WEBVTT
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In this segment we will go
very fast through a few
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definitions and facts
that remain true in
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the continuous case.
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Everything is completely
analogous to
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the discrete case.
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And there are absolutely
no surprises here.
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So, for example, we have defined
joint PMFs for the
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case of more than two discrete
random variables.
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And we have a bunch of
facts about them.
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In a similar manor, we can
define joint PDFs for more
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than two random variables.
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And if you have understood the
material so far, you can guess
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how such a joint PDF
will be used.
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For example, you can calculate
the probability of a three
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dimensional set by integrating
the joint PDF over that three
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dimensional set.
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And there are analogs off all of
the other formulas that we
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have here where we follow
the usual recipe.
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Sums become integrals, and PMFs
are replaced by PDFs.
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Finally, when you deal with a
random variable, which is
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defined as a function of jointly
continuous random
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variables, we can use an
expected value rule that takes
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the same form as in
the discrete case.
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And using the expected value
rule, we can establish, once
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more, the usual linearity
properties of expectations.
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So absolutely no
surprises here.
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The derivations are either
completely straightforward.
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Or they follow exactly the same
line of argument as in
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the discrete case, with just
minor changes in notation.