WEBVTT
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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.