WEBVTT
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Let us now consider
a simple example.
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Let random variables X and Y
be described by a joint PMF
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which is the one shown
in this table.
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Question--
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are X and Y independent?
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We can try to answer this
question by using the
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definition of independence.
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But it is actually more
instructive to proceed in a
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somewhat more intuitive way.
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We look at this table, and we
observe that the value of one
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is possible for X. In
particular, the probability
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that X takes the value of one,
this is the marginal
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probability, this can be found
by adding the probabilities of
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all of the outcomes in this
column, which, in this case,
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is 3 over 20.
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Suppose now that somebody tells
you the value of Y. For
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example, I tell you that Y
happens to be equal to one, in
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which case you are transported
into this universe.
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In this universe, the
conditional probability that X
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takes a value of one, given that
Y takes a value of one,
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what is it?
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In this universe, there's
zero probability
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associated to this outcome.
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So this probability
is zero, which is
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different than 3 over 20.
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And since these two numbers are
different, this means that
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information from Y changes our
beliefs about what's going to
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happen to X. And so, we do
not have independence.
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So again, intuitively, in the
beginning, we thought that X
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equal to one was possible.
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But information given by Y,
namely that Y is equal to one,
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tells us that, actually, X
equals to one is impossible.
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Information about Y changed our
beliefs about X, so X and
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Y are dependent.
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Now, when we first introduced
the notion of independence
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some time ago, we also
introduced the notion of
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conditional independence.
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And we said that conditional
independence is the same as
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ordinary independence, except
that it would be applied to a
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conditional universe.
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Something similar can be done
for the case of random
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variables as well.
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So suppose, for example, that
someone tells us that the
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outcome of the experiment
was such that it belongs
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to this blue set.
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This is the set where X is less
than or equal to 2, and Y
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is larger than or
equal to three.
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So we're given this information,
and this is now
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our new conditional model.
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The question is, within this
new conditional model are
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random variables X and
Y independent?
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Let's just right down the
conditional model, where I'm
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only showing the four possible
outcomes that are allowed in
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the conditional model.
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All the others, of course, will
have zero probability in
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the conditional model.
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So in the conditional model,
probabilities will keep the
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same proportions as in the
unconditional model--
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and the proportions
are 1, 2, 2, 4--
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but then they need to be scaled,
or normalized, so that
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they add to 1.
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And to make them add to 1, we
need to divide them all by 9.
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In this conditional model, this
is the joint PMF of the
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two random variables X and Y.
Let us find the marginal PMFs.
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To find the marginal PMF
of X, we add the
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entries in this column.
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And we get here 1/3,
and here 2/3.
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And to find the marginal
PMF of y, we add the
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entries in this [row]
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to find 2/3.
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And we adds the entries
in that [row]
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to find 1/3.
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So this is the marginal
PMF of x.
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That's the marginal PMF of Y.
And now we notice that this
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entry of the joint PMF
is 1/3 times 1/3, the
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product of the marginals.
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This entry is the product of 1/3
times 2/3, the product of
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the marginals, and so on for
the remaining entries.
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So each entry of the joint PMF
is equal to the product of the
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corresponding entries of
the marginal PMFs.
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And this is the definition
of independence.
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So in this conditional blue
universe, we do have
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independence.
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And the way that this was
established was to check that
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the joint PMF factors as a
product of marginal PMFs.