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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.