In this segment we develop some consequences
of the independence assumption that we
have made on the trials that constitute a Bernoulli process.
These properties will be pretty intuitive,
but they play an important role.
They're helpful in solving problems,
and they're also quite helpful in understanding
the continuous time version of the Bernoulli process, namely
the Poisson process that we will be studying later.
So here's the story.
We start with a Bernoulli processes
with some parameter p.
The process starts.
A friend of yours watches the processes,
and they observe the results of the different trials,
let's say for five time steps.
And at this time, right after time five,
they call you into the room, and you
start watching the rest of the process.
What will you see?
The first random variable that you will see
is the result of whatever happens in this time
slot, which is the sixth slot of the original process.
The second random variable that you will see
is the result of the seventh random variable
in the original process, and so on.
So the process that you get to see
is the process Yi, where i ranges
over the non-negative integers.
What properties does this process have?
Because of the assumption that the different trials are
independent, this means that the first five trials
are independent from the trials that happen after time five.
So one property is that the process
is Yi is independent of whatever happens
in the past, which is X1 up to X5.
Second, the random variable that you see, X6, X7, and so on,
are independent Bernoulli random variables with parameter p.
So the random variables Yi constitute also
a Bernoulli process with parameter p.
So the process that you get to see,
which is the sequence of trials after time five,
is identical, probabilistically, to a Bernoulli process
with parameter p like the process Xi.
So it's as if a Bernoulli process was just
starting fresh at this particular time.
And because of this, we say that the process
has a fresh-start property after a certain time.
In this example, we used 5 as the certain time,
but instead of 5, we could have any particular integer
little n, in which case our process Y1 starts
with Xn plus 1, continues with Xn plus 2 and so on.
And here, instead of X5, we would have written Xn.
So after a deterministic time n, what you see
is the same as if we had a Bernoulli process that
was starting fresh at this particular time,
and which is also independent from whatever
happened in the past.
Let us now complicate the story a little bit.
Suppose that your friend watches the Bernoulli process,
and they keep watching it until a success is
observed for the first time.
Right at that time, they call you into the room,
and you started watching the rest of the process.
This is the length of time that we
have called T1, the number of trials until the first success.
So what is it that you will be watching?
The first random variable that you will see
is what happens in slot T1 plus 1.
The second random variable that you will see
is what happened in slot T1 plus 2, and so on.
And this defines, again, a process,
the sequence of the Yi's This is what
you will see starting from this particular time.
What kind of process is it?
Well, these trials happened in the past.
We know what they were.
But no matter what they were, the future trials
will still be independent of the past,
and each one of the trials will have
probability p of being a success.
So the properties that we have, again,
is that the trials that you see are
independent of the past, which in this case
is everything from x1 up to time xT1.
And what you see is a Bernoulli process.
We describe the situation by saying
that the process starts fresh after time T1.
And by this, again, we mean that if you
start watching the process right after T1, what you will see
will be a Bernoulli process which
is independent from whatever happened in the past.
Having just argued that the process starts
fresh at the time T1 of the first success,
we can now ask why whether such a property is also true
more generally.
That is, if we start watching the process at some random time
n, will the process start fresh at that time?
Let us look at some additional examples.
Suppose that capital N is the time of the third success.
So your friend watches the Bernoulli process,
and each time, they say, did the third success occur?
Not yet.
Yes, the third success just occurred.
And at that time, they call you into the room
and you start to watching what happens from that time on.
What will you be seeing?
After that time, there will be independent Bernoulli trials
that take place.
And these refer to the future of the process, looking at [it]
from this particular point in time.
And the future is independent from whatever
happened in the past.
So what you actually see is, indeed, a fresh Bernoulli
process that starts here and which
is independent from the random variables that
occurred in the past.
Let us look at another example.
Let capital N be the first time that three successes in a row
have been recorded.
So your friend, again, watches the process.
And they ask each time, did we see three success in a row?
Not yet.
Yes.
I just saw three successes in a row.
And now your friend calls you in,
and you start watching the process
from this point in time.
By the same argument as before, whatever
happens in the future is going to be Bernoulli trials that
are independent from the past, so you will, again,
have a fresh-start property starting from this time.
So in both cases, formally, what we have
is that the process that you get to observe starting after time
capital N, after the time that you're called and asked
to start watching, what you will see
is going to be a sequence of independent Bernoulli trials,
that is, a Bernoulli process.
And this sequence of future trials
is independent from whatever happened
in the past of the process.
What both of these examples have in common
is that the random variable N, the time at which you're
called in, is causally determined
from the history of the process.
What does that mean?
It means that somebody is watching the process,
and at each point in time, based on what they have seen so far,
they decide whether to call you in or not.
What would be an example of a non-causal time N?
Here it is.
N could be the time just before the first occurrence of 1,
1, 1.
So in this example here, you would be called into the room
and start watching at this time.
So your friend somehow knows that a sequence of 1,1, 1
is going to occur and calls you just before it happens.
How could that be?
Well, imagine that the Bernoulli process actually
was run yesterday.
It was recorded in a movie.
Your friend has seen the movie, so knows
everything that's going to happen.
And so, when the movie is replayed today,
your friend can call you in at this time and tell you,
you know, something very interesting is about happen.
Come in and start watching.
Now, what will you be watching?
What you will watch will be 1, 1, 1, with certainty.
You're certain that the first three trials that you will see
will be 1's.
And, well, the subsequent one's will be random.
But since you know that the first three trials will be 1,
this means that statistically, they're
not described by the statistics of a Bernoulli process.
In a Bernoulli process, each trial
has a probability of being 1 and the probability of being 0.
But since, in your case, you're certain to watch
1's in the beginning, this means that the random variables
that you see do not conform to the description of a Bernoulli
process.
So this is an example in which N is not causally determined.
And in this example, you do not to get
to see a Bernoulli process.
We do not have the fresh-start property.
What happened here is more generally true.
We do have a fresh-start property, but not always.
We have it under the assumption that the time at which you're
asked to start watching is determined
from the past history of the process in some causal manner.
This is a general fact that can be established rigorously.
However we will not go through a formal mathematical derivation.
The formal argument for the most general case
involves somewhat tedious symbol manipulations
that do not provide any additional insight.
However, the intuition behind this result
should be fairly clear, and we will use it freely
whenever we need it.