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In what kind of situations
does the Poisson process arise?

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In general, it
arises whenever we

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have events like arrivals
that are somewhat rare,

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and which happen in a
completely uncoordinated manner,

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so that they can show up
at any particular time.

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In such situations,
the number of arrivals

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will be often described by a
certain distribution called

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the Poisson
distribution, and which

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is named after the person who
first studied this situation,

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who is a famous
French mathematician

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by the name of
Simon Denis Poisson.

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An early example
where the data seems

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to fit the description
of the Poisson process

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is a curious one.

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It had to do with
deaths from horse

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kicks, that is, accidental
deaths, in the Prussian army.

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The idea here is that
a death by horse kick

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can happen pretty
much at any time.

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And different arrivals,
that is, different accidents

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are completely uncoordinated
from each other.

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So the process is sort
of completely random.

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For more scientific
applications,

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it was realized that
certain physical phenomena

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obey the Poisson process.

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Examples are the following.

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You have some radioactive
body which decays,

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and the decaying
happens once in awhile,

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emitting various particles.

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Different particles get emitted
at completely random times

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in a completely
uncoordinated manner

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and, therefore, this
process is actually

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described as a Poisson process.

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Conversely, if you have
a photo detector who

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looks at a very
weak light source.

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So photons arrive from that
weak light source one at a time.

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And you look at the time at
which photons hit the detector.

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Then, the process
of photon arrivals

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is very well-modeled
by the Poisson process.

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For more modern
applications, if you

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look at the financial markets
and the times at which

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certain very unexpected events,
like certain market shocks,

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occur, a model that
is commonly employed

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is to use a Poisson
process model.

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Although this is not an
entirely accurate model,

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it provides a first approach
to situations like this.

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But these days, the
predominant source

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of applications for
the Poisson process

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is in various
service operations.

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You are the phone company.

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Phone calls get placed
at random times.

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And because there are
several people involved

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who are uncoordinated
with each other,

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those calls get placed at
completely random times.

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And the same story
goes about, let's

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say, service requests
to a web server,

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service requests to
any kind of company.

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So, many applications that
are being studied these days

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and which rest on Poisson models
involve service operations

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of this type.