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In this lecture, we look in
some depth into various
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properties of Poisson
processes.
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These properties would be quite
hard to study if one
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were to proceed just
analytically by
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manipulating formulas.
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However, by using memorylessness
and our
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intuitive understanding of
what the Poisson process
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really is, they become
quite simple.
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And the mathematical
manipulations can be avoided
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almost entirely.
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We will start by arguing that
the sum of independent Poisson
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random variables is Poisson.
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But we will then establish the
much stronger statement that
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if we merge two independent
Poisson processes, we, again,
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obtain a Poisson process.
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We will see that we can exploit
this fact to solve
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some problems that would be
quite difficult otherwise.
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Once more, intuitive reasoning
is the key.
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Finally, we will spend some time
discussing a phenomenon
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that goes under the name
of random incidence.
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The Poisson process
has been running.
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You show up at a certain time.
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You look at the size of the
inter-arrival interval during
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which you show up.
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It turns out that this
inter-arrival interval that
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you get to observe is
not a typical one.
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It tends to be larger
than the typical
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inter-arrival interval.
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We will understand what exactly
is going on, build
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some intuition, and realize
that this is a general
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phenomenon that also shows up
in many other contexts.