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In this segment, we discuss
splitting a Poisson process

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into two different streams.

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So we have a Poisson process
with some arrival rate lambda,

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and whenever there
is an arrival,

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we decide to send it either
to one or another stream.

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And for example, this arrival
might be sent to that stream,

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this arrival might be
sent to this stream,

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this arrival might be
sent to this stream,

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that arrival might be
sent to that stream.

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How do we decide where
to send these arrivals?

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We make those decisions
using independent coin

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flips with a coin that has a
certain fixed bias equal to q.

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So when this arrival
comes, there's

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probability q that it will
be sent that direction

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or probability 1
minus q that it will

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be sent in the other direction.

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All of these coin
flips are independent,

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so the decision on where
to send the second arrival

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is going to be independent
from the decision of where

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to send the first arrival.

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And furthermore, we
make one more assumption

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that these coin flips are
independent from everything

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else.

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They're independent
from the time history

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of the original Poisson process.

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For example, the coin flip
that decides the destination

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of this first arrival
can not depend

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on how long it took for
this arrival to occur.

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We will argue now that this
stream is a Poisson process

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and by symmetry,
therefore, this stream

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is also a Poisson process.

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We need to verify
two assumptions.

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One has to do with independence.

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The argument here is entirely
analogous to arguments

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that we have already
carried out in the past,

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namely disjoint time intervals
in the original process

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are independent.

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Coin flips that happened during
those disjoint time intervals

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are also independent
of each other,

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and for this reason, whatever
happens during disjoint time

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intervals in that stream
will also be independent.

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The other property
that we need to verify

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has to do with probabilities
of small intervals.

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If we take a little interval
here of length delta,

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what is going to happen
during that interval?

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Well, we look at what happens
during the corresponding

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interval in the original stream.

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In the original
stream, the probability

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of having two or more arrivals--
this probability is order

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of delta squared,
so there's no way

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of having two or more arrivals
during that little interval,

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or to be more precise,
the probability

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of two or more arrivals here
is going to be negligible,

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order of delta squared.

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What is the probability
of having one arrival

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during that little interval?

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We will have one
arrival here if we've

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had one arrival in this
time interval, which

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happens with probability
lambda times delta,

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and also the coin
flip sent the arrival

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in this direction,
which is something

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that happens with probability
q, and the remaining probability

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is assigned to the
event of having

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zero arrivals during
that interval.

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So the probability of two or
more arrivals is negligible,

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and the probability
of one arrival

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is proportional to delta.

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And that's what we need in
order to have a Poisson process.

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The factor of proportionality
that multiplies delta

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is equal to lambda times q.

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Therefore, this is a Poisson
process with parameter,

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or arrival rate, equal
to lambda times q.

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And by a similar argument,
this process here

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is going to be a Poisson
process with parameter

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equal to lambda times 1 minus q.

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So by splitting a Poisson
process using independent coin

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flips, we obtain two
different Poisson streams.

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Are these Poisson
streams independent?

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For the case of the
Bernoulli process,

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we had seen that the resulting
streams were not independent.

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The reason for the
Bernoulli process

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was that if I tell you
that at a certain slot

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we had an arrival
in this stream,

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that would tell you that
in the corresponding time

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slot of the other process
you could not have an arrival

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and that was a
source of dependence.

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It turns out that for
the Poisson process,

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because it runs in
continuous time,

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telling you that we had an
arrival at this particular time

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instant does not give you
any substantial or any

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nontrivial information
about the other process,

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and the two processes
remain independent.

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This result is surprising in
some ways, but it is true.

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A mathematical
derivation proceeds

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along a line that's a little
different from the intuitive

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argument that I just outlined.

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And we will not go
through that derivation,

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but it's a useful fact to know.