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We have said that the Bernoulli
process is the simplest

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stochastic processes there is.

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But what is a stochastic
process anyway?

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A stochastic process
can be thought

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of as a sequence of
random variables.

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Now, how is this different
from what we have doing before,

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where we have dealt with
multiple random variables?

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Well, one difference
is that here we're

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talking about an infinite
sequence of random variables.

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And that complicates
things to a certain extent.

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Now, what does it take to
describe a stochastic process?

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We should specify the
properties of each one

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of those random variables.

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For example, we might be
interested in the mean,

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variance, or PMF of
those random variables.

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For the case of the
Bernoulli process,

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this would be easy to do.

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We know what the
expected value is.

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We have a formula
for the variance.

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And we have a fairly simple PMF.

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There's probability p that X
is equal to 1 and probability 1

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minus p that X equals to 0.

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But this is not enough.

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We also need to know how the
different random variables are

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related to each other.

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And this is done by specifying,
directly or indirectly,

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the joint distribution,
the joint PMF or PDF,

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of the random
variables involved.

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And because we have an infinite
number of random variables,

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it's not enough to
do this, let's say,

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for the first n of them.

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We need to be able to specify
this joint distribution

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no matter what the number n is.

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For the case of the
Bernoulli process,

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we have specified this joint
PMF in an indirect way,

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because we have said that
the random variables are

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independent of each other.

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So the joint factors as a
product of the marginals.

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And we already know
what the marginals are.

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So we do, indeed, have a
specification of the joint PMF,

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and we have that
for all values of n.

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Of course, for more complicated
stochastic processes,

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this calculation might be
somewhat more difficult.

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Now, there is a second view
of a stochastic process

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which rests on the following.

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It's not just a collection
of random variables,

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but they are a
collection that's indexed

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by an index that
keeps increasing.

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And quite often, we think of
this index as corresponding

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to time.

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And so we have a
mental picture that

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involves a process that
keeps evolving in time.

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What is this picture?

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This picture is
best developed if we

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think in terms of
the sample space.

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Although we have an infinite
sequence of random variables,

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we are dealing with
a single experiment.

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And that single
experiment runs in time.

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And when we carry
out the experiment,

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we might to get an outcome
such as the following.

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For the Bernoulli process, we
might get a 0, 0, 1, 0, 1, 1,

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0, and so on.

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And we continue.

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So an infinite
sequence of that kind

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is one possible outcome of this
infinitely long experiment,

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one particular outcome of
the stochastic process.

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If we carry out the
process once more,

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we might get a
different outcome.

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For example, we might get a 0,
1, 1, 0, 0, 0, 1, 1, and so on,

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and continuing.

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And in general, any time
function of this kind

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is one possible outcome
of the experiment.

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Overall, the sample space
that we're dealing with

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is the set of all infinite
sequences of 0s and 1s.

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This point of view
emphasizes the fact

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that we have a phenomenon
that evolves over time

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and can be used to
answer questions that

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have to do with the long-term
evolution of this process.

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Here's one particular
kind of question

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we might want one answer.

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What is the probability that all
of the Xi's turn out to be 1?

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Notice that this is an event
that involves all of the Xi's

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not just a finite
number of them.

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So this is not a
probability that we

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can calculate right away
by using this joint pmf.

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We need to do a
little more work.

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What is the work
that we want to do?

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Instead of calculating
this quantity,

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we will calculate a
somewhat related quantity.

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Let us look at the
event that the first n

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results were equal to 1.

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How is this event
related to this event?

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Well, this event here implies
that this event has happened.

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So this is a smaller event.

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This is more difficult
to obtain than this one.

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And this gives us an inequality
for the probabilities

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that go this way.

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Now, we know that
this probability

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is equal to p to the n.

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And this inequality
here is true for all n.

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No matter how large n
we take, this quantity

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is smaller than that.

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But now, since p has been
assumed to be less than 1,

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when we take n
larger and larger,

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this number becomes
arbitrarily small.

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So this quantity is
less than or equal

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to an arbitrarily small number.

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So this quantity can
only be equal to 0.

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And this is a simple example
of how we calculate properties

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of the stochastic process as it
evolves over the infinite time

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horizon and how we can
sometimes calculate them using

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these so-called finite
dimensional joint probabilities

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that tell us what
the process is doing

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over a finite amount of time.

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Throughout, we will sometimes
view stochastic processes

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in this manner, in terms of
probability distributions.

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But sometimes we will
also want to reason

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in terms of the behavior
of the stochastic process

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as a time function, as a
process that evolves in time.