WEBVTT
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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.