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PROFESSOR: OK, we're
all ready to go.

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This is Discrete Stochastic
Processes as you are know.

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It is--

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want to get it where
I can read it, too.

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We're going to try to deal
with a bunch of different

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topics today, some of which are
a little bit philosophical

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saying, what is probability
really?

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You are supposed to have taken
a course in probability, but

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unfortunately courses in
probability are almost always

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courses in how to solve
well-posed problems.

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The big problem in probability
theory, and particularly

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stochastic processes is not
so much how do you solve

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well-posed problems.

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Anybody can do that.

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Or anybody who has a little bit
of background can do it.

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The hard problem is finding
the right models for a

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real-world problem.

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I will call these real-world
problems.

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I hate people who call things
real-world because they sound

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like they dislike theory
or something.

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It's the only word I can think
of though, because physical

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world is no longer adequate
because so much of the

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applications of probability are
to all sorts of different

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things, not having to do with
the physical world very much,

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but having to do with things
like the business world, or

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the economic world, or the
biological world, or all of

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these things.

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So real-world is just a code
word we'll use to distinguish

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theory from anything real that
you might have to deal with.

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Theory is very nice
because theory--

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everything is specified.

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There's a right answer
to everything.

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There is a wrong
answer usually.

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But there's at least
one right answer.

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And most of us like those
things because

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they're very specific.

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People go into engineering and
science very often because

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they don't like the
uncertainty of a

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lot of other fields.

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The problem is as soon as you
go into probability theory,

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you're moving away from that
safe region where everything

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is specified, and you're moving
into a region where

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things, in fact, are not very
well-specified and you have to

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be careful about it.

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OK, so first we're going to talk
about probability in the

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real world and probability as
a branch of mathematics.

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Then we're going to
say what discrete

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stochastic processes are.

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Then we're going to talk just a
very, very little bit about

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the processes we're
going to study.

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If you want to see more
of that, you have two

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chapters of the notes.

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You can look at them.

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You can look at the
table of contents.

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And more than that, if you look
at my website, you will

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see the notes for all the other
chapters if you want to

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read ahead or if you want to
really find out what kinds of

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things we're going to talk about
and what kinds of things

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we're not going to talk about.

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Then we're going to talk
about when, where,

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and how is this useful?

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The short answer to that is
it's useful everywhere.

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But we'll have to
see why that is.

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Then we're going to talk
about the axioms

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of probability theory.

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You cannot take any elementary
course in probability, or even

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in statistics, without seeing
the axioms of probability.

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And in almost all of those
cases, and in almost all of

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the graduate courses I've
seen, you see them, they

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disappear, and suddenly you're
solving problems in whatever

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way you can, and the
axioms have nothing

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to do with it anymore.

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So we're going to see that, in
fact, the axioms do have

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something to do with this.

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Those of you who want to be
real engineers and not

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mathematicians, you'll
find this a little

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uncomfortable at times.

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We are going to be
proving things.

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And you will have to
get used to that.

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And I'll try to convince you
of why it's important to be

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able to prove things.

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Then we're going on to a
review of probability,

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independent events,

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experiments, and random variables.

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So that's what we'll do today.

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Incidentally, this course
started about--

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must've been 25 years
ago or so.

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I started it because we had a
huge number of students at MIT

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who had been interested in
communication and control, and

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who were suddenly starting to
get interested in networks.

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And there were all sorts of
queuing problems that they had

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to deal with every day.

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And they started to read
about queuing theory.

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and it was the most disjointed,
crazy theory in

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the world where there were
1,000 different kinds of

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queues and each one
of them had to be

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treated in its own way.

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And we realized that stochastic
processes was the

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right way to tie all of
that together, so we

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started this course.

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And we made it mostly discrete
so it would deal primarily

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with network type
applications.

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As things have grown, it now
deals with a whole lot more

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applications.

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And we'll see how that
works later on.

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OK, how did probability get
started in the real world?

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Well, there were games of chance
that everybody was

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interested in.

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People really like to gamble.

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I don't know why.

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I don't like to gamble
that much.

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I would rather be certain
about things.

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But most people love
to gamble.

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And most people have an
intuitive sense of what

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probability is about.

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I mean, eight-year-old kids,
when they start to learn to

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play games of chance--

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and there are all sorts
of board games

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that involve chance.

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These kids, if they're
bright--

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and I'm sure you people fall
into that category--

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they immediately start to figure
out what the odds are.

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I mean, how many of you have
never thought about what the

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odds are in some
gambling game?

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OK, that makes my point.

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So all of you understand this
at an intuitive level.

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But what makes games of chance
easier to deal with than all

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the other issues where we have
uncertainty in life?

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Well, games of chance are
inherently repeatable.

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You play a game of chance and
you play many, many hands, or

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many, many throws, or
many, many trials.

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And after many, many trials of
what's essentially the same

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experiment, you start to get
a sense of what relative

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frequencies are.

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You start to get a sense of what
the odds are because of

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doing this repeatedly.

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So games of chance are easy to
use probability on because

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they are repeatable.

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You have essentially the same
thing going on each time, but

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each time there's a
different answer.

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You flip a coin and sometimes
it comes up heads and

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sometimes it comes up tails.

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So in fact, we have to figure
out how to deal with that fact

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that there is uncertainty
there.

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I'll talk about that in
just another minute.

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But anyway, most of life's
decisions involve uncertainty.

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I mean, for all of you, when you
go into a PhD program, you

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have two problems.

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Am I going to enjoy this?

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And you don't know whether
you're going to enjoy it

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because not until you really
get into it do you have a

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sense of whether this set of
problems you're dealing with

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is something that you
like to deal with.

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And the only way you can do
that is to make guesses.

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You come up with likelihoods.

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There's some likelihood.

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There's a risk cost-benefit
that you deal with.

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And in life, risk cost-benefits
are always based

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on some sense of what the
likelihood of something is.

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Now, what is a likelihood?

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A likelihood is just
a probability.

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It's a synonym for
probability.

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When you get into the
mathematics of probability,

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likelihood has a special
meaning to it.

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But in the real world,
likelihood is just a word you

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use when you don't want to let
people know that you're really

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talking about probabilities.

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OK, so that's where we are.

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But on the last slide, you saw
the word "essentially,

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essentially, essentially." If
you read the notes, and I hope

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you read the notes, because I
spent the last three years

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doing virtually nothing
but trying to

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make these notes clear.

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I would appreciate it if any
of you, with whatever

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background you have, when you
read these nodes, if you read

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them twice and you still don't
understand something, tell me

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you don't understand it.

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If you know why you don't
understand it, I'd appreciate

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it knowing that.

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But just saying "help" is enough
to let me know that I

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still haven't made something
as clear as it should be.

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At least as clear as it should
be for some of the people who

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I think should be taking
this course.

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One of the problems we have at
MIT now, and at every graduate

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school in the world I think,
is that human knowledge has

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00:10:00,980 --> 00:10:05,390
changed and grown so much
in the last 50 years.

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So when you study something
general, like probability,

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00:10:09,010 --> 00:10:12,600
there's just an enormous
mass of stuff you

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have to deal with.

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And because of that, when you
try to write notes for a

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00:10:16,360 --> 00:10:18,290
course, you don't know
what anybody's

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background is anymore.

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I mean, it used to be that when
you saw a graduate course

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at MIT, people would know
what limits were.

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00:10:27,320 --> 00:10:30,830
People would know what basic
mathematics is all about.

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They would know what
continuity means.

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00:10:33,000 --> 00:10:35,110
They would know some
linear algebra.

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They would know all sorts
of different things.

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Many people still do
know those things.

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Many other people have studied
all sorts of other fascinating

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00:10:44,830 --> 00:10:46,110
and very interesting things.

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They're just as smart, but they
have a very different

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background.

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So if your background is
different, it's not your fault

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that you don't have the kind
of background that makes

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probability easy.

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Just yell.

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Or yell in class.

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Please ask questions.

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00:11:01,440 --> 00:11:04,200
The fact that we're videotaping
this makes it far

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00:11:04,200 --> 00:11:08,340
more interesting for anybody
who's using OpenCourseWare to

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00:11:08,340 --> 00:11:12,710
see some kinds of questions
going on, so I very much

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00:11:12,710 --> 00:11:15,760
encourage that.

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00:11:15,760 --> 00:11:19,690
I'm fairly old at this point,
and my memory is getting shot.

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So if you ask a question and I
don't remember what it's all

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00:11:23,500 --> 00:11:26,590
about, just be patient
with it.

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I will come back the next time,
or I'll send you an

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00:11:28,870 --> 00:11:31,500
email straightening it out.

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But I will often get confused
doing something, and that's

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just because of my age.

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It's what we call "senior
moments." It's not that I

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don't understand the subject.

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00:11:42,110 --> 00:11:43,630
I think I understand it, I just

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don't remember it anymore.

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Important point about
probability.

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Think about flipping a coin.

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I'm going to talk about
flipping coins a

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great deal this term.

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It's an absolutely trivial
topic, but it's important

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because when you understand
deep things about a large

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subject, the best way to
understand them is to

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understand them in terms of the
most trivial examples you

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00:12:13,420 --> 00:12:14,880
can think of.

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00:12:14,880 --> 00:12:18,770
Now, when you flip a coin, the
outcome-- heads or tails--

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really depends on the initial
velocity, the orientation of

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the person flipping it, or the
machine flipping it, the coin

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surfaces, the ground surface.

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And after you put all of those
things into a careful

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equation, you will know whether
the coin is going to

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come up heads or tails.

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I don't think quantum theory
has anything to do with

249
00:12:41,860 --> 00:12:43,620
something as big as a coin.

250
00:12:43,620 --> 00:12:45,210
I might be wrong.

251
00:12:45,210 --> 00:12:46,630
I've never looked into it.

252
00:12:46,630 --> 00:12:48,200
And frankly, I don't care.

253
00:12:48,200 --> 00:12:50,870
Because the point that I'm
trying to make is that

254
00:12:50,870 --> 00:12:53,410
flipping a coin, and many of
the things that you view as

255
00:12:53,410 --> 00:12:57,170
random, when you look at them
in a slightly different way,

256
00:12:57,170 --> 00:12:59,020
are not random.

257
00:12:59,020 --> 00:13:01,390
There's a big field in
communication called data

258
00:13:01,390 --> 00:13:02,810
compression.

259
00:13:02,810 --> 00:13:07,270
And data compression is based on
random models for the data,

260
00:13:07,270 --> 00:13:09,480
which is going to
be compressed.

261
00:13:09,480 --> 00:13:13,070
Now, what I'm saying here today
is by no means random.

262
00:13:13,070 --> 00:13:14,900
Or maybe it is partly random.

263
00:13:14,900 --> 00:13:17,910
Maybe it's coming out of a
random mind, I don't know.

264
00:13:17,910 --> 00:13:23,725
But all of the data we try to
compress to the people who

265
00:13:23,725 --> 00:13:27,260
have created that data, it's
not random at all.

266
00:13:27,260 --> 00:13:30,710
If you study the data
carefully enough--

267
00:13:30,710 --> 00:13:33,510
I mean, code breakers and people
like that are extremely

268
00:13:33,510 --> 00:13:37,550
good at sorting out what the
meaning is in something, which

269
00:13:37,550 --> 00:13:41,230
cannot be done by data
compression techniques at all.

270
00:13:41,230 --> 00:13:45,280
So the point is, when you're
doing data compression, you

271
00:13:45,280 --> 00:13:48,010
model the data as being random
and having certain

272
00:13:48,010 --> 00:13:49,570
characteristics.

273
00:13:49,570 --> 00:13:51,800
But it really isn't.

274
00:13:51,800 --> 00:13:54,130
So the model is no good.

275
00:13:54,130 --> 00:13:56,550
When you get to more important
questions--

276
00:13:56,550 --> 00:13:59,950
well, data compression is
an important question.

277
00:13:59,950 --> 00:14:02,530
When you ask, what's the
probability of another

278
00:14:02,530 --> 00:14:05,790
catastrophic oil spill
in the next year?

279
00:14:05,790 --> 00:14:08,040
Or you ask the question, what's
the probability that

280
00:14:08,040 --> 00:14:10,140
Google stock will double
in five years?

281
00:14:10,140 --> 00:14:12,580
That's less important,
but it's still

282
00:14:12,580 --> 00:14:15,630
important to many people.

283
00:14:15,630 --> 00:14:16,880
How do you model that?

284
00:14:19,410 --> 00:14:22,300
Understanding probability
theory, understanding all the

285
00:14:22,300 --> 00:14:26,500
mathematics of this is not going
to help you model this.

286
00:14:26,500 --> 00:14:29,880
Now, why do I make such
a big deal about this?

