WEBVTT
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We now come to a very important
concept, the concept
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of independence of
random variables.
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We are already familiar with the
notion of independence of
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two events.
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We have the mathematical
definition, and the
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interpretation is that
conditional probabilities are
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the same as unconditional
ones.
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Intuitively, when you are told
that B occurred, this does not
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change your beliefs about A,
and so the conditional
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probability of A is the same
as the unconditional
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probability.
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We have a similar definition
of independence of a random
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variable and an event A. The
mathematical definition is
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that event A and the event that
X takes on a specific
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value, that these two events
are independent in the
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ordinary sense.
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So the probability of both of
these events happening is the
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product of their individual
probabilities.
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But we require this to be true
for all values of little x.
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Intuitively, if I tell you that
A occurred, this is not
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going to change the distribution
of the random
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variable x.
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This is one interpretation of
what independence means in
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this context.
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And this has to be true for all
values of little x, that
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is, when [the]
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event occurs, the probabilities
of any
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particular little x [are]
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going to be the same as the
original unconditional
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probabilities.
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We also have a symmetrical
interpretation.
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If I tell you the value of
X, then the conditional
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probability of event A is
not going to change.
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It's going to be the same as the
unconditional probability.
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And again, this is going to be
the case for all values of X.
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So, no matter what they tell
you about X, your beliefs
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about A are not going
to change.
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We can now move and define the
notion of independence of two
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random variables.
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The mathematical definition is
that the event that X takes on
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a value little x and the event
that Y takes on a value little
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y, these two events are
independent, and this is true
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for all possible values of
little x and little y.
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In PMF notation, this
relation here can be
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written in this form.
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And basically, the joint PMF
factors out as a product of
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the marginal PMFs of the
two random variables.
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Again, this relation has to be
true for all possible little x
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and little y.
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What does independence mean?
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When I tell you the value of y,
and no matter what value I
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tell you, your beliefs about
X will not change.
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So that the conditional PMF of
X given Y is going to be the
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same as the unconditional PMF of
X. And this has to be true
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for any values of the arguments
of these PMFs.
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There is also a symmetric
interpretation, which is that
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the conditional PMF of Y given
X is going to be the same as
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the unconditional PMF of Y.
We have the symmetric
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interpretation because, as we
can see from this definition,
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X and Y have symmetric roles.
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Finally, we can define the
notion of independence of
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multiple random variables
by a similar relation.
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Here, the definition is for
the case of three random
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variables, but you can imagine
how the definition for any
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finite number of random
variables will go.
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Namely, the joint PMF of all
the random variables can be
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expressed as the product of the
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corresponding marginal PMFs.
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What is the intuitive
interpretation of
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independence here?
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It means that information
about some of the random
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variables will not change your
beliefs, the probabilities,
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about the remaining
random variables.
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Any conditional probabilities
and any conditional PMFs will
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be the same as the unconditional
ones.
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In the real world, independence
models situations
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where each of the random
variables is generated in a
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decoupled manner, in a separate
probabilistic
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experiment.
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And these probabilistic
experiments do not interact
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with each other and have no
common sources of uncertainty.