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We will now study
what happens when

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we merge independent
Poisson processes.

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The story as well as
the final conclusion

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is going to be similar to
what happened for the case

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where we merged independent
Bernoulli processes.

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In particular, we will see that
the merged process will also

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be Poisson.

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What is the story?

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Suppose that you
have two light bulbs.

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One of them is red and
flashes at random times that

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are described according
to a Poisson process

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with a certain rate, lambda1.

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The other light bulb is green,
flashes also as a Poisson

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process with a certain rate.

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Furthermore, we assume
that the two light bulbs

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are independent of each other.

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If you're color blind so that
the only thing that you see

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is flashes but without being
able to tell the color,

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what kind of process
are you going to see?

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Well, you will see a
Poisson process also,

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and at this point,
you can probably

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guess what the
arrival rate is going

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to be for this Poisson process.

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It should be the sum
of these two arrival

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rates of the processes
that you started with.

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So this will be our
final conclusion,

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but we want to verify
that this is indeed

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the correct conclusion.

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So let us look at the
situation in some more detail.

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We have the two processes, two
arrival processes-- the red one

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and the green one--
and the merged process

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is formed by recording an
arrival at any time where

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either of the two
processes that you

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started with records an arrival.

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Let us now look at
the time interval

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and think about the number of
arrivals in the merged process

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during this time interval.

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What is that the number?

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That number is equal to
the number of arrivals

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that you have in the
first process plus

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the number of arrivals that
you have in the second process.

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Let's call those numbers N1 and
N2 so that what we have here

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is N1 plus N2.

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Now, N1 is a Poisson
random variable

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because this is a
Poisson process.

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Similarly, N2 is a
Poisson random variable.

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We assume that these two
processes are independent.

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Therefore, N1 plus N2 is the
sum of independent Poisson

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random variables, and
therefore, N1 plus N2

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is also a Poisson
random variable.

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This is reassuring.

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It's a good piece of evidence
that the blue process

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is a Poisson process,
but this is not enough.

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To argue that it is
a Poisson process,

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we need to check the defining
properties of a Poisson

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process.

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One defining property is
the independence property.

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If we take disjoint intervals,
the number of arrivals

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here is independent, or
should be independent,

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from the number
of arrivals there.

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The argument here is exactly the
same as for the Bernoulli case,

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so we will not go
through it in any detail.

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We just notice that whatever
happens during that time

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has to do with whatever happens
during those times in the two

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processes that we started with.

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And similarly, what
happens in these times

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has to do with what happens
in these two processes

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during those times.

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Because for each one of the two
processes that we start with,

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we have the Poisson assumption
so that this interval

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is independent from that
interval in the sense

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that arrivals here and
arrivals there are independent.

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So because of this, whatever
happens during those

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times has nothing
to do with whatever

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happens in those times,
so number of arrivals

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here is independent from the
number of arrivals there.

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The other property
that we need to check

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is that the probability
of recording an arrival

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during a small time
interval of length delta,

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that this probability
has the right scaling

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properties, that it
is linear in delta,

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in the length of this interval,
and that the probability of two

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or more arrivals
here is negligible.

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To see what happens during a
typical interval in the merged

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process, we need
to consider what

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is going to happen during
that typical interval

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in the other two
processes and consider

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all the possible combinations.

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During a little
interval, the red process

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is going to have zero arrivals
with this probability,

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one arrival with
this probability,

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and two or more arrivals
with this probability, which

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is negligible.

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Actually here, we're ignoring
terms of order delta squared.

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These are the
correct expressions

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if we only focus on terms that
are either constants or linear

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in delta.

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We are ignoring terms
that are of order delta

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square or higher.

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And similarly for
the green process,

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we have these probabilities
for the number of arrivals

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that may happen during
a small interval.

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For the merged process, we will
have zero arrivals if and only

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if we have zero arrivals
in the red process

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and zero arrivals in
the green process.

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The probability of these
two events happening,

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because we assume that the two
processes that we started with

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are independent, is
going to be the product

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of the probabilities of zero
arrivals in one process times

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zero arrivals in
the other process.

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We multiply those two
terms, and if we throw away

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the term delta squared,
which is negligible,

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we see that this event is going
to happen with probability 1

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minus lambda1 plus
lambda2 times delta.

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What's the probability
that we get one arrival?

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This is an event that
can happen in two ways.

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We could have one arrival
in the red process

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and zero arrivals in
the green process,

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and this combination happens
with this probability.

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Alternatively, we could have
one arrival in the green process

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and zero arrivals
in the red process.

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This is this event and it
happens with this probability.

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Having one arrival
in the blue process

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can happen either
this way or that way,

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so the probability
of one arrival

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will be the sum of
these two probabilities.

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And if we throw
away terms that are

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order of delta squared,
what we're left with

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is just lambda1 plus
lambda2 times delta.

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Finally, there's the possibility
that the blue process

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is going to have two
or more arrivals.

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This happens if we have one
red and one green arrival,

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which happens with
this probability,

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or if anyone of the processes
has two or more arrivals, which

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would be terms here, here, and
these would be the scenarios.

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But we notice that each
one of these scenarios

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has probability that's
order of delta squared.

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This term also has probability
of order delta squared,

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so overall, the possibility
that the blue process has

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two or more arrivals-- this is
something that has probability

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that's of order delta squared.

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So during a typical
small interval,

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there is probability
proportional

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to the length of the interval
of having one arrival,

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and lambda1 plus lambda2 is the
factor of this proportionality,

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and the remaining probability
is assigned to the event

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that there are zero arrivals.

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There's essentially
negligible probability

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of having two or more
arrivals, but this

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together with the
independence assumption

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is exactly what comes in the
definition of a Poisson process

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with an arrival rate
equal to this number.

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And so we have established
that the merged process

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is a Poisson process
whose rate is

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the sum of the rates of the
processes that we started from.