WEBVTT

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In this segment we
develop some consequences

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of the independence
assumption that we

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have made on the trials that
constitute a Bernoulli process.

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These properties will
be pretty intuitive,

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but they play an important role.

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They're helpful in
solving problems,

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and they're also quite
helpful in understanding

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the continuous time version of
the Bernoulli process, namely

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the Poisson process that
we will be studying later.

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So here's the story.

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We start with a
Bernoulli processes

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with some parameter p.

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The process starts.

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A friend of yours
watches the processes,

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and they observe the results
of the different trials,

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let's say for five time steps.

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And at this time,
right after time five,

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they call you into
the room, and you

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start watching the
rest of the process.

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What will you see?

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The first random variable
that you will see

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is the result of whatever
happens in this time

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slot, which is the sixth
slot of the original process.

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The second random
variable that you will see

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is the result of the
seventh random variable

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in the original
process, and so on.

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So the process
that you get to see

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is the process
Yi, where i ranges

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over the non-negative integers.

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What properties does
this process have?

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Because of the assumption
that the different trials are

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independent, this means
that the first five trials

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are independent from the trials
that happen after time five.

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So one property is
that the process

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is Yi is independent
of whatever happens

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in the past, which
is X1 up to X5.

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Second, the random variable
that you see, X6, X7, and so on,

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are independent Bernoulli random
variables with parameter p.

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So the random variables
Yi constitute also

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a Bernoulli process
with parameter p.

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So the process that
you get to see,

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which is the sequence of
trials after time five,

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is identical, probabilistically,
to a Bernoulli process

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with parameter p
like the process Xi.

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So it's as if a Bernoulli
process was just

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starting fresh at
this particular time.

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And because of this,
we say that the process

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has a fresh-start property
after a certain time.

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In this example, we used
5 as the certain time,

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but instead of 5, we could
have any particular integer

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little n, in which case
our process Y1 starts

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with Xn plus 1, continues
with Xn plus 2 and so on.

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And here, instead of X5,
we would have written Xn.

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So after a deterministic
time n, what you see

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is the same as if we had
a Bernoulli process that

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was starting fresh at
this particular time,

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and which is also
independent from whatever

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happened in the past.

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Let us now complicate
the story a little bit.

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Suppose that your friend
watches the Bernoulli process,

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and they keep watching
it until a success is

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observed for the first time.

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Right at that time, they
call you into the room,

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and you started watching
the rest of the process.

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This is the length
of time that we

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have called T1, the number of
trials until the first success.

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So what is it that
you will be watching?

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The first random variable
that you will see

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is what happens
in slot T1 plus 1.

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The second random
variable that you will see

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is what happened in slot
T1 plus 2, and so on.

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And this defines,
again, a process,

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the sequence of the
Yi's This is what

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you will see starting
from this particular time.

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What kind of process is it?

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Well, these trials
happened in the past.

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We know what they were.

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But no matter what they
were, the future trials

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will still be
independent of the past,

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and each one of the
trials will have

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probability p of
being a success.

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So the properties
that we have, again,

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is that the trials
that you see are

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independent of the
past, which in this case

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is everything from
x1 up to time xT1.

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And what you see is
a Bernoulli process.

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We describe the
situation by saying

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that the process starts
fresh after time T1.

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And by this, again,
we mean that if you

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start watching the process right
after T1, what you will see

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will be a Bernoulli
process which

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is independent from whatever
happened in the past.

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Having just argued
that the process starts

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fresh at the time T1
of the first success,

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we can now ask why whether
such a property is also true

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more generally.

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That is, if we start watching
the process at some random time

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n, will the process
start fresh at that time?

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Let us look at some
additional examples.

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Suppose that capital N is the
time of the third success.

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So your friend watches
the Bernoulli process,

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and each time, they say,
did the third success occur?

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Not yet.

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Not yet.

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Not yet.

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Not yet.

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Yes, the third
success just occurred.

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And at that time, they
call you into the room

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and you start to watching what
happens from that time on.

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What will you be seeing?

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After that time, there will be
independent Bernoulli trials

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that take place.

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And these refer to the future
of the process, looking at [it]

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from this particular
point in time.

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And the future is
independent from whatever

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happened in the past.

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So what you actually see is,
indeed, a fresh Bernoulli

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process that starts
here and which

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is independent from the
random variables that

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occurred in the past.

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Let us look at another example.

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Let capital N be the first time
that three successes in a row

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have been recorded.

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So your friend, again,
watches the process.

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And they ask each time, did
we see three success in a row?

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Not yet.

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Not yet.

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Not yet.

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Not yet.

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Not yet.

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Yes.

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I just saw three
successes in a row.

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And now your friend
calls you in,

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and you start
watching the process

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from this point in time.

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By the same argument
as before, whatever

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happens in the future is going
to be Bernoulli trials that

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are independent from the
past, so you will, again,

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have a fresh-start property
starting from this time.

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So in both cases,
formally, what we have

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is that the process that you get
to observe starting after time

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capital N, after the time
that you're called and asked

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to start watching,
what you will see

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is going to be a sequence of
independent Bernoulli trials,

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that is, a Bernoulli process.

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And this sequence
of future trials

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is independent from
whatever happened

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in the past of the process.

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What both of these
examples have in common

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is that the random variable
N, the time at which you're

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called in, is
causally determined

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from the history of the process.

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What does that mean?

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It means that somebody
is watching the process,

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and at each point in time, based
on what they have seen so far,

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they decide whether
to call you in or not.

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What would be an example
of a non-causal time N?

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Here it is.

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N could be the time just before
the first occurrence of 1,

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1, 1.

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So in this example here, you
would be called into the room

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and start watching at this time.

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So your friend somehow knows
that a sequence of 1,1, 1

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is going to occur and calls
you just before it happens.

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How could that be?

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Well, imagine that the
Bernoulli process actually

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was run yesterday.

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It was recorded in a movie.

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Your friend has seen
the movie, so knows

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everything that's
going to happen.

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And so, when the movie
is replayed today,

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your friend can call you in
at this time and tell you,

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you know, something very
interesting is about happen.

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Come in and start watching.

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Now, what will you be watching?

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What you will watch will
be 1, 1, 1, with certainty.

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You're certain that the first
three trials that you will see

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will be 1's.

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And, well, the subsequent
one's will be random.

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But since you know that the
first three trials will be 1,

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this means that
statistically, they're

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not described by the statistics
of a Bernoulli process.

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In a Bernoulli
process, each trial

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has a probability of being 1
and the probability of being 0.

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But since, in your case,
you're certain to watch

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1's in the beginning, this
means that the random variables

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that you see do not conform to
the description of a Bernoulli

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process.

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So this is an example in which
N is not causally determined.

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And in this example,
you do not to get

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to see a Bernoulli process.

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We do not have the
fresh-start property.

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What happened here is
more generally true.

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We do have a fresh-start
property, but not always.

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We have it under the assumption
that the time at which you're

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asked to start
watching is determined

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from the past history of the
process in some causal manner.

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This is a general fact that
can be established rigorously.

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However we will not go through a
formal mathematical derivation.

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The formal argument for
the most general case

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involves somewhat tedious
symbol manipulations

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that do not provide
any additional insight.

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However, the intuition
behind this result

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should be fairly clear,
and we will use it freely

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whenever we need it.