287
00:14:29,880 --> 00:14:33,910
Well, there have been a number
of times in the last 10 or 15

288
00:14:33,910 --> 00:14:38,480
years when the whole financial
system of the world has almost

289
00:14:38,480 --> 00:14:43,200
been destroyed by very,
very bright PhDs.

290
00:14:43,200 --> 00:14:47,680
Many of them coming from
electrical engineering.

291
00:14:47,680 --> 00:14:50,010
Most of whom are really
superb at

292
00:14:50,010 --> 00:14:52,580
understanding probability theory.

293
00:14:52,580 --> 00:14:55,490
And they have used their
probability theory to analyze

294
00:14:55,490 --> 00:14:58,640
risk and other things
in investments.

295
00:14:58,640 --> 00:15:00,630
And what has happened?

296
00:15:00,630 --> 00:15:03,100
They do very well for a while.

297
00:15:03,100 --> 00:15:06,050
Suddenly they do so well that
they think they can borrow all

298
00:15:06,050 --> 00:15:08,750
sorts of money and risk other
people's money as

299
00:15:08,750 --> 00:15:09,610
well as their own.

300
00:15:09,610 --> 00:15:13,080
In fact, they try to do that
right from the beginning.

301
00:15:13,080 --> 00:15:16,030
And then suddenly, the whole
thing collapses.

302
00:15:16,030 --> 00:15:18,240
Because their models
are no damn good.

303
00:15:18,240 --> 00:15:21,220
There's nothing wrong with their
mathematics, it's that

304
00:15:21,220 --> 00:15:22,770
their models are no good.

305
00:15:22,770 --> 00:15:26,710
So please, especially if you're
going to do something

306
00:15:26,710 --> 00:15:28,940
important in your lives--

307
00:15:28,940 --> 00:15:32,410
if you're just going to write
papers in engineering

308
00:15:32,410 --> 00:15:36,000
journals, maybe it's
all right.

309
00:15:36,000 --> 00:15:39,940
But if you're going to make
decisions about things, please

310
00:15:39,940 --> 00:15:43,060
spend some time thinking about
the probability models that

311
00:15:43,060 --> 00:15:47,890
you come up because this
is vitally important.

312
00:15:47,890 --> 00:15:49,110
OK, what's probability?

313
00:15:49,110 --> 00:15:50,630
It's a branch of mathematics.

314
00:15:50,630 --> 00:15:53,870
Now we're into something
that's more familiar,

315
00:15:53,870 --> 00:15:57,420
something that's simpler,
something we can deal with.

316
00:16:00,070 --> 00:16:02,340
You might be uncomfortable
with what

317
00:16:02,340 --> 00:16:04,280
probability really means.

318
00:16:04,280 --> 00:16:08,660
And all probability books, all
stochastic process books are

319
00:16:08,660 --> 00:16:10,080
uncomfortable with this.

320
00:16:10,080 --> 00:16:14,250
Feller is the best book
in probability

321
00:16:14,250 --> 00:16:17,260
there's ever been written.

322
00:16:17,260 --> 00:16:21,090
Any question you have, he
probably has the answer to it.

323
00:16:21,090 --> 00:16:26,960
When you look at what he says
about real-world probability,

324
00:16:26,960 --> 00:16:31,150
the modeling issues, he's an
extraordinarily bright guy.

325
00:16:31,150 --> 00:16:33,900
And he spent some time
thinking about this.

326
00:16:33,900 --> 00:16:35,360
But you read it and you realize

327
00:16:35,360 --> 00:16:38,630
that it's pure nonsense.

328
00:16:38,630 --> 00:16:40,710
So please, take my
word for it.

329
00:16:43,590 --> 00:16:46,780
Don't assume that real-world
probability is something

330
00:16:46,780 --> 00:16:49,070
you're going to learn about from
other people because you

331
00:16:49,070 --> 00:16:51,120
can't trust what any
of them say.

332
00:16:51,120 --> 00:16:53,800
It's something you have to think
through for yourselves,

333
00:16:53,800 --> 00:16:56,410
and we'll talk more about
this as we go.

334
00:16:56,410 --> 00:16:59,200
But now, when we get into
mathematics, that's fine.

335
00:16:59,200 --> 00:17:00,440
We just create models.

336
00:17:00,440 --> 00:17:03,090
And once we have the model,
we just use it.

337
00:17:03,090 --> 00:17:06,680
We have standard models for
all sorts of different

338
00:17:06,680 --> 00:17:08,260
standard problems.

339
00:17:08,260 --> 00:17:11,890
When you talk about coin
tossing, what almost everyone

340
00:17:11,890 --> 00:17:16,339
means is not this crazy thing I
was just talking about where

341
00:17:16,339 --> 00:17:20,800
you have an initial angular
momentum when you flip a coin

342
00:17:20,800 --> 00:17:22,300
and all of that stuff.

343
00:17:22,300 --> 00:17:25,990
It's a purely mathematical model
where a coin is flipped

344
00:17:25,990 --> 00:17:28,440
and with probability one half
it comes up heads and with

345
00:17:28,440 --> 00:17:32,070
probability one half
it comes up tails.

346
00:17:32,070 --> 00:17:36,050
OK, students are given a
well-specified model, and they

347
00:17:36,050 --> 00:17:37,840
calculate various things.

348
00:17:37,840 --> 00:17:40,370
This is in mathematical
probability.

349
00:17:40,370 --> 00:17:43,710
Heads and tails are equiprobable
in that system.

350
00:17:43,710 --> 00:17:46,510
Subsequent tosses
are independent.

351
00:17:46,510 --> 00:17:49,630
Here's a little bit
of cynicism.

352
00:17:49,630 --> 00:17:52,700
I apologize for insulting
you people with it.

353
00:17:52,700 --> 00:17:57,430
I apologize to any faculty
member who later reads this.

354
00:17:57,430 --> 00:18:01,300
And I particularly apologize to
businessmen and government

355
00:18:01,300 --> 00:18:03,070
people who might read it.

356
00:18:03,070 --> 00:18:06,840
Students compute, professors
write papers, business and

357
00:18:06,840 --> 00:18:10,290
government leaders obtain
questionable models and data

358
00:18:10,290 --> 00:18:12,540
on which they can
blame failures.

359
00:18:12,540 --> 00:18:16,070
Most cynical towards business
leaders because business

360
00:18:16,070 --> 00:18:19,220
leaders often hire
consultants.

361
00:18:19,220 --> 00:18:23,300
Not so much to learn what to do,
but so they have excuses

362
00:18:23,300 --> 00:18:26,430
when what they do doesn't
work out right.

363
00:18:26,430 --> 00:18:30,220
When I say the students compute,
what I mean is this

364
00:18:30,220 --> 00:18:34,030
in almost all the courses you've
taken up until now--

365
00:18:34,030 --> 00:18:36,390
and in this course also--

366
00:18:36,390 --> 00:18:38,220
what you're going
to be doing is

367
00:18:38,220 --> 00:18:39,820
solving well-posed problems.

368
00:18:39,820 --> 00:18:43,730
You solve well-posed exercises
because that's a good way to

369
00:18:43,730 --> 00:18:46,340
understand what the
mathematics of

370
00:18:46,340 --> 00:18:48,410
the subject is about.

371
00:18:48,410 --> 00:18:51,780
Don't think that that's
the only part of it.

372
00:18:51,780 --> 00:18:54,690
If that's the only thing you're
doing, you might as

373
00:18:54,690 --> 00:18:55,830
well not waste your time.

374
00:18:55,830 --> 00:18:57,600
You might as well do
something else.

375
00:18:57,600 --> 00:19:00,730
You might as well go out and
shovel snow today instead of

376
00:19:00,730 --> 00:19:02,860
trying to learn about
probability theory.

377
00:19:02,860 --> 00:19:06,960
It's more pleasant to learn
about probability theory.

378
00:19:06,960 --> 00:19:11,010
OK, the use of probability
models has two major

379
00:19:11,010 --> 00:19:13,300
problems with it.

380
00:19:13,300 --> 00:19:16,400
The first problem is, how do
you make a model for a

381
00:19:16,400 --> 00:19:18,600
real-world problem?

382
00:19:18,600 --> 00:19:23,930
And a partial answer is, learn
about estimation and decisions

383
00:19:23,930 --> 00:19:26,150
in the context of
standard models.

384
00:19:26,150 --> 00:19:31,870
In other words, decisions and
estimation inside a completely

385
00:19:31,870 --> 00:19:34,880
mathematical framework.

386
00:19:34,880 --> 00:19:36,480
Then you learn a great
deal about the

387
00:19:36,480 --> 00:19:39,140
real-world problem itself.

388
00:19:39,140 --> 00:19:43,880
Not about the mathematics of it,
but about how you actually

389
00:19:43,880 --> 00:19:46,270
understand what's going on.

390
00:19:46,270 --> 00:19:49,740
If you talk to somebody
who is a superb

391
00:19:49,740 --> 00:19:53,030
architect in any field--

392
00:19:53,030 --> 00:19:57,420
networks, computer systems,
control systems, anything--

393
00:19:57,420 --> 00:19:59,500
what are you going to find?

394
00:19:59,500 --> 00:20:03,630
You're not going to find huge,
involved sets of equations

395
00:20:03,630 --> 00:20:06,700
that they're going to use to
explain something to you.

396
00:20:06,700 --> 00:20:09,780
They're going to pick at-- if
there any good, they're going

397
00:20:09,780 --> 00:20:12,480
to take this big problem, and
they're going to take your

398
00:20:12,480 --> 00:20:14,560
issue with this big problem.

399
00:20:14,560 --> 00:20:17,700
And they're going to find the
one or two really important

400
00:20:17,700 --> 00:20:21,790
things that tell you something
that you have to know.

401
00:20:21,790 --> 00:20:24,350
And that's what you want to
get out of this course.

402
00:20:24,350 --> 00:20:28,150
You want to get the ability to
take all of the chat, put it

403
00:20:28,150 --> 00:20:31,920
all together, and be able to
say one or two important

404
00:20:31,920 --> 00:20:35,220
things which is really
necessary.

405
00:20:35,220 --> 00:20:37,670
That's where you're going to.

406
00:20:37,670 --> 00:20:41,300
Before you get there, you'll
take low-level jobs in various

407
00:20:41,300 --> 00:20:43,570
companies and you'll compute
a lot of things.

408
00:20:43,570 --> 00:20:45,310
You'll simulate a
lot of things.

409
00:20:45,310 --> 00:20:47,540
You'll deal with a
lot of detail.

410
00:20:47,540 --> 00:20:50,170
Eventually, you're going to get
to the point where you've

411
00:20:50,170 --> 00:20:51,920
got to make major decisions.

412
00:20:51,920 --> 00:20:53,800
And you want to be
ready for it.

413
00:20:53,800 --> 00:20:55,580
OK, that's enough philosophy.

414
00:20:55,580 --> 00:20:58,830
I will try to give no more
philosophy today, except when

415
00:20:58,830 --> 00:21:00,080
I get pushed into it.

416
00:21:09,776 --> 00:21:15,930
OK, one of the problems in this
problem of finding a good

417
00:21:15,930 --> 00:21:18,880
model is that no model
is perfect.

418
00:21:18,880 --> 00:21:22,130
Namely, what happens is you
keep finding more and more

419
00:21:22,130 --> 00:21:25,050
complicated models, which
deal with more

420
00:21:25,050 --> 00:21:26,990
and more of the issues.

421
00:21:26,990 --> 00:21:31,100
And as you deal with them,
things get more complicated.

422
00:21:31,100 --> 00:21:34,380
You're more down in the
level of details and

423
00:21:34,380 --> 00:21:35,640
you're finding out less.

424
00:21:35,640 --> 00:21:39,850
So you want to find some sort of
match between a model that

425
00:21:39,850 --> 00:21:42,880
tells you something and a model
which is complicated

426
00:21:42,880 --> 00:21:45,180
enough to deal with
the issues.

427
00:21:45,180 --> 00:21:49,740
There's a beautiful quote by
Alfred North Whitehead.

428
00:21:49,740 --> 00:21:51,960
I don't know whether you've
ever heard of Whitehead.

429
00:21:51,960 --> 00:21:55,160
You've probably heard of
Bertrand Russell, who was both

430
00:21:55,160 --> 00:22:01,000
a great logician and a great
philosopher, and had a lot to

431
00:22:01,000 --> 00:22:04,590
do with the origins
of set theory.

432
00:22:04,590 --> 00:22:08,570
Whitehead and Russell together,
wrote this massive

433
00:22:08,570 --> 00:22:14,160
book around the turn of the last
century between the 1900s

434
00:22:14,160 --> 00:22:18,630
and the 2000s called Principia
Mathematica where they try to

435
00:22:18,630 --> 00:22:22,220
resolve all of the paradoxes
which were coming up in

436
00:22:22,220 --> 00:22:23,420
mathematics.

