We will now consider an operation
that is, in some sense, the opposite of merging.
We have a node, and traffic arrives to that node.
And each time that we have an arrival, we flip a coin.
And with probability q, we send that arrival
to one particular stream.
And with probability 1 minus q, we
send the arrival to another stream.
So we get two streams that are formed
by taking the original stream and splitting it
into two pieces.
And as an example, these might be
arrivals at a department store.
And one stream corresponds to the people
who go to the clothes section of the department store,
whereas the other stream corresponds to the people that
go to all of the other sections of the store.
So let us now make our model a little more precise.
We have a Bernoulli process, which
is independent across time.
We also use an independent coin flip
to deal with each one of the arrivals.
But we will also make one additional assumption, namely
that the Bernoulli process is also
independent from the process of coin flips.
With this assumption in place, let us now
continue, and let us draw a picture.
We have a Bernoulli process with parameter p,
and arrivals get recorded at certain times.
Each time that there is an arrival, we will flip a coin.
And with probability q, the arrival
will be sent to that stream.
With probability 1 minus q, the arrival
will be sent to the other stream.
So one possible outcome of the experiment
might be this one, where these two arrivals were
sent to this stream and these two arrivals were
sent to the top stream.
And we have these probabilities q and 1 minus q
of sending the arrivals to one or the other stream.
What kind of process is this one?
We argue it is a Bernoulli process.
First, we need to check independence.
Here, the argument is more or less the same as in the case
when we studied the merging of processes.
For example, if we look at two different slots and we ask,
how is the event at that slot and at that slot determined?
Well, what happens in this slot is determined
by whether we had an arrival here
and what happened to the outcome of the coin flip at that time.
What happens in this slot is determined
by whether we had an arrival here
and what happened to the coin flip at that time.
Now, the coin flips are independent
from the original Bernoulli process.
And for either the coin flips or the Bernoulli process,
we have independence across time.
So all of the four random variables involved here
that determine what happens in these two slots
are independent of each other.
So what happens in this slot is a function
of two random variables here, which
are independent from the two random variables that
determined what happens in that slot.
So what happens in these two slots are independent events.
And this argument goes through more generally
when we consider multiple distinct slots.
So this is the argument for the independence
of the different slots in this particular process.
And then during each slot, what happens
is that we will have an arrival if and only
if this process records an arrival, which happens
with probability p, and the corresponding coin
flip happens to send the arrival in this direction, which
happens with probability q.
And so the conclusion is that this process
is a Bernoulli process with parameter p times q.
By a similar argument, the other process that we obtain
will also be Bernoulli but with probability p times 1 minus q.
And a final question-- are these two processes
that we get after the splitting independent of each other?
This is a question that we can answer
by reasoning intuitively.
If I tell you that there was an arrival in this slot,
what can you infer from this?
Well, it tells me that there was an arrival
in the original stream, which was sent here.
But since it was sent in this direction,
it means that it was not sent in the other direction.
And so we do not have an arrival in this slot.
Knowing that we have an arrival here
means that we do not have an arrival there.
So information about one of the streams
gives us information about what happened in the other stream.
And therefore, we do not have independence.
So this is what happens when we split two Bernoulli processes.
And earlier we saw what happens when
we merge two independent Bernoulli processes.
These two operations of merging and splitting
are quite common in constructing more complex models
using Bernoulli processes as the elements of those models.
They are often useful models either in transportation
systems, where you have streams of traffic
that merge or split, also in models of computer networks
or any other kind of queueing system.
And these same operations of merging and splitting
will also show up when we study the continuous time
analog of the Bernoulli process, namely the Poisson process.