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Independence is one of the
central concepts of

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probability theory, because it
allows us to build large

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models from simpler ones.

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How should we define
independence in

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the continuous case?

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Our guide comes from the
discrete definition.

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By analogy with the discrete
case, we will say that two

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jointly continuous random
variables are independent if

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the joint PDF is equal to the
product of the marginal PDFs.

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We can now compare with the
multiplication rule, which is

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always true as long as the
density of Y is positive.

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So this is always true.

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In the case of independence,
this is true.

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So in the case of independence,
we must have

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that this term is equal to that
term, at least whenever

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this quantity--

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the marginal of Y--

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is positive.

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So to restate it, independence
is equivalent to having the

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conditional, given Y, be the
same as the unconditional PDF

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of X. And this has to be true
whenever Y has a positive

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density so that this quantity
is well defined, and it also

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has to be true for all xs.

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Now, what does this
really mean?

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The conditional PDF, as we have
discussed, in terms of

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pictures, is a slice
of the joint PDF.

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Therefore, independence is the
same as requiring that all of

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the slices of the joint have the
same shape, and it is the

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shape of the marginal PDF.

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For a more intuitive
interpretation, no matter what

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value of Y you observe,
the distribution

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of X does not change.

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In this sense, Y does not convey
any information about

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X. Notice also that this
definition is symmetric as far

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as X and Y are concerned.

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So by symmetry, when we have
independence, it also means

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that X does not convey any
information about Y, and that

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the conditional density of Y,
given X, has to be the same as

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the unconditional
density of Y.

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We can also define independence
of multiple

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random variables.

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The definition is
the obvious one.

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The joint PDF of all the random
variables involved must

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be equal to the product
of the marginal PDFs.

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Intuitively, what that means is
that knowing the values of

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some of the random variables
does not affect our beliefs

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about the remaining
random variables.

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Finally, let us note some
consequences of independence,

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which are identical to the
corresponding properties that

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we had in the discrete case,
and the proofs are also

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exactly the same.

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So the expectation of the
product of independent random

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variables is the product of the
expectations, the variance

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of the sum of independent random
variables is the sum of

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the variances, and functions
of independent random

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variables are also independent,
which, in

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particular, implies, using the
previous rule, that the

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expected value of a product of
this kind is going to be the

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product of these expectations.

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So independence of continuous
random variables is pretty

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much the same as independence
of discrete random variables

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as far as mathematics are
concerned, and the intuitive

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content of the independence
assumption is the same as in

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the discrete case.

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One random variable does
not provide any

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information about the other.