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In this segment, we consider
the sum of independent Poisson

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random variables, and we
establish a remarkable fact,

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namely that the sum
is also Poisson.

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This is a fact that
we can establish

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by using the
convolution formula.

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The PMF of the sum of
independent random variables

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is the convolution
of their PMFs.

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So we can take two Poisson
PMFs, convolve them, carry out

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the algebra, and find
out that in the end,

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you obtain again a Poisson PMF.

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However, such a derivation
is completely unintuitive,

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and does not give
you any insight.

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Instead, we will
derive this fact

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by using our
intuition about what

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Poisson random variables
really represent.

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We will work with
a Poisson process

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of rate lambda equal to 1.

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But let us remind ourselves
of the general Poisson PMF

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if we have a more
general rate lambda.

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This is the PMF of
the number of arrivals

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in a Poisson process with
rate lambda during a time

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interval of length tau.

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And this Poisson PMF has a
mean equal to lambda tau.

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And you can think of
lambda tau as being

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the parameter of
this Poisson PMF.

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So we say that this is a
Poisson PMF with parameter

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equal to lambda times tau.

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Now, let us consider two
consecutive time intervals

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in this processes that
have length mu and nu.

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And let us consider
the numbers of arrivals

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during each one of
these intervals.

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So we have M arrivals
here and N arrivals there.

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Of course, M and N
are random variables.

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What kind of random
variables are they?

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Well, the number of arrivals in
the Poisson process of rate 1,

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over a period of
duration mu is going

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to have a Poisson PMF in
which lambda is one, tau,

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the time interval
is equal to mu,

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so it's going to be a Poisson
random variable with parameter,

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or mean, equal to mu.

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Similarly for N, it's going to
be a Poisson random variable

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with parameter equal to nu.

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Are these two random
variables independent?

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Of course they are.

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In a Poisson
process, the numbers

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of arrivals in
disjoint time intervals

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are independent
random variables.

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What kind of random
variable is their sum?

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Their sum is the total
number of arrivals

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during an interval
of length mu plus nu,

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and therefore this is a
Poisson random variable

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with mean equal to mu plus nu.

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So, what do we have here?

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We have the sum of two
independent Poisson random

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variables, and that
sum turns out also

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to be a Poisson random variable.

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More generally, if
somebody gives you

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two independent Poisson
random variables,

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you can always think of
them as representing numbers

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of arrivals in disjoint
time intervals,

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and therefore by
following this argument,

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their sum is going to be
a Poisson random variable.

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And this is the conclusion
that we wanted to establish.

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It's a remarkable fact.

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It's similar to the fact
that we had established

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for normal random variables.

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The sum of independent
normal random variables

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is also normal, so Poisson
and normal distributions

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are special in this respect.

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This is a property that most
other distributions do not

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have, with very few exceptions.