437
00:22:23,420 --> 00:22:27,920
And Whitehead's general
philosophical comment was,

438
00:22:27,920 --> 00:22:33,580
"Seek simplicity and
distrust it."

439
00:22:33,580 --> 00:22:37,025
Now, every time I look at
that, I say, why in hell

440
00:22:37,025 --> 00:22:41,640
didn't he say, seek simplicity
and question it?

441
00:22:41,640 --> 00:22:43,850
I mean, you all hear about
questioning authority, of

442
00:22:43,850 --> 00:22:46,520
course, and that's
important to do.

443
00:22:46,520 --> 00:22:50,310
Why when you find a simple model
for something should you

444
00:22:50,310 --> 00:22:52,420
distrust it?

445
00:22:52,420 --> 00:22:55,670
Well, the reason is
psychological.

446
00:22:55,670 --> 00:22:59,660
If you find a simple model for
something and you question it,

447
00:22:59,660 --> 00:23:03,260
you have an enormous
psychological bias towards not

448
00:23:03,260 --> 00:23:04,550
giving up the simple model.

449
00:23:04,550 --> 00:23:07,340
You want to keep that
simple model.

450
00:23:07,340 --> 00:23:11,270
And therefore, it takes an
enormous amount of evidence

451
00:23:11,270 --> 00:23:13,770
before you're going to
give something out.

452
00:23:13,770 --> 00:23:15,530
Whitehead said something
more than that.

453
00:23:15,530 --> 00:23:19,400
He said, "Seek simplicity
and distrust it."

454
00:23:19,400 --> 00:23:21,940
Now, why do I talk about the
philosophy of science when

455
00:23:21,940 --> 00:23:24,110
we're trying to learn about
probability theory?

456
00:23:24,110 --> 00:23:27,480
Well, probability theory is
a mathematical theory.

457
00:23:27,480 --> 00:23:29,960
It's the basis for a great
deal of science.

458
00:23:29,960 --> 00:23:35,050
And it's the place where
modeling is most difficult.

459
00:23:35,050 --> 00:23:37,600
Scientific questions in most
areas, if there's no

460
00:23:37,600 --> 00:23:41,140
probability or uncertainty
involved, you just do an

461
00:23:41,140 --> 00:23:44,170
experiment that tells
you the answer.

462
00:23:44,170 --> 00:23:46,330
You might not do it carefully
enough and then 10 other

463
00:23:46,330 --> 00:23:47,380
people do it.

464
00:23:47,380 --> 00:23:49,670
And finally, everybody
agrees, this is the

465
00:23:49,670 --> 00:23:51,140
answer to that problem.

466
00:23:51,140 --> 00:23:53,930
In probability, it ain't
that simple.

467
00:23:53,930 --> 00:23:57,720
And that's why one has to focus
on this a little more

468
00:23:57,720 --> 00:23:59,740
than usual.

469
00:23:59,740 --> 00:24:04,730
The second problem is, how do
you make a probability model

470
00:24:04,730 --> 00:24:06,430
that has no hidden
paradoxes in it?

471
00:24:06,430 --> 00:24:11,130
In other words, when you make
a mathematical model, how do

472
00:24:11,130 --> 00:24:14,930
you make sure that it really
is well-posed?

473
00:24:14,930 --> 00:24:17,490
How do you make sure that when
you solve a problem in that

474
00:24:17,490 --> 00:24:20,400
mathematical model that you
don't come to something that

475
00:24:20,400 --> 00:24:22,820
doesn't make any sense?

476
00:24:22,820 --> 00:24:27,060
Well, everyone's answer to that
is you use Kolmogorov's

477
00:24:27,060 --> 00:24:29,030
axioms of probability.

478
00:24:29,030 --> 00:24:33,190
Because back in 1933, Kolmogorov
published this

479
00:24:33,190 --> 00:24:35,410
little thin book.

480
00:24:35,410 --> 00:24:37,370
Those of you who are interested
in the history of

481
00:24:37,370 --> 00:24:40,620
science probably ought
to read it.

482
00:24:40,620 --> 00:24:43,530
You will find you only
understand the first five

483
00:24:43,530 --> 00:24:46,330
pages the first time
you look at it.

484
00:24:46,330 --> 00:24:49,170
But it's worthwhile doing that
because here was one of the

485
00:24:49,170 --> 00:24:53,930
truly great minds of the
early 20th century.

486
00:24:53,930 --> 00:24:58,260
And he took everything he knew
about probability, which was a

487
00:24:58,260 --> 00:25:01,630
whole lot more than I know
certainly, and a whole lot

488
00:25:01,630 --> 00:25:04,740
more than anybody else at the
time knew, and he collapsed it

489
00:25:04,740 --> 00:25:07,100
into these very simple axioms.

490
00:25:07,100 --> 00:25:12,330
And he said, if you obey these
axioms in a model that you use

491
00:25:12,330 --> 00:25:16,580
in probability, those axioms
will keep you out of any

492
00:25:16,580 --> 00:25:18,270
paradoxes at all.

493
00:25:18,270 --> 00:25:21,690
And then, he showed why that
was and he showed how the

494
00:25:21,690 --> 00:25:23,845
axioms could be used
and so forth.

495
00:25:23,845 --> 00:25:25,990
So we're going to spend a little
bit of time talking

496
00:25:25,990 --> 00:25:27,790
about them today.

497
00:25:27,790 --> 00:25:31,850
OK, quickly, what is a discrete
stochastic process?

498
00:25:31,850 --> 00:25:34,180
Well, a stochastic process--

499
00:25:34,180 --> 00:25:36,100
you've been talking
about probability.

500
00:25:36,100 --> 00:25:41,690
And you might be getting the
idea that I'm just using the

501
00:25:41,690 --> 00:25:45,670
name "stochastic processes" as
a foil for talking about what

502
00:25:45,670 --> 00:25:48,210
I really love, which
is the probability.

503
00:25:48,210 --> 00:25:51,210
And there's a certain amount
of truth to that.

504
00:25:51,210 --> 00:25:56,360
But stochastic processes are
special types of probability

505
00:25:56,360 --> 00:26:02,680
models where the sample points
represent functions in time.

506
00:26:02,680 --> 00:26:05,420
In other words, when we're
dealing with a probability

507
00:26:05,420 --> 00:26:07,270
model, the basis of
a probability

508
00:26:07,270 --> 00:26:08,750
model is a sample space.

509
00:26:08,750 --> 00:26:12,220
It's the set of possible things
that might happen.

510
00:26:12,220 --> 00:26:15,550
And you can reduce that to the
sample points, which are the

511
00:26:15,550 --> 00:26:20,010
indivisible, little, tiny crumbs
of what happens when

512
00:26:20,010 --> 00:26:20,790
you do an experiment.

513
00:26:20,790 --> 00:26:24,080
It's the thing which specifies
everything that can be

514
00:26:24,080 --> 00:26:29,040
specified in that model
of that experiment.

515
00:26:29,040 --> 00:26:31,670
OK, when you get to a stochastic
process, what

516
00:26:31,670 --> 00:26:35,600
you're doing is you're looking
at a situation in which these

517
00:26:35,600 --> 00:26:40,560
sample points, the solutions to
what happens is, in fact, a

518
00:26:40,560 --> 00:26:44,550
whole sequence of random
variables in time.

519
00:26:49,475 --> 00:26:53,410
And what that means is instead
of looking at just a vector of

520
00:26:53,410 --> 00:26:56,070
random variables, you're going
to be looking at a whole

521
00:26:56,070 --> 00:26:59,040
sequence of random variables.

522
00:26:59,040 --> 00:27:03,890
Now, what is different about a
vector of a very large number

523
00:27:03,890 --> 00:27:06,580
of random variables
and an infinite

524
00:27:06,580 --> 00:27:09,280
sequence of random variables?

525
00:27:09,280 --> 00:27:14,150
Well, from an engineering
standpoint, not very much.

526
00:27:14,150 --> 00:27:18,060
I mean, there's nothing you can
do to actually look at an

527
00:27:18,060 --> 00:27:22,160
infinite sequence of
random variables.

528
00:27:22,160 --> 00:27:26,380
If you start out at the Big Bang
and you carry it on to

529
00:27:26,380 --> 00:27:29,220
what you might imagine is the
time when the sun explodes or

530
00:27:29,220 --> 00:27:32,380
something, that's a finite
amount of time.

531
00:27:32,380 --> 00:27:37,330
And if you imagine how fast
you can observe things,

532
00:27:37,330 --> 00:27:39,530
there's a finite number
of random

533
00:27:39,530 --> 00:27:41,280
variables you might observe.

534
00:27:41,280 --> 00:27:44,670
All these models we're going to
be dealing with get outside

535
00:27:44,670 --> 00:27:47,960
or that realm, and they deal
with something that starts

536
00:27:47,960 --> 00:27:50,200
infinitely far in the
past and goes

537
00:27:50,200 --> 00:27:52,760
infinitely far in the future.

538
00:27:52,760 --> 00:27:55,860
It doesn't make much
sense, does it?

539
00:27:55,860 --> 00:27:58,610
But then look at the
alternative.

540
00:27:58,610 --> 00:28:01,800
You built a device which
you're going to sell to

541
00:28:01,800 --> 00:28:04,840
people, and which they're
going to use.

542
00:28:04,840 --> 00:28:06,660
And you know they're only going
to use it three or four

543
00:28:06,660 --> 00:28:09,970
year until something
better comes along.

544
00:28:09,970 --> 00:28:13,410
But do you want to build in to
everything you're doing the

545
00:28:13,410 --> 00:28:15,790
idea that it's going to be
obsolete in three years?

546
00:28:15,790 --> 00:28:17,390
No.

547
00:28:17,390 --> 00:28:21,510
You want to design this thing
so, in fact, it will work for

548
00:28:21,510 --> 00:28:24,300
an essentially arbitrary
amount of time.

549
00:28:24,300 --> 00:28:27,160
And therefore, you make a
mathematical model of it.

550
00:28:27,160 --> 00:28:30,860
You look at what happens over
an infinite span of time.

551
00:28:30,860 --> 00:28:34,330
So whenever we get into
mathematics, we always go to

552
00:28:34,330 --> 00:28:36,290
an infinite number of
things rather than a

553
00:28:36,290 --> 00:28:39,130
finite number of things.

554
00:28:39,130 --> 00:28:42,420
Now, discrete stochastic
processes are those where the

555
00:28:42,420 --> 00:28:45,560
random variables are
discrete in time.

556
00:28:45,560 --> 00:28:47,820
Namely, a finite number
of possible outcomes

557
00:28:47,820 --> 00:28:49,880
from each of them.

558
00:28:49,880 --> 00:28:55,340
Or the set of possible sample
values is discrete.

559
00:28:55,340 --> 00:28:57,000
What does that mean?

560
00:28:57,000 --> 00:29:00,090
It doesn't mean a whole lot when
you really start asking

561
00:29:00,090 --> 00:29:02,680
detailed questions about this.

562
00:29:02,680 --> 00:29:07,420
What it means is, I want to talk
about a particular kind

563
00:29:07,420 --> 00:29:08,760
of stochastic processes.

564
00:29:11,290 --> 00:29:15,740
And it's a class of processes
which will be more than we can

565
00:29:15,740 --> 00:29:17,600
deal with in one term.

566
00:29:17,600 --> 00:29:20,900
And I want to exclude certain
processes, like noise

567
00:29:20,900 --> 00:29:24,890
processes, because we don't have
time to do both of them.

568
00:29:24,890 --> 00:29:27,980
So don't worry too much about
exactly what a discrete

569
00:29:27,980 --> 00:29:29,230
stochastic process is.

570
00:29:31,720 --> 00:29:34,930
It's whatever we want to call
it when we deal with it.

571
00:29:38,220 --> 00:29:39,630
Oops.

572
00:29:39,630 --> 00:29:43,252
Oh, where am I?

573
00:29:43,252 --> 00:29:47,180
Oh, I wanted to talk about the
different processes we're

574
00:29:47,180 --> 00:29:49,540
going to study.

575
00:29:49,540 --> 00:29:51,310
The first kind of process
is what we

576
00:29:51,310 --> 00:29:54,310
call a counting process.

577
00:29:54,310 --> 00:29:57,230
The sample points in the
process-- remember, a sample

578
00:29:57,230 --> 00:30:00,660
point specifies everything
about an experiment.

579
00:30:00,660 --> 00:30:03,660
It tells you every
little detail.

580
00:30:03,660 --> 00:30:07,010
And the sample points here
in counting processes are

581
00:30:07,010 --> 00:30:08,980
sequences of arrivals.

582
00:30:08,980 --> 00:30:13,140
This is a very useful idea in
dealing with queuing systems

583
00:30:13,140 --> 00:30:15,260
because queuing systems
have arrivals.

584
00:30:15,260 --> 00:30:16,670
They have departures.

585
00:30:16,670 --> 00:30:20,120
They have rules for how the
arrivals get processed before

586
00:30:20,120 --> 00:30:22,300
they get spit out.

587
00:30:22,300 --> 00:30:26,220
And a big part of that is
studying first the arrival

588
00:30:26,220 --> 00:30:29,420
process, then we study the
departure process.

589
00:30:29,420 --> 00:30:32,330
We study how to put
them together.

590
00:30:32,330 --> 00:30:35,730
And when we get to chapter 2 of
the notes, we're going to

591
00:30:35,730 --> 00:30:39,400
be studying Poisson processes,
which are in a sense, the

592
00:30:39,400 --> 00:30:43,020
perfect discrete stochastic
process.

593
00:30:43,020 --> 00:30:45,660
It's like coin tossing
in probability.

594
00:30:45,660 --> 00:30:47,930
Everything that might
be true with a

595
00:30:47,930 --> 00:30:50,530
Poisson process is true.

596
00:30:50,530 --> 00:30:52,830
The only things that aren't
true are the things that

597
00:30:52,830 --> 00:30:55,130
obviously can't be true.

598
00:30:55,130 --> 00:30:57,280
And we'll find out why
that is and how that

599
00:30:57,280 --> 00:30:59,150
works a little later.

600
00:30:59,150 --> 00:31:00,750
We're then going to
study renewal

601
00:31:00,750 --> 00:31:03,950
processes in chapter 4.

602
00:31:03,950 --> 00:31:07,510
We're going to put Markov
chains in between.

603
00:31:07,510 --> 00:31:09,800
And you'll see why
when we do it.

604
00:31:09,800 --> 00:31:13,960
And renewal processes are a more
complicated kind of thing

605
00:31:13,960 --> 00:31:15,810
than Poisson processes.

606
00:31:15,810 --> 00:31:18,440
And there's no point confusing
you at this point saying what

607
00:31:18,440 --> 00:31:20,800
the difference is, so I won't.

608
00:31:20,800 --> 00:31:25,690
Markov processes
are processes.

609
00:31:25,690 --> 00:31:29,590
In other words, the sequences
in time of things where what

610
00:31:29,590 --> 00:31:34,270
happens in the future depends
on what happens in the past,

611
00:31:34,270 --> 00:31:36,820
only through the state
at the present.

612
00:31:36,820 --> 00:31:39,140
In other words, if you can
specify the state in the

613
00:31:39,140 --> 00:31:41,760
present, you can forget about
everything in the past.

614
00:31:44,460 --> 00:31:49,750
If you have those kinds of
processes around, you don't

615
00:31:49,750 --> 00:31:54,030
have to study history at all,
which would be very nice.

616
00:31:54,030 --> 00:31:58,110
But unfortunately, not all
processes behave that way.

617
00:31:58,110 --> 00:32:00,860
When you do the modeling to try
to find out what the state

618
00:32:00,860 --> 00:32:03,650
is, which is what you have to
know at the present, you find

619
00:32:03,650 --> 00:32:06,050
out there's a lot of
history involved.

620
00:32:06,050 --> 00:32:08,470
OK, finally, we're going to talk
about random wa;ls and

621
00:32:08,470 --> 00:32:09,640
martingales.

622
00:32:09,640 --> 00:32:12,030
I'm not going to even say
what a random walks or

623
00:32:12,030 --> 00:32:13,410
a martingale is.

624
00:32:13,410 --> 00:32:15,580
We will find out about that
soon enough, but I want to

625
00:32:15,580 --> 00:32:18,760
tell you that's what's in
chapter 7 of the notes.

626
00:32:18,760 --> 00:32:21,420
That's the last topic
we will deal with.

627
00:32:21,420 --> 00:32:24,690
We'll study all sorts of
mixtures of these.

628
00:32:24,690 --> 00:32:26,990
Things which involve a
little bit of each.

629
00:32:26,990 --> 00:32:28,790
We'll start out working
on one thing and

630
00:32:28,790 --> 00:32:30,070
we'll find out another.

631
00:32:30,070 --> 00:32:34,000
One of these other topics is the
right way to look at it.

632
00:32:34,000 --> 00:32:37,390
If you want to know more about
that, please go look at the

633
00:32:37,390 --> 00:32:40,490
notes, and you'll find out
as much as you want.

634
00:32:40,490 --> 00:32:46,210
But it's not appropriate to
talk about it right now.

635
00:32:46,210 --> 00:32:49,400
OK, when, where, and
how is this useful?

636
00:32:49,400 --> 00:32:52,410
You see, I'm almost at the
point where we'll start

637
00:32:52,410 --> 00:32:55,130
actually talking about
real stuff.

638
00:32:55,130 --> 00:32:58,440
And when I say real stuff, I
mean mathematical stuff, which

639
00:32:58,440 --> 00:32:59,780
is not real stuff.

640
00:33:04,120 --> 00:33:05,040
Broad answer--

641
00:33:05,040 --> 00:33:09,160
probability in stochastic
processes are an important

642
00:33:09,160 --> 00:33:12,470
adjunct to rational thought
about all human

643
00:33:12,470 --> 00:33:14,160
and scientific endeavor.

644
00:33:14,160 --> 00:33:16,500
That's a very strong
statement.

645
00:33:16,500 --> 00:33:18,210
I happen to believe it.

646
00:33:18,210 --> 00:33:20,250
You might not believe it.

647
00:33:20,250 --> 00:33:22,720
And you're welcome to
not believe it.

648
00:33:22,720 --> 00:33:26,220
It won't be on a quiz or
anything, believe me.

649
00:33:26,220 --> 00:33:34,010
But almost anything you have to
deal with is dealing with

650
00:33:34,010 --> 00:33:35,860
something in the future.

651
00:33:35,860 --> 00:33:38,890
I mean, you have to plan for
things which are going to

652
00:33:38,890 --> 00:33:41,180
happen in the future.

653
00:33:41,180 --> 00:33:44,180
When you look at the future,
there's a lot of uncertainty

654
00:33:44,180 --> 00:33:45,740
involved with it.

655
00:33:45,740 --> 00:33:48,080
One of the ways is dealing
with uncertainty.

656
00:33:48,080 --> 00:33:50,670
And probably the only scientific
way of dealing with

657
00:33:50,670 --> 00:33:53,180
uncertainty is through
the mechanism

658
00:33:53,180 --> 00:33:56,660
of probability models.

659
00:33:56,660 --> 00:34:01,760
So anything you want to deal
with, which is important,

660
00:34:01,760 --> 00:34:04,000
you're probably better off
knowing something about

661
00:34:04,000 --> 00:34:06,050
probability than not.

662
00:34:06,050 --> 00:34:09,810
A narrow answer is probability
in stochastic processes are

663
00:34:09,810 --> 00:34:13,989
essential components of
the following areas.

664
00:34:13,989 --> 00:34:17,889
Now, I must confess I made up
this list in about 10 minutes

665
00:34:17,889 --> 00:34:20,860
without thinking about
it very seriously.

666
00:34:20,860 --> 00:34:22,880
And these things are related
to each other.

667
00:34:22,880 --> 00:34:25,690
Some of them are parts
of others.

668
00:34:25,690 --> 00:34:26,360
Let me read them.

669
00:34:26,360 --> 00:34:28,330
Communication systems
and networks.

670
00:34:28,330 --> 00:34:31,960
That's where I got involved
in this question, and very

671
00:34:31,960 --> 00:34:33,350
important there.

672
00:34:33,350 --> 00:34:34,820
Computer systems.

673
00:34:34,820 --> 00:34:38,420
I also got involved in it
because of computer systems.

674
00:34:38,420 --> 00:34:39,940
Queuing in all areas.

675
00:34:39,940 --> 00:34:42,400
Well, I got involved in queuing
because of being

676
00:34:42,400 --> 00:34:44,409
interested in networks.

677
00:34:44,409 --> 00:34:46,820
risk management in all areas.

678
00:34:46,820 --> 00:34:49,370
I got interested in that because
I started to get

679
00:34:49,370 --> 00:34:52,790
disturbed about civilization
destroying itself because

680
00:34:52,790 --> 00:34:56,199
people who have a great deal of
power don't know anything

681
00:34:56,199 --> 00:34:57,449
about taking risks.

682
00:35:00,600 --> 00:35:02,560
OK, catastrophe management.

683
00:35:02,560 --> 00:35:04,860
How do you prevent oil spills
and things like that?

684
00:35:04,860 --> 00:35:08,960
How do you prevent nuclear
plants from going off?

685
00:35:08,960 --> 00:35:11,660
How do you prevent nuclear
weapons from falling in the

686
00:35:11,660 --> 00:35:14,320
hands of the wrong people?

687
00:35:14,320 --> 00:35:17,050
These again, are probability
issues.

688
00:35:17,050 --> 00:35:19,760
These are important probability
issues because

689
00:35:19,760 --> 00:35:24,430
most people don't regard them
as probability issues.

690
00:35:24,430 --> 00:35:28,520
If you say there is one chance
in a billion that something

691
00:35:28,520 --> 00:35:33,510
will happen, 3/4 of the
population will say, that's

692
00:35:33,510 --> 00:35:34,460
not acceptable.

693
00:35:34,460 --> 00:35:37,400
I don't want any risk.

694
00:35:37,400 --> 00:35:39,050
And these people are fools.

695
00:35:39,050 --> 00:35:44,950
But unfortunately, these fools
outnumber those of us who have

696
00:35:44,950 --> 00:35:46,440
studied these issues.

697
00:35:46,440 --> 00:35:48,460
So we have to deal with it.

698
00:35:48,460 --> 00:35:51,340
We have to understand
it if nothing else.

699
00:35:51,340 --> 00:35:53,890
OK, failures in all
types of systems--

700
00:35:53,890 --> 00:35:56,790
operations research, biology,
medicine, optical systems, and

701
00:35:56,790 --> 00:35:58,070
control system.

702
00:35:58,070 --> 00:35:59,500
Name your own favorite thing.

703
00:35:59,500 --> 00:36:01,550
You can put it all in.

704
00:36:01,550 --> 00:36:04,410
Probability gets used
everywhere.

705
00:36:04,410 --> 00:36:05,660
OK, let's go to the axioms.

706
00:36:09,460 --> 00:36:13,480
Probability models have three
components to them.

707
00:36:13,480 --> 00:36:16,020
There's a sample space.

708
00:36:16,020 --> 00:36:18,810
Now, here we're in mathematics
again.

709
00:36:18,810 --> 00:36:24,050
The sample space is just
a set of things.

710
00:36:24,050 --> 00:36:26,660
You don't have to be at all
specific about what those

711
00:36:26,660 --> 00:36:28,970
things are.

712
00:36:28,970 --> 00:36:32,400
I mean, at this point we're
right in to set theory, which

713
00:36:32,400 --> 00:36:36,400
is the most basic part
of mathematics again.

714
00:36:36,400 --> 00:36:40,430
And a set contains elements.

715
00:36:40,430 --> 00:36:41,890
And that's what we're
talking about here.

716
00:36:41,890 --> 00:36:43,700
So there's a sample space.

717
00:36:43,700 --> 00:36:45,770
There are the elements
in that sample space.

718
00:36:48,310 --> 00:36:52,120
There's also a collection
of things called events.

719
00:36:52,120 --> 00:36:56,650
Now, the events are subsets
of the sample space.

720
00:36:56,650 --> 00:37:02,700
And if you're dealing with a
finite set of things, there's

721
00:37:02,700 --> 00:37:06,740
no reason why the events should
not be all subsets of

722
00:37:06,740 --> 00:37:09,830
that countable collection
of things.

723
00:37:09,830 --> 00:37:13,870
If you have a deck of cards,
there are 52 factorial ways of

724
00:37:13,870 --> 00:37:17,310
arranging the cards in
that deck of cards.

725
00:37:17,310 --> 00:37:19,590
Very large number.

726
00:37:19,590 --> 00:37:23,290
But when you talk about subsets
of that, you might as

727
00:37:23,290 --> 00:37:29,530
well talk about all combinations
of those

728
00:37:29,530 --> 00:37:31,550
configurations of the deck.

729
00:37:31,550 --> 00:37:34,770
You can talk about, what's the
probability that the first

730
00:37:34,770 --> 00:37:38,770
five cards in that deck happen
to contain 4 aces?

731
00:37:38,770 --> 00:37:40,870
That's an easy thing
to compute.

732
00:37:40,870 --> 00:37:44,010
I'm sure you've all computed
it at some point or other.

733
00:37:44,010 --> 00:37:46,640
Those who like to play poker,
of course do this.

734
00:37:46,640 --> 00:37:48,960
It's fun.

735
00:37:48,960 --> 00:37:51,540
But it's a straightforward
problem.

736
00:37:51,540 --> 00:37:55,660
When you have these countable
sets of things, there's no

737
00:37:55,660 --> 00:38:01,010
reason at all for not having the
set of events consist of

738
00:38:01,010 --> 00:38:04,200
all possible subsets.

739
00:38:04,200 --> 00:38:06,150
Well, people believed that
for a long time.

740
00:38:06,150 --> 00:38:09,640
One of the things that forced
Kolmogorov to start dealing

741
00:38:09,640 --> 00:38:13,420
with these axioms was the
realization that when you had

742
00:38:13,420 --> 00:38:18,010
much more complicated sets,
where in fact you had the set

743
00:38:18,010 --> 00:38:21,910
of real numbers as possible
outcomes, or sequences of

744
00:38:21,910 --> 00:38:26,460
things which go from 0 to
infinity, and all of these

745
00:38:26,460 --> 00:38:32,000
sets, which are uncountable,
you really can't make sense

746
00:38:32,000 --> 00:38:37,960
out of probability models where
all subsets of sample

747
00:38:37,960 --> 00:38:41,240
points are called events.

748
00:38:41,240 --> 00:38:45,020
So in terms of measure theory,
you're forced to restrict the

749
00:38:45,020 --> 00:38:46,980
set of things you call events.

750
00:38:46,980 --> 00:38:48,860
Now, we're not going to
deal with measure

751
00:38:48,860 --> 00:38:51,200
theory in this subject.

752
00:38:51,200 --> 00:38:54,710
But every once in a while, we
will have to mention it

753
00:38:54,710 --> 00:38:59,130
because the reason why a lot of
things are the way they are

754
00:38:59,130 --> 00:39:01,260
is because of measure theory.

755
00:39:01,260 --> 00:39:03,900
So you'll have to be at
least conscious of it.

756
00:39:03,900 --> 00:39:08,040
If you really want to be
serious, as far as your study

757
00:39:08,040 --> 00:39:11,310
of mathematical probability
theory, you really have to

758
00:39:11,310 --> 00:39:13,570
take a course in measure
theory at some point.

759
00:39:13,570 --> 00:39:16,240
But you don't have
to do it now.

760
00:39:16,240 --> 00:39:20,640
In fact, I would almost urge
most of you not to do it now.

761
00:39:20,640 --> 00:39:26,440
Because once you get all the
way into measure theory,

762
00:39:26,440 --> 00:39:29,310
you're so far into measure
theory that you can't come

763
00:39:29,310 --> 00:39:31,790
back and think about real
problems anymore.

764
00:39:31,790 --> 00:39:35,460
You're suddenly stuck in the
world of mathematics, which

765
00:39:35,460 --> 00:39:38,080
happens to lots of people.

766
00:39:38,080 --> 00:39:41,100
So anyway, some of you should
learn about all this

767
00:39:41,100 --> 00:39:42,720
mathematics.

768
00:39:42,720 --> 00:39:43,670
Some of you shouldn't.

769
00:39:43,670 --> 00:39:45,585
Some of you should learn
about it later.

770
00:39:45,585 --> 00:39:48,150
So you can do whatever
you want.

771
00:39:48,150 --> 00:39:52,590
OK, the axioms about events
is that if you

772
00:39:52,590 --> 00:39:53,950
have a set of events.

773
00:39:53,950 --> 00:39:58,260
In other words, a set of
subsets, and it's a countable

774
00:39:58,260 --> 00:40:01,310
set, then the union
of all of those--

775
00:40:01,310 --> 00:40:05,190
the union from n equals
1 to infinity of a sub

776
00:40:05,190 --> 00:40:08,750
n is also an event.

777
00:40:08,750 --> 00:40:10,510
I've gone for 50 minutes
and nobody has

778
00:40:10,510 --> 00:40:13,660
asked a question yet.

779
00:40:13,660 --> 00:40:14,910
Who has a question?

780
00:40:17,240 --> 00:40:19,990
Who thinks that all of
this is nonsense?

781
00:40:19,990 --> 00:40:21,050
How many of you?

782
00:40:21,050 --> 00:40:22,300
I do.

783
00:40:26,104 --> 00:40:28,830
OK, I'll come back in
another 10 minutes.

784
00:40:28,830 --> 00:40:31,320
And if nobody has a question
by then, I'm just going to

785
00:40:31,320 --> 00:40:33,560
stop and wait.

786
00:40:33,560 --> 00:40:35,530
OK, so anyway.

787
00:40:35,530 --> 00:40:38,990
If you look at a union
of events.

788
00:40:38,990 --> 00:40:44,410
Now, remember, that an event
is a subset of points.

789
00:40:44,410 --> 00:40:46,920
We're just talking about
set theory now.

790
00:40:46,920 --> 00:40:51,360
So the union of this
union here--

791
00:40:51,360 --> 00:40:52,620
excuse me.

792
00:40:52,620 --> 00:40:58,960
This union here is A1, all the
points in A1, and all the

793
00:40:58,960 --> 00:41:02,230
points in A2, and all the points
in A3, all the way up

794
00:41:02,230 --> 00:41:02,970
to infinity.

795
00:41:02,970 --> 00:41:05,060
That's what we're talking
about here.

796
00:41:05,060 --> 00:41:09,020
And one of the axioms of
probability theory is that if

797
00:41:09,020 --> 00:41:12,650
each of these things are events,
then this union is

798
00:41:12,650 --> 00:41:13,420
also an event.

799
00:41:13,420 --> 00:41:14,590
That's just an axiom.

800
00:41:14,590 --> 00:41:18,930
You can't define events
if that's not true.

801
00:41:18,930 --> 00:41:22,640
And if you try to define events
where this isn't true,

802
00:41:22,640 --> 00:41:25,400
you eventually come into the
most god awful problems you

803
00:41:25,400 --> 00:41:27,090
might imagine.

804
00:41:27,090 --> 00:41:31,110
And suddenly, nothing
make sense anymore.

805
00:41:31,110 --> 00:41:36,700
Most of the time when we define
a set of events in a

806
00:41:36,700 --> 00:41:40,170
probability model, each
singleton event--

807
00:41:40,170 --> 00:41:44,070
namely, each single point has
a set, which contains only

808
00:41:44,070 --> 00:41:47,060
that element, is taken
as an event.

809
00:41:47,060 --> 00:41:49,930
There's no real reason
to not do that.

810
00:41:49,930 --> 00:41:54,890
If you don't do that, you might
as well just put those

811
00:41:54,890 --> 00:41:59,430
points together and not regard
them as separate points.

812
00:41:59,430 --> 00:42:02,910
We will see an example in a
little bit where, in fact, you

813
00:42:02,910 --> 00:42:03,950
might want to do that.

814
00:42:03,950 --> 00:42:07,080
But let's hold that off
for a little bit.

815
00:42:07,080 --> 00:42:09,320
OK, not all subsets
need to be events.

816
00:42:09,320 --> 00:42:14,190
Usually, each sample point is
taken to be a singleton event.

817
00:42:14,190 --> 00:42:17,870
And then non-events are
truly weird things.

818
00:42:17,870 --> 00:42:20,320
I mean, as soon as you take
all sample points to be

819
00:42:20,320 --> 00:42:24,460
events, all countable unions of
sample points are events.

820
00:42:27,150 --> 00:42:30,490
And then intersections of events
are events, and so

821
00:42:30,490 --> 00:42:32,330
forth, and so forth,
and so forth.

822
00:42:32,330 --> 00:42:35,610
So most things are events.

823
00:42:35,610 --> 00:42:38,820
And just because of measure
theory, you can't make all

824
00:42:38,820 --> 00:42:39,580
things events.

825
00:42:39,580 --> 00:42:43,610
And I'm not going to give you
any example of that because

826
00:42:43,610 --> 00:42:47,030
examples are horrendous.

827
00:42:47,030 --> 00:42:51,120
OK, the empty set has
to be an event.

828
00:42:51,120 --> 00:42:54,850
Why does the empty set
have to be an event.

829
00:42:54,850 --> 00:42:56,293
If we're going to believe
these axioms--

830
00:42:58,950 --> 00:43:01,990
I'm in a real bind here because
every one of you

831
00:43:01,990 --> 00:43:05,230
people has seen these
axioms before.

832
00:43:05,230 --> 00:43:10,320
And you've all gone on and said,
I can get an A in any

833
00:43:10,320 --> 00:43:13,630
probability class in the world
without having any idea of

834
00:43:13,630 --> 00:43:15,900
what these axioms
are all about.

835
00:43:15,900 --> 00:43:18,620
And therefore, it's
unimportant.

836
00:43:18,620 --> 00:43:21,180
So you see something
that says, the

837
00:43:21,180 --> 00:43:23,370
empty set is an event.

838
00:43:23,370 --> 00:43:24,450
And you say, well, of
course that has

839
00:43:24,450 --> 00:43:25,780
nothing to do with anything.

840
00:43:25,780 --> 00:43:28,450
Why should I worry about whether
the empty set is an

841
00:43:28,450 --> 00:43:28,960
event or not?

842
00:43:28,960 --> 00:43:31,730
The empty set can't happen,
so how can it be an event?

843
00:43:31,730 --> 00:43:35,380
Well, because of these axioms,
it has to be an event.

844
00:43:35,380 --> 00:43:40,750
The axioms say that if A is an
event, and that's the whole

845
00:43:40,750 --> 00:43:43,640
sample space, then the
complement has

846
00:43:43,640 --> 00:43:45,380
to be an event also.

847
00:43:45,380 --> 00:43:48,320
So that says that the empty
set has to be an event.

848
00:43:48,320 --> 00:43:51,320
And that just follows
from the axioms.

849
00:43:51,320 --> 00:43:53,910
If all sample points are
singleton events, then all

850
00:43:53,910 --> 00:43:59,250
finite and countable
sets are events.

851
00:43:59,250 --> 00:44:01,790
And finally, deMorgan's law.

852
00:44:01,790 --> 00:44:05,700
Is there anyone who isn't
familiar with deMorgan's law?

853
00:44:05,700 --> 00:44:09,210
Anyone who hasn't seen
even that small

854
00:44:09,210 --> 00:44:12,140
amount of set theory?

855
00:44:12,140 --> 00:44:16,570
If not, look it up on--

856
00:44:16,570 --> 00:44:23,120
what's the name of
the computer--

857
00:44:23,120 --> 00:44:25,560
Wikipedia.

858
00:44:25,560 --> 00:44:27,260
Most of you will think
that things on

859
00:44:27,260 --> 00:44:29,470
Wikipedia are not reliable.

860
00:44:29,470 --> 00:44:34,070
Strangely enough, in terms of
probability theory and a lot

861
00:44:34,070 --> 00:44:37,930
of mathematics, Wikipedia does
things a whole lot better than

862
00:44:37,930 --> 00:44:40,150
most textbooks do.

863
00:44:40,150 --> 00:44:43,520
So any time you're unfamiliar
with what a word means or

864
00:44:43,520 --> 00:44:46,170
something, you can look
it up in your

865
00:44:46,170 --> 00:44:49,300
old probability textbook.

866
00:44:49,300 --> 00:44:51,270
If you've used [INAUDIBLE]

867
00:44:51,270 --> 00:44:53,310
and [INAUDIBLE], you will
probably find the

868
00:44:53,310 --> 00:44:54,670
right answer there.

869
00:44:54,670 --> 00:44:57,630
Other textbooks, maybe
the right answer.

870
00:44:57,630 --> 00:45:00,520
Wikipedia's more reliable
than most of them.

871
00:45:00,520 --> 00:45:02,920
And it's also clearer
than most of them.

872
00:45:02,920 --> 00:45:08,240
So I highly recommend using
Wikipedia whenever you get

873
00:45:08,240 --> 00:45:10,750
totally confused by something.

874
00:45:10,750 --> 00:45:17,020
OK, so probability measure
and events satisfies

875
00:45:17,020 --> 00:45:18,440
the following axioms.

876
00:45:18,440 --> 00:45:21,350
We've said what things
are events.

877
00:45:21,350 --> 00:45:24,710
The only things that have
probabilities are events.

878
00:45:24,710 --> 00:45:27,920
So the entire set has
a probability.

879
00:45:27,920 --> 00:45:30,630
When you do the experiment,
something has to happen.

880
00:45:30,630 --> 00:45:32,560
So one of the sample
points occurs.

881
00:45:32,560 --> 00:45:34,820
That's the whole idea
of probability.

882
00:45:34,820 --> 00:45:37,960
And therefore, omega
has probability 1.

883
00:45:37,960 --> 00:45:39,540
Capital Omega.

884
00:45:39,540 --> 00:45:44,010
If A is an event, then the
probability of A has to be

885
00:45:44,010 --> 00:45:46,370
greater than or equal to 0.

886
00:45:46,370 --> 00:45:49,670
You can probably see without too
much trouble why it has to

887
00:45:49,670 --> 00:45:51,790
be less than or equal
to 1 also.

888
00:45:51,790 --> 00:45:53,700
But that's not one
of the axioms.

889
00:45:53,700 --> 00:45:59,670
You see, when you state a set of
axioms for something, you'd

890
00:45:59,670 --> 00:46:03,970
like to use the minimum set of
axioms you can, so that you

891
00:46:03,970 --> 00:46:07,590
don't have to verify too many
things before you say, yes,

892
00:46:07,590 --> 00:46:10,120
this satisfies all the axioms.

893
00:46:10,120 --> 00:46:12,680
So the second one is the
probability of A has to be

894
00:46:12,680 --> 00:46:14,940
greater than or equal to 0.

895
00:46:14,940 --> 00:46:18,880
The third one says that if you
have a sequence of disjoint

896
00:46:18,880 --> 00:46:23,450
events, incidentally when I say
a sequence, I will almost

897
00:46:23,450 --> 00:46:25,885
always mean a countably
infinite sequence--

898
00:46:25,885 --> 00:46:29,740
A1, A2, A3, all the way up.

899
00:46:29,740 --> 00:46:32,980
If I'm talking about what
most of you would

900
00:46:32,980 --> 00:46:35,130
call a finite sequence--

901
00:46:35,130 --> 00:46:37,610
and I like the word "finite
sequence," but I like to be

902
00:46:37,610 --> 00:46:39,450
able to talk about sequences.

903
00:46:39,450 --> 00:46:42,180
I'm talking about a finite
sequence I will usually call

904
00:46:42,180 --> 00:46:45,280
it an n-tuple of random
variables or

905
00:46:45,280 --> 00:46:47,400
an n-tuple of things.

906
00:46:47,400 --> 00:46:51,380
So sequence really means you
go the whole way out.

907
00:46:51,380 --> 00:46:55,800
OK, if A1, A2, all the way
up are disjoint events--

908
00:46:55,800 --> 00:46:56,630
disjoint.

909
00:46:56,630 --> 00:47:00,460
Disjoint means if omega is only
in one, it can't be in

910
00:47:00,460 --> 00:47:01,690
any of the others.

911
00:47:01,690 --> 00:47:05,990
Then the probability of this
countable union is going to be

912
00:47:05,990 --> 00:47:08,830
equal to the sum of the
probabilities of the

913
00:47:08,830 --> 00:47:10,980
individual event.

914
00:47:10,980 --> 00:47:13,910
Anyone who has ever done a
probability problem knows all

915
00:47:13,910 --> 00:47:15,200
of these things.

916
00:47:15,200 --> 00:47:17,570
The only thing you don't know
and you probably haven't

917
00:47:17,570 --> 00:47:19,760
thought about is why
everything else

918
00:47:19,760 --> 00:47:21,730
follows from this.

919
00:47:21,730 --> 00:47:25,610
But this is the whole
mathematical theory.

920
00:47:25,610 --> 00:47:27,040
Why should we study
it anymore?

921
00:47:27,040 --> 00:47:28,210
We're done.

922
00:47:28,210 --> 00:47:29,340
We have the axioms.

923
00:47:29,340 --> 00:47:34,430
Everything else follows, it's
just a matter of computation.

924
00:47:34,430 --> 00:47:38,180
Just sit down and do it.

925
00:47:38,180 --> 00:47:41,480
Not quite that simple.

926
00:47:41,480 --> 00:47:45,990
Anyway, a few consequences of
the probability of the empty

927
00:47:45,990 --> 00:47:51,030
set is 0, which says when you
do an experiment something's

928
00:47:51,030 --> 00:47:51,690
going to happen.

929
00:47:51,690 --> 00:47:55,360
And therefore, the probability
that nothing happens is 0

930
00:47:55,360 --> 00:47:57,440
because that's what
the model says.

931
00:47:57,440 --> 00:48:02,320
The probability of the
complement of an event is 1

932
00:48:02,320 --> 00:48:05,380
minus the probability
of that event.

933
00:48:05,380 --> 00:48:08,260
Which, in fact, is what's says
that all events have to have

934
00:48:08,260 --> 00:48:11,240
probabilities less than
or equal to 1.

935
00:48:11,240 --> 00:48:15,250
And if the event A is contained
in the event B--

936
00:48:15,250 --> 00:48:17,870
remember when we talk about
events, we're talking about

937
00:48:17,870 --> 00:48:20,780
two different things,
both simultaneously.

938
00:48:20,780 --> 00:48:26,150
One of them is this beautiful
idea with measure theory

939
00:48:26,150 --> 00:48:28,340
worked into it and
everything else.

940
00:48:28,340 --> 00:48:31,910
And the other is just a simple
set theoretic idea.

941
00:48:31,910 --> 00:48:34,170
And all we need to be
familiar with is a

942
00:48:34,170 --> 00:48:36,870
set theoretical idea.

943
00:48:36,870 --> 00:48:42,190
Within that set theoretical
idea, A contained in B means

944
00:48:42,190 --> 00:48:46,400
that every sample point that's
in A is also in B. It means

945
00:48:46,400 --> 00:48:49,650
that when you do an experiment,
and the event A

946
00:48:49,650 --> 00:48:53,360
occurs, the event B has to
occur because one of the

947
00:48:53,360 --> 00:48:57,610
things that compose
A has to occur.

948
00:48:57,610 --> 00:49:01,255
And that thing has to be in B
because A is contained in B.

949
00:49:01,255 --> 00:49:04,390
So the probability of A has to
be less than or equal to the

950
00:49:04,390 --> 00:49:08,170
probability of B. That has to
be less than or equal to 1.

951
00:49:08,170 --> 00:49:11,310
These are things you all know.

952
00:49:11,310 --> 00:49:13,230
Another consequence is
the union bound.

953
00:49:13,230 --> 00:49:16,690
Many of you have probably
seen the union bound.

954
00:49:16,690 --> 00:49:22,830
We will use it probably almost
every day in this course.

955
00:49:22,830 --> 00:49:26,600
So it's good to have that as one
of the things you remember

956
00:49:26,600 --> 00:49:28,740
at the highest level.

957
00:49:28,740 --> 00:49:31,780
If you have a set of events--

958
00:49:31,780 --> 00:49:34,360
A1, A2, and so forth--

959
00:49:34,360 --> 00:49:36,270
the probability of
that union--

960
00:49:36,270 --> 00:49:38,950
namely, the event that consists
of all of them--

961
00:49:38,950 --> 00:49:43,360
is less than or equal to the
sum of the individual event

962
00:49:43,360 --> 00:49:45,010
probabilities.

963
00:49:45,010 --> 00:49:52,540
I give a little proof here for
just two events, A1 and A2.

964
00:49:52,540 --> 00:49:54,790
So you see why this is true.

965
00:49:54,790 --> 00:49:57,950
I hope you can extend
this to 3 and 4.

966
00:49:57,950 --> 00:50:00,400
I can't draw a picture of
it very easily for 3

967
00:50:00,400 --> 00:50:01,780
and 4 and so forth.

968
00:50:01,780 --> 00:50:03,980
But here's the event A1.

969
00:50:03,980 --> 00:50:05,830
Here's the event A2.

970
00:50:05,830 --> 00:50:10,320
Visualize this as a set of
sample points, which are just

971
00:50:10,320 --> 00:50:12,720
in the two-dimensional
space here.

972
00:50:12,720 --> 00:50:15,730
So all these points
here are in A1.

973
00:50:15,730 --> 00:50:18,790
All these points are in A2.

974
00:50:18,790 --> 00:50:20,750
This set of points here
are the points that

975
00:50:20,750 --> 00:50:23,770
are in A1 and A2.

976
00:50:23,770 --> 00:50:27,610
I will use just writing things
next to each other to mean

977
00:50:27,610 --> 00:50:28,630
intersection.

978
00:50:28,630 --> 00:50:33,950
And sometimes I'll use a big
cap to main intersection.

979
00:50:33,950 --> 00:50:37,810
So all of these things are
both in A1 and A2.

980
00:50:37,810 --> 00:50:40,550
This is A2, but not A1.

981
00:50:40,550 --> 00:50:45,860
So the probability of this whole
event, A1 union A2, is

982
00:50:45,860 --> 00:50:49,810
the probability of this thing
and this thing together.

983
00:50:49,810 --> 00:50:53,300
So it's the probability
of this plus the

984
00:50:53,300 --> 00:50:56,040
probability of this.

985
00:50:56,040 --> 00:50:59,890
The probability of this is less
than the probability of

986
00:50:59,890 --> 00:51:05,630
A2 because this is contained
in that whole rectangle.

987
00:51:05,630 --> 00:51:09,680
And therefore, the probability
of the union of A1 and A2 is

988
00:51:09,680 --> 00:51:13,160
less than or equal to the
probability of A1 plus

989
00:51:13,160 --> 00:51:15,060
probability of A2.

990
00:51:15,060 --> 00:51:18,700
Now, the classy way to extend
this to a countably infinite

991
00:51:18,700 --> 00:51:21,860
set is to use induction.

992
00:51:21,860 --> 00:51:26,340
And I leave that as something
that you can all play with

993
00:51:26,340 --> 00:51:30,090
some time when it's not between
9:30 and 11:00 in the

994
00:51:30,090 --> 00:51:32,050
morning and you're struggling
to stay awake.

995
00:51:35,880 --> 00:51:38,410
And if you don't want to do that
on your own, you can look

996
00:51:38,410 --> 00:51:40,430
at it in the notes.

997
00:51:40,430 --> 00:51:45,550
OK, these axioms look
ho-hum to you.

998
00:51:45,550 --> 00:51:47,970
And you've always ignored them
before, and you think you're

999
00:51:47,970 --> 00:51:50,440
going to be able to
ignore them now.

1000
00:51:50,440 --> 00:51:53,670
Partly you can, but partly you
can't because every once in a

1001
00:51:53,670 --> 00:51:58,660
while we'll start doing things
where you really need to

1002
00:51:58,660 --> 00:52:02,510
understand what the
axioms say.

1003
00:52:02,510 --> 00:52:06,770
OK, one other thing which you
might not have noticed.

1004
00:52:06,770 --> 00:52:10,570
When you studied elementary
probability, wherever you

1005
00:52:10,570 --> 00:52:15,040
studied it, what do you spend
most of your time doing?

1006
00:52:15,040 --> 00:52:19,160
You spent most of your time
talking about random variables

1007
00:52:19,160 --> 00:52:22,740
and talking about
expectations.

1008
00:52:22,740 --> 00:52:28,370
The axioms don't have random
variables in them.

1009
00:52:28,370 --> 00:52:30,850
They don't have expectations
in them.

1010
00:52:30,850 --> 00:52:33,560
All they have in them
is events and

1011
00:52:33,560 --> 00:52:36,230
probabilities of events.

1012
00:52:36,230 --> 00:52:39,380
So these axioms say that the
really important things in

1013
00:52:39,380 --> 00:52:43,480
probability are the events and
the probabilities of events.

1014
00:52:43,480 --> 00:52:46,820
And the random variables and the
expectations are derived

1015
00:52:46,820 --> 00:52:51,480
quantities, which we'll now
start to talk about.

1016
00:52:51,480 --> 00:52:54,470
OK, so we're now down to
independent events and

1017
00:52:54,470 --> 00:52:56,980
experiments.

1018
00:52:56,980 --> 00:53:02,350
Two events, A1 and A2, are
independent if the probability

1019
00:53:02,350 --> 00:53:05,310
of the two of them is equal
to the product of their

1020
00:53:05,310 --> 00:53:06,290
probabilities.

1021
00:53:06,290 --> 00:53:08,400
You've all seen this.

1022
00:53:08,400 --> 00:53:09,970
I'm sure you've all seen it.

1023
00:53:09,970 --> 00:53:13,090
If you haven't at least seen it,
you probably shouldn't be

1024
00:53:13,090 --> 00:53:17,240
in this class because even
though the text does

1025
00:53:17,240 --> 00:53:21,920
everything in detail that has to
be done, you need to have a

1026
00:53:21,920 --> 00:53:24,660
little bit of insight from
having dealt with these

1027
00:53:24,660 --> 00:53:26,410
subjects before.

1028
00:53:26,410 --> 00:53:28,830
If you don't have that, you're
just going to get lost very,

1029
00:53:28,830 --> 00:53:31,570
very quickly.

1030
00:53:31,570 --> 00:53:35,440
So the probability is the
intersection of the event A1

1031
00:53:35,440 --> 00:53:40,160
and A2 is the product
of the two.

1032
00:53:40,160 --> 00:53:43,590
Now, in other words, you have
a red die and a white die.

1033
00:53:43,590 --> 00:53:49,900
You flip the dice, what's the
probability that you get a 1

1034
00:53:49,900 --> 00:53:52,360
for the red die and a
1 for the white die?

1035
00:53:52,360 --> 00:53:57,630
Well, the probability you get
a 1 for the red die is 1/6.

1036
00:53:57,630 --> 00:54:00,360
Just by symmetry, there are only
6 possible things that

1037
00:54:00,360 --> 00:54:01,470
can happen.

1038
00:54:01,470 --> 00:54:04,040
Probability of white
die comes up as 1.

1039
00:54:04,040 --> 00:54:07,530
Probability is 1/6 for that.

1040
00:54:07,530 --> 00:54:12,720
And the probability of the two
things, they're independent.

1041
00:54:12,720 --> 00:54:16,970
There's a sense of real-world
independence and probability

1042
00:54:16,970 --> 00:54:18,420
theory independence.

1043
00:54:18,420 --> 00:54:21,290
Real-world independence says the
two things are isolated,

1044
00:54:21,290 --> 00:54:23,290
they don't interfere
with each other.

1045
00:54:23,290 --> 00:54:28,020
Probability theory says just
by definition, well, the

1046
00:54:28,020 --> 00:54:31,800
real-world idea of them not
interfering with each other

1047
00:54:31,800 --> 00:54:33,390
should say--

1048
00:54:33,390 --> 00:54:38,820
and I'm waving my hands here
because this is so elementary,

1049
00:54:38,820 --> 00:54:39,690
you all know it.

1050
00:54:39,690 --> 00:54:43,620
And I would bore you if I
talked about it more.

1051
00:54:43,620 --> 00:54:45,305
But I probably should
talk about it, but

1052
00:54:45,305 --> 00:54:46,280
I'm not going to.

1053
00:54:46,280 --> 00:54:50,060
Anyway, this is the definition
of independence.

1054
00:54:50,060 --> 00:54:54,000
If you don't have any idea of
how this corresponds to being

1055
00:54:54,000 --> 00:54:58,510
unconnected in the real-world,
then go to Wikipedia.

1056
00:54:58,510 --> 00:54:59,330
Read the notes.

1057
00:54:59,330 --> 00:55:00,760
Well, you should read
the notes anyway.

1058
00:55:00,760 --> 00:55:03,980
I hope you will read the notes
because I'm not going to say

1059
00:55:03,980 --> 00:55:07,140
everything in class that
needs to be said.

1060
00:55:07,140 --> 00:55:10,450
And you will get a better
feeling for it.

1061
00:55:10,450 --> 00:55:12,820
Now, here's something
important.

1062
00:55:12,820 --> 00:55:17,410
Given two probability models,
a combined model can be

1063
00:55:17,410 --> 00:55:22,460
defined in which, first, the
sample space, omega, is the

1064
00:55:22,460 --> 00:55:26,820
Cartesian product of omega
1 and omega 2.

1065
00:55:26,820 --> 00:55:28,950
Namely, it's the Cartesian
product of

1066
00:55:28,950 --> 00:55:30,820
the two sample spaces.

1067
00:55:30,820 --> 00:55:34,390
Think of rolling the red
die and the white die.

1068
00:55:34,390 --> 00:55:36,740
Rolling a red die is
an experiment.

1069
00:55:36,740 --> 00:55:39,460
There are 6 possible
outcomes, a 1 to 6.

1070
00:55:39,460 --> 00:55:42,940
Rolled a white die, there
are 6 possible

1071
00:55:42,940 --> 00:55:44,940
outcomes, a 1 to a 6.

1072
00:55:44,940 --> 00:55:48,250
You roll the two dice together,
and you really need

1073
00:55:48,250 --> 00:55:49,890
to have some way of
putting these

1074
00:55:49,890 --> 00:55:51,510
two experiments together.

1075
00:55:51,510 --> 00:55:52,650
How do you put them together?

1076
00:55:52,650 --> 00:55:57,770
You talk about the outcome for
the two dice, number for one

1077
00:55:57,770 --> 00:55:59,020
and number for the other.

1078
00:55:59,020 --> 00:56:03,200
The Cartesian product simply
means you have the set made up

1079
00:56:03,200 --> 00:56:08,680
of 1 to 6 Cartesian product
with 1 to 6.

1080
00:56:08,680 --> 00:56:10,065
So you have 36 possibilities.

1081
00:56:16,350 --> 00:56:19,010
It's an interesting thing,
which comes from

1082
00:56:19,010 --> 00:56:20,790
Kolmogorov's axioms.

1083
00:56:20,790 --> 00:56:27,420
That, in fact, you can take in
any two probability models for

1084
00:56:27,420 --> 00:56:29,100
two different experiments.

1085
00:56:29,100 --> 00:56:33,440
You can take this Cartesian
product of sample points.

1086
00:56:33,440 --> 00:56:36,830
You can assume that what happens
here is independent of

1087
00:56:36,830 --> 00:56:38,500
what happens here.

1088
00:56:38,500 --> 00:56:41,600
And when you do this, you will,
in fact, get something

1089
00:56:41,600 --> 00:56:45,180
for the two experiments put
together which satisfies

1090
00:56:45,180 --> 00:56:47,850
Kolmogorov's axioms.

1091
00:56:47,850 --> 00:56:53,590
That is neither trivial nor
very hard to prove.

1092
00:56:53,590 --> 00:56:58,020
I mean, for the case of two
dice, you can see it almost

1093
00:56:58,020 --> 00:56:58,990
immediately.

1094
00:56:58,990 --> 00:57:00,720
I mean, you see what the
sample space is.

1095
00:57:04,050 --> 00:57:06,220
It's this Cartesian product.

1096
00:57:06,220 --> 00:57:08,460
And you see what the
probabilities have to be

1097
00:57:08,460 --> 00:57:12,370
because the probability of, say,
1 and 2 for the red die

1098
00:57:12,370 --> 00:57:18,070
and 1 and 2 for the white
is 2, 6 times 2, 6.

1099
00:57:18,070 --> 00:57:24,180
So with probability 1/9, you're
going to get a 1 and a

1100
00:57:24,180 --> 00:57:28,360
2 combined with a 1 and a 2.

1101
00:57:28,360 --> 00:57:31,440
I'm going to talk a little
bit later about something

1102
00:57:31,440 --> 00:57:34,240
that you all know.

1103
00:57:34,240 --> 00:57:37,650
What happens if you roll
two white dice?

1104
00:57:37,650 --> 00:57:40,380
This is something you all ought
to think about a little

1105
00:57:40,380 --> 00:57:44,420
bit because it really isn't
as simple as it sounds.

1106
00:57:44,420 --> 00:57:49,840
If you roll two dice, what's
the probability that you'll

1107
00:57:49,840 --> 00:57:52,440
get a 1 and a 2?

1108
00:57:52,440 --> 00:57:56,440
And how can you justify that?

1109
00:57:56,440 --> 00:57:59,660
First, what's a sample space
when you roll two white dice?

1110
00:58:05,350 --> 00:58:07,630
Well, if you look at the
possible things that might

1111
00:58:07,630 --> 00:58:15,640
happen, you can get 1, 1; 2,
2; 3, 3; 4, 4; 5, 5; 6, 6.

1112
00:58:15,640 --> 00:58:18,140
You can also get 1, 2 or 2, 1.

1113
00:58:18,140 --> 00:58:21,930
But you can't tell them apart,
so there's one sample point,

1114
00:58:21,930 --> 00:58:25,170
you might say, which
is 1, 2, and 2, 1.

1115
00:58:25,170 --> 00:58:29,410
Another sample point which is
2, 3; 3, 2, and so forth.

1116
00:58:29,410 --> 00:58:33,740
If you count them up, there are
21 separate sample points

1117
00:58:33,740 --> 00:58:35,990
that you might have.

1118
00:58:35,990 --> 00:58:41,150
And when you look at what the
probabilities ought to be, the

1119
00:58:41,150 --> 00:58:44,410
probabilities of the
pairs are 136 each.

1120
00:58:44,410 --> 00:58:48,830
And the probabilities of the
I J, where I is unequal

1121
00:58:48,830 --> 00:58:53,380
to J is 1/18 each.

1122
00:58:53,380 --> 00:58:56,080
That's awful.

1123
00:58:56,080 --> 00:58:58,400
So what do you do?

1124
00:58:58,400 --> 00:59:00,540
When you're rolling two dice,
you do the same thing that

1125
00:59:00,540 --> 00:59:02,860
everybody else does.

1126
00:59:02,860 --> 00:59:05,800
You say, well, even though it's
two white dice, I'm going

1127
00:59:05,800 --> 00:59:08,850
to think of it as if it's a
white die and a red die.

1128
00:59:08,850 --> 00:59:12,060
I'm going to think of it as if
the two are indistinguishable.

1129
00:59:12,060 --> 00:59:15,950
My sample space is going to be
these 36 different things.

1130
00:59:15,950 --> 00:59:18,100
I will never be able
to distinguish a

1131
00:59:18,100 --> 00:59:19,340
1, 2 from a 2, 1.

1132
00:59:19,340 --> 00:59:21,690
But I don't care because
I now know the

1133
00:59:21,690 --> 00:59:24,450
probability of each of them.

1134
00:59:24,450 --> 00:59:30,850
What I'm trying to say by this,
is this a very, very

1135
00:59:30,850 --> 00:59:36,100
trivial example of where you
really have to think through

1136
00:59:36,100 --> 00:59:40,000
the question of what kind of
mathematical model do you want

1137
00:59:40,000 --> 00:59:44,220
of the most simple situation
you can think of almost.

1138
00:59:44,220 --> 00:59:48,290
When you combine two different
experiments together and you

1139
00:59:48,290 --> 00:59:52,930
lose distinguishability,
then what do you do?

1140
00:59:52,930 --> 00:59:55,390
Well, the sensible thing to
do is assume that the

1141
00:59:55,390 --> 00:59:58,010
distinguishability is still
there, but it's not

1142
00:59:58,010 --> 00:59:59,260
observable.

1143
01:00:01,120 --> 01:00:05,760
But that makes it hard to make
a correspondence between the

1144
01:00:05,760 --> 01:00:08,070
real world and the probability
world.

1145
01:00:08,070 --> 01:00:10,660
So we'll come back
to that later.

1146
01:00:10,660 --> 01:00:13,770
But for the most part, you don't
have to worry about it

1147
01:00:13,770 --> 01:00:15,810
because this is something
you've dealt

1148
01:00:15,810 --> 01:00:17,060
with all of your lives.

1149
01:00:19,710 --> 01:00:22,340
I mean, you've done probability

1150
01:00:22,340 --> 01:00:24,730
problems with dice.

1151
01:00:24,730 --> 01:00:27,200
You've done probability problems
with all sorts of

1152
01:00:27,200 --> 01:00:29,140
other things where things are

1153
01:00:29,140 --> 01:00:31,230
indistinguishable from each other.

1154
01:00:31,230 --> 01:00:36,140
And after doing a few of these
problems, you are used to

1155
01:00:36,140 --> 01:00:38,630
being schizophrenic about it.

1156
01:00:38,630 --> 01:00:41,300
And on one hand, thinking
that these things are

1157
01:00:41,300 --> 01:00:43,090
distinguishable to figure
out what all the

1158
01:00:43,090 --> 01:00:44,600
probabilities are.

1159
01:00:44,600 --> 01:00:47,460
And then you go back to saying,
well, they aren't

1160
01:00:47,460 --> 01:00:51,600
really distinguishable, and
you find the right answer.

1161
01:00:51,600 --> 01:00:54,010
So you don't have to
worry about it.

1162
01:00:54,010 --> 01:00:59,390
All I'm trying to say here is
that you should understand it.

1163
01:00:59,390 --> 01:01:03,460
Because when you get the
complicated situations, this

1164
01:01:03,460 --> 01:01:07,540
is one of the main things which
will cause confusion.

1165
01:01:07,540 --> 01:01:10,260
It's one of the main things
where people write papers and

1166
01:01:10,260 --> 01:01:14,180
other people say that paper is
wrong because they're both

1167
01:01:14,180 --> 01:01:18,420
thinking of different
models for it.

1168
01:01:18,420 --> 01:01:23,130
Important thing is if you
satisfy Kolmogorov's axioms in

1169
01:01:23,130 --> 01:01:27,140
each of a set of models, and
most important thing is where

1170
01:01:27,140 --> 01:01:30,200
each of these models are
exactly the same.

1171
01:01:30,200 --> 01:01:34,080
And then you make them each
independent of each other,

1172
01:01:34,080 --> 01:01:36,960
Kolmogorov's axioms are going
to be satisfied for the

1173
01:01:36,960 --> 01:01:40,950
combination, as well as
the individual model.

1174
01:01:40,950 --> 01:01:44,030
Why do I care about that?

1175
01:01:44,030 --> 01:01:46,600
Because we're studying
stochastic processes.

1176
01:01:46,600 --> 01:01:51,370
We're studying an infinite
sequence of random variables.

1177
01:01:51,370 --> 01:01:56,890
And I don't want to generate a
complete probability model for

1178
01:01:56,890 --> 01:01:59,660
an infinite set of random
variables every time I talk

1179
01:01:59,660 --> 01:02:02,410
about an infinite set
of random variables.

1180
01:02:02,410 --> 01:02:04,850
If I'm talking about an infinite
sequence of flipping

1181
01:02:04,850 --> 01:02:10,040
a coin, I want to do what you
do, which is say, for each

1182
01:02:10,040 --> 01:02:16,700
coin, the coin is equiprobably
a head or a tail.

1183
01:02:16,700 --> 01:02:19,830
And the coin tosses are
independent of each other.

1184
01:02:19,830 --> 01:02:23,070
And I want to know that I can go
from that to thinking about

1185
01:02:23,070 --> 01:02:24,060
this sequence.

1186
01:02:24,060 --> 01:02:26,990
Strange things will happen
in these sequences when

1187
01:02:26,990 --> 01:02:28,250
we go to the limit.

1188
01:02:28,250 --> 01:02:32,720
But still, we don't want to have
to worry about a model

1189
01:02:32,720 --> 01:02:34,770
for the whole infinite
sequence.

1190
01:02:34,770 --> 01:02:39,650
So that's one of the things
we should deal with.

1191
01:02:39,650 --> 01:02:40,900
Finally, random variables.

1192
01:02:44,330 --> 01:02:45,580
Definition.

1193
01:02:47,950 --> 01:02:51,810
Three years ago, I taught this
course and I asked people to

1194
01:02:51,810 --> 01:02:55,880
write down definition of what
a random variable was.

1195
01:02:55,880 --> 01:03:00,140
And almost no one really had
any idea of what it was.

1196
01:03:00,140 --> 01:03:01,870
They said it was something
that had a probability

1197
01:03:01,870 --> 01:03:06,620
density, or something that had
a probability mass function,

1198
01:03:06,620 --> 01:03:10,390
or something that had a
distribution function, or

1199
01:03:10,390 --> 01:03:12,570
something like that.

1200
01:03:12,570 --> 01:03:17,190
What it is, if you want to get
a definition which fits in

1201
01:03:17,190 --> 01:03:20,640
with the axioms, the only thing
we know from the axioms

1202
01:03:20,640 --> 01:03:22,750
is there's a sample space.

1203
01:03:22,750 --> 01:03:25,500
There are events and there
are probabilities.

1204
01:03:25,500 --> 01:03:29,910
So a random variable, what it
really is, is it's a function

1205
01:03:29,910 --> 01:03:37,370
from the set of sample points
to the set of real values.

1206
01:03:37,370 --> 01:03:41,150
And as you get into this, you
will realize that the set of

1207
01:03:41,150 --> 01:03:43,300
real values does not
include minus

1208
01:03:43,300 --> 01:03:45,250
infinity or plus infinity.

1209
01:03:45,250 --> 01:03:47,800
It says that every sample
point gets mapped into a

1210
01:03:47,800 --> 01:03:49,800
finite value.

1211
01:03:49,800 --> 01:03:52,630
This happens, of course,
when you flip a coin.

1212
01:03:52,630 --> 01:03:54,560
Well, flipping a coin,
the outcome is

1213
01:03:54,560 --> 01:03:56,140
not a random variable.

1214
01:03:56,140 --> 01:03:59,440
But you'd like to make it a
random variable, so you say,

1215
01:03:59,440 --> 01:04:02,810
OK, I'm going to model
tails as 0 and heads

1216
01:04:02,810 --> 01:04:05,640
as 1, or vice versa.

1217
01:04:05,640 --> 01:04:07,200
And then what happens?

1218
01:04:07,200 --> 01:04:11,860
Your model for coin tossing,
a sequence of coin tosses

1219
01:04:11,860 --> 01:04:17,550
becomes the same as your
model for data.

1220
01:04:17,550 --> 01:04:20,690
So that what you know about coin
tossing, you can apply to

1221
01:04:20,690 --> 01:04:22,920
data compression.

1222
01:04:22,920 --> 01:04:27,040
You see, when you think about
these things mathematically,

1223
01:04:27,040 --> 01:04:29,500
then you can make all sorts
of connections you

1224
01:04:29,500 --> 01:04:31,250
couldn't make otherwise.

1225
01:04:31,250 --> 01:04:37,010
So random variables have to
satisfy the constraint that--

1226
01:04:45,670 --> 01:04:51,340
they have to satisfy the
constraint that the set of

1227
01:04:51,340 --> 01:04:56,970
sample points, such that x, x
of omega, which is a real

1228
01:04:56,970 --> 01:05:01,080
number, is less than or equal
to some given real number.

1229
01:05:01,080 --> 01:05:04,050
That this set has
to be an event.

1230
01:05:04,050 --> 01:05:05,680
Because those are the only
things that have

1231
01:05:05,680 --> 01:05:07,310
probabilities.

1232
01:05:07,310 --> 01:05:09,970
So if we want to be able to talk
about the probabilities

1233
01:05:09,970 --> 01:05:14,080
of these random variables lying
in certain ranges, or

1234
01:05:14,080 --> 01:05:16,850
things like this, or having
PMFs, or anything that you

1235
01:05:16,850 --> 01:05:19,750
like to do, you need this
constraint on it.

1236
01:05:19,750 --> 01:05:24,470
It's an event for all A in
the set of real numbers.

1237
01:05:24,470 --> 01:05:29,790
Also, if this set
of things here

1238
01:05:29,790 --> 01:05:31,100
are each random variables.

1239
01:05:31,100 --> 01:05:33,900
In other words, if each of them
are functions from the

1240
01:05:33,900 --> 01:05:42,600
sample space to the real line,
then the set of omega such

1241
01:05:42,600 --> 01:05:46,620
that x1 of omega is less than
or equal to A1, up to An of

1242
01:05:46,620 --> 01:05:50,940
omega is less than or equal
to A n is an event also.

1243
01:05:50,940 --> 01:05:54,860
You might recognize this as the
distribution function, the

1244
01:05:54,860 --> 01:05:58,190
joint distribution function
for n random variables.

1245
01:05:58,190 --> 01:06:01,920
You might recognize this as
the distribution function

1246
01:06:01,920 --> 01:06:05,830
evaluated at A for a single
random variable.

1247
01:06:05,830 --> 01:06:12,770
So you define a random
variable.

1248
01:06:12,770 --> 01:06:14,750
And what we're doing here is--

1249
01:06:14,750 --> 01:06:17,880
it's kind of funny because we
already have these axioms.

1250
01:06:17,880 --> 01:06:20,770
But now when we want to define
things in the context of these

1251
01:06:20,770 --> 01:06:24,905
axioms, we need extra things
in the definitions.

1252
01:06:28,840 --> 01:06:32,840
This is a distribution function,
a distribution

1253
01:06:32,840 --> 01:06:35,990
function of the random variable
x is the probability

1254
01:06:35,990 --> 01:06:39,860
that the random variable x is
less than or equal to x, which

1255
01:06:39,860 --> 01:06:45,850
means that x is a mapping from
omega into real numbers.

1256
01:06:45,850 --> 01:06:48,670
It says that with this mapping
here, you're mapping this

1257
01:06:48,670 --> 01:06:51,730
whole sample space into
the real line.

1258
01:06:51,730 --> 01:06:55,690
Some omegas get mapped into
things less than or equal to a

1259
01:06:55,690 --> 01:06:56,930
real number x.

1260
01:06:56,930 --> 01:06:59,510
Some of them get mapped into
things greater than

1261
01:06:59,510 --> 01:07:01,590
the real number x.

1262
01:07:01,590 --> 01:07:04,820
And the set that gets mapped
into something less than or

1263
01:07:04,820 --> 01:07:08,400
equal to x, according to the
definition of a random

1264
01:07:08,400 --> 01:07:10,230
variable, has to be an event.

1265
01:07:10,230 --> 01:07:13,000
Therefore, it has to
have a probability.

1266
01:07:13,000 --> 01:07:18,390
And these probabilities
increase as we go.

1267
01:07:18,390 --> 01:07:24,440
It is totally immaterial for all
purposes whether we have a

1268
01:07:24,440 --> 01:07:28,690
less than or equal to here
or a less than here.

1269
01:07:28,690 --> 01:07:32,450
And everyone follows the
convention of using a less

1270
01:07:32,450 --> 01:07:36,420
than or equal to here rather
than a less than here.

1271
01:07:36,420 --> 01:07:41,360
The importance of that is
that when you look at a

1272
01:07:41,360 --> 01:07:44,240
distribution function,
the distribution

1273
01:07:44,240 --> 01:07:46,820
function often has jumps.

1274
01:07:46,820 --> 01:07:53,100
And the distribution will have
a jump whenever there's a

1275
01:07:53,100 --> 01:07:57,130
nonzero probability that the
random variable takes on a

1276
01:07:57,130 --> 01:08:00,270
particular value x here.

1277
01:08:00,270 --> 01:08:04,160
It takes on this particular
value with something more than

1278
01:08:04,160 --> 01:08:05,520
probabilities here.

1279
01:08:05,520 --> 01:08:09,330
If you have a probability
density for a random variable,

1280
01:08:09,330 --> 01:08:12,680
this curve just moves
up continuously.

1281
01:08:12,680 --> 01:08:14,400
And the derivative
of this curve is

1282
01:08:14,400 --> 01:08:16,310
the probability density.

1283
01:08:16,310 --> 01:08:19,970
If you have a probability mass
function, this is a staircase

1284
01:08:19,970 --> 01:08:21,220
type of function.

1285
01:08:24,510 --> 01:08:27,279
Because of the fact that we
define the distribution

1286
01:08:27,279 --> 01:08:31,410
function with a less than or
equal to rather than a less

1287
01:08:31,410 --> 01:08:37,819
than means that in every one of
these jumps, the value here

1288
01:08:37,819 --> 01:08:40,439
is the upper value
of the jump.

1289
01:08:40,439 --> 01:08:45,180
Value here is the upper value
of the jump, and so forth.

1290
01:08:45,180 --> 01:08:46,500
Now, I'm going to--

1291
01:08:54,616 --> 01:08:57,649
I've already said
half of this.

1292
01:08:57,649 --> 01:09:02,120
Affects maps only until finite
or countable set of values.

1293
01:09:02,120 --> 01:09:03,370
It's discrete.

1294
01:09:05,350 --> 01:09:08,139
And it has a probability
mass function--

1295
01:09:08,139 --> 01:09:09,880
this notation.

1296
01:09:09,880 --> 01:09:13,640
If the derivative exists, then
you say that the random

1297
01:09:13,640 --> 01:09:18,370
variable is continuous
and it has a density.

1298
01:09:18,370 --> 01:09:22,319
And most problems that you do in
probability theory, you're

1299
01:09:22,319 --> 01:09:23,689
dealing with random variables.

1300
01:09:23,689 --> 01:09:25,979
And they either have a
probability mass function if

1301
01:09:25,979 --> 01:09:28,970
they're discrete or they have
a density if they're

1302
01:09:28,970 --> 01:09:30,140
continuous.

1303
01:09:30,140 --> 01:09:33,689
And this is just saying some
things are one way, some

1304
01:09:33,689 --> 01:09:36,069
things are the other way.

1305
01:09:36,069 --> 01:09:37,529
And some things are neither.

1306
01:09:37,529 --> 01:09:40,180
And we'll see lots of things
that are neither.

1307
01:09:40,180 --> 01:09:43,330
And you need the distribution
function to talk about things

1308
01:09:43,330 --> 01:09:44,529
that are neither.

1309
01:09:44,529 --> 01:09:46,770
We will find that the
distribution function, which

1310
01:09:46,770 --> 01:09:50,120
you've hardly ever used in the
past, is extraordinarily

1311
01:09:50,120 --> 01:09:55,440
important, both for theoretical
purposes and for

1312
01:09:55,440 --> 01:09:56,230
other purposes.

1313
01:09:56,230 --> 01:10:02,080
You really need that as a way
of solving problems, as well

1314
01:10:02,080 --> 01:10:06,040
as keeping yourself
out of trouble.

1315
01:10:06,040 --> 01:10:10,530
For every random variable, the
distribution function exists.

1316
01:10:10,530 --> 01:10:11,580
Why?

1317
01:10:11,580 --> 01:10:13,990
Anybody know why this
has to exist for

1318
01:10:13,990 --> 01:10:15,240
every random variable?

1319
01:10:21,450 --> 01:10:22,215
Yeah.

1320
01:10:22,215 --> 01:10:23,670
AUDIENCE: Because
the [INAUDIBLE].

1321
01:10:31,230 --> 01:10:31,740
PROFESSOR: