In this segment, we discuss splitting a Poisson process
into two different streams.
So we have a Poisson process with some arrival rate lambda,
and whenever there is an arrival,
we decide to send it either to one or another stream.
And for example, this arrival might be sent to that stream,
this arrival might be sent to this stream,
that arrival might be sent to that stream.
How do we decide where to send these arrivals?
We make those decisions using independent coin
flips with a coin that has a certain fixed bias equal to q.
So when this arrival comes, there's
probability q that it will be sent that direction
or probability 1 minus q that it will
be sent in the other direction.
All of these coin flips are independent,
so the decision on where to send the second arrival
is going to be independent from the decision of where
to send the first arrival.
And furthermore, we make one more assumption
that these coin flips are independent from everything
else.
They're independent from the time history
of the original Poisson process.
For example, the coin flip that decides the destination
of this first arrival can not depend
on how long it took for this arrival to occur.
We will argue now that this stream is a Poisson process
and by symmetry, therefore, this stream
is also a Poisson process.
We need to verify two assumptions.
One has to do with independence.
The argument here is entirely analogous to arguments
that we have already carried out in the past,
namely disjoint time intervals in the original process
are independent.
Coin flips that happened during those disjoint time intervals
are also independent of each other,
and for this reason, whatever happens during disjoint time
intervals in that stream will also be independent.
The other property that we need to verify
has to do with probabilities of small intervals.
If we take a little interval here of length delta,
what is going to happen during that interval?
Well, we look at what happens during the corresponding
interval in the original stream.
In the original stream, the probability
of having two or more arrivals-- this probability is order
of delta squared, so there's no way
of having two or more arrivals during that little interval,
or to be more precise, the probability
of two or more arrivals here is going to be negligible,
order of delta squared.
What is the probability of having one arrival
during that little interval?
We will have one arrival here if we've
had one arrival in this time interval, which
happens with probability lambda times delta,
and also the coin flip sent the arrival
in this direction, which is something
that happens with probability q, and the remaining probability
is assigned to the event of having
zero arrivals during that interval.
So the probability of two or more arrivals is negligible,
and the probability of one arrival
is proportional to delta.
And that's what we need in order to have a Poisson process.
The factor of proportionality that multiplies delta
is equal to lambda times q.
Therefore, this is a Poisson process with parameter,
or arrival rate, equal to lambda times q.
And by a similar argument, this process here
is going to be a Poisson process with parameter
equal to lambda times 1 minus q.
So by splitting a Poisson process using independent coin
flips, we obtain two different Poisson streams.
Are these Poisson streams independent?
For the case of the Bernoulli process,
we had seen that the resulting streams were not independent.
The reason for the Bernoulli process
was that if I tell you that at a certain slot
we had an arrival in this stream,
that would tell you that in the corresponding time
slot of the other process you could not have an arrival
and that was a source of dependence.
It turns out that for the Poisson process,
because it runs in continuous time,
telling you that we had an arrival at this particular time
instant does not give you any substantial or any
nontrivial information about the other process,
and the two processes remain independent.
This result is surprising in some ways, but it is true.
A mathematical derivation proceeds
along a line that's a little different from the intuitive
argument that I just outlined.
And we will not go through that derivation,
but it's a useful fact to know.