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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.