WEBVTT
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An important reason why
conditional probabilities are
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very useful is that they allow
us to divide and conquer.
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They allow us to split
complicated probability modes
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into simpler submodels
that we can then
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analyze one at a time.
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Let me remind you of the Total
Probability Theorem that has
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his particular flavor.
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We divide our sample space into
three disjoint events--
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A1, A2, and A3.
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And these events form a
partition of the sample space,
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that is, they exhaust
all possibilities.
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They correspond to three
alternative scenarios, one of
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which is going to occur.
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And then we may be interested
in a certain event B. That
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event B may occur under
either scenario.
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And the Total Probability
Theorem tells us that we can
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calculate the probability of
event B by considering the
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probability that it occurs under
any given scenario and
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weigh those probabilities
according to the probabilities
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of the different scenarios.
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Now, let us bring random
variables into the picture.
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Let us fix a particular
value--
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little x--
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and let the event B be the event
that the random variable
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takes on this particular
value.
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Let us now translate the
Total Probability
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Theorem to this situation.
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First, the picture will look
slightly different.
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Our event B has been replaced
by the particular event that
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we're now considering.
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Now, what is this probability?
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The probability that event B
occurs, having fixed the
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particular choice of little x,
is the value of PMF at that
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particular x.
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How about this probability
here?
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This is the probability that the
random variable, capital
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X, takes on the value
little x--
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that's what a PMF is--
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but in the conditional
universe.
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So we're dealing with
a conditional PMF.
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And so on with the
other terms.
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So this equation here is just
the usual Total Probability
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Theorem but translated
into PMF notation.
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Now this version of the Total
Probability Theorem, of
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course, is true for all
values of little x.
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This means that we can now
multiply both sides of this
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equation by x and them
sum over all
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possibles choices of x.
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We recognize that here we have
the expected value of the
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random variable X.
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Now, we do the same thing
to the right hand side.
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We multiply by x.
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And then we sum over all
possible values of x.
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This is going to be
the first term.
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And then we will have
similar terms.
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Now, what do we have here?
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This expression is just the
conditional expectation of the
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random variable X under
the scenario that
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event A1 has occurred.
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So what we have established is
this particular formula, which
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is called the Total Expectation
Theorem.
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It tells us that the expected
value of a random variable can
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be calculated by considering
different scenarios.
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Finding the expected value under
each of the possible
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scenarios and weigh them.
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Weigh the scenarios according
to their respective
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probabilities.
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The picture is like this.
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Under each scenario, the
random variable X has a
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certain conditional
expectation.
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We take all these
into account.
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We weigh them according
to their corresponding
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probabilities.
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And we add them up to find
the expected value of X.
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So we can divide and conquer.
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We can replace a possibly
complicated calculation of an
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expected value by hopefully
simpler calculations under
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each one of possible
scenarios.
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Let me illustrate the idea
by a simple example.
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Let us consider this PMF, and
let us try to calculate the
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expected value of the associated
random variable.
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One way to divide and conquer
is to define an event, A1,
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which is that our random
variable takes values in this
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set, and another event, A2,
which is that the random
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variable takes values
in that set.
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Let us now apply the Total
Expectations Theorem.
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Let us calculate all the terms
that are required.
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First, we find the
probabilities of
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the different scenarios.
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The probability of event
A1 is 1/9 plus 1/9 plus
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1/9 which is 1/3.
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And the probability of event
A2 is 2/9 plus 2/9 plus 2/9
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which adds up to 2/3.
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How about conditional
expectations?
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In a universe where event A1 one
has occurred, only these
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three values are possible.
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They had equal probabilities,
so in the conditional model,
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they will also have equal
probabilities.
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So we will have a uniform
distribution over
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the set {0, 1, 2}.
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By symmetry, the expected
value is going
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to be in the middle.
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So this expected value
is equal to 1.
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And by a similar argument, the
expected value of X under the
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second scenario is going to be
the midpoint of this range,
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which is equal to 7.
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And now we can apply the Total
Probability Theorem and write
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that the expected value of X is
equal to the probability of
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the first scenario times the
expected value under the first
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scenario plus the probability
of the second scenario times
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the expected value under
the second scenario.
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In this case, by breaking down
the problem in these two
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subcases, the calculations
that were required were
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somewhat simpler than if you
were to proceed directly.
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Of course, this is a rather
simple example.
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But as we go on with this
course, we will apply the
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Total Probability Theorem in
much more interesting and
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complicated situations.