.
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It's exactly the same formula
as we had in the discrete
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case, except that now X is a
continuous random variable.
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Let us now look at a simple
example that involves a model
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with different scenarios.
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Bill wakes up in the morning
and wants to go to the
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supermarket.
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There are two scenarios.
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With probability one third,
a first scenario occurs.
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And under that scenario, Bill
will go at a time that's
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uniformly distributed between
0 and 2 hours from now.
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So the conditional PDF of X,
in this case, is uniform on
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the interval from 0 to 2.
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There's a second scenario that
Bill will take long nap and
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will go later in the day.
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That scenario has a probability
of 2/3.
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And under that case, the
conditional PDF of X is going
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to be uniform on the range
between 6 and 8.
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By the total probability theorem
for densities, the
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density of X, of the
random variable--
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the time at which he goes
to the supermarket--
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consists of two pieces.
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One piece is a uniform
between 0 and 2.
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This uniform ordinarily would
have a height or 1/2.
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On the other hand, it gets
weighted by the corresponding
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probability, which is 1/3.
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So we obtain a piece here that
has a height of 1/6.
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Under the alternative scenario,
the conditional
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density is a uniform on the
interval between 6 and 8.
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This uniform has a height of
1/2 again, but it gets
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multiplied by a factor of 2/3.
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And this results in a height
for this term that we have
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here, which is 1/3.
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And this is the form of the PDF
of the time at which Bill
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will go to the supermarket.
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We can now finally use the total
expectation theorem.
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The conditional expectation
under the two scenarios can be
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found as follows.
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Under one scenario, we have
a uniform between 0 and 2.
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And so the conditional
expectation is 1, and it gets
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weighted by the corresponding
probability, which is 1/3.
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Under the second scenario, which
has probability 2/3, the
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conditional expectation is the
midpoint of this uniform,
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which is 7.
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And this gives us the expected
value of the
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time at which he goes.
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So this is a simple example, but
it illustrates nicely how
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we can construct a model
that involves a number
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of different scenarios.
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And by knowing the probability
distribution under each one of
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the scenarios, we can
find the probability
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distribution overall.
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And we can also find the
expected value for the overall
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experiment.