In this segment, we consider the sum of independent Poisson
random variables, and we establish a remarkable fact,
namely that the sum is also Poisson.
This is a fact that we can establish
by using the convolution formula.
The PMF of the sum of independent random variables
is the convolution of their PMFs.
So we can take two Poisson PMFs, convolve them, carry out
the algebra, and find out that in the end,
you obtain again a Poisson PMF.
However, such a derivation is completely unintuitive,
and does not give you any insight.
Instead, we will derive this fact
by using our intuition about what
Poisson random variables really represent.
We will work with a Poisson process
of rate lambda equal to 1.
But let us remind ourselves of the general Poisson PMF
if we have a more general rate lambda.
This is the PMF of the number of arrivals
in a Poisson process with rate lambda during a time
interval of length tau.
And this Poisson PMF has a mean equal to lambda tau.
And you can think of lambda tau as being
the parameter of this Poisson PMF.
So we say that this is a Poisson PMF with parameter
equal to lambda times tau.
Now, let us consider two consecutive time intervals
in this processes that have length mu and nu.
And let us consider the numbers of arrivals
during each one of these intervals.
So we have M arrivals here and N arrivals there.
Of course, M and N are random variables.
What kind of random variables are they?
Well, the number of arrivals in the Poisson process of rate 1,
over a period of duration mu is going
to have a Poisson PMF in which lambda is one, tau,
the time interval is equal to mu,
so it's going to be a Poisson random variable with parameter,
or mean, equal to mu.
Similarly for N, it's going to be a Poisson random variable
with parameter equal to nu.
Are these two random variables independent?
Of course they are.
In a Poisson process, the numbers
of arrivals in disjoint time intervals
are independent random variables.
What kind of random variable is their sum?
Their sum is the total number of arrivals
during an interval of length mu plus nu,
and therefore this is a Poisson random variable
with mean equal to mu plus nu.
So, what do we have here?
We have the sum of two independent Poisson random
variables, and that sum turns out also
to be a Poisson random variable.
More generally, if somebody gives you
two independent Poisson random variables,
you can always think of them as representing numbers
of arrivals in disjoint time intervals,
and therefore by following this argument,
their sum is going to be a Poisson random variable.
And this is the conclusion that we wanted to establish.
It's a remarkable fact.
It's similar to the fact that we had established
for normal random variables.
The sum of independent normal random variables
is also normal, so Poisson and normal distributions
are special in this respect.
This is a property that most other distributions do not
have, with very few exceptions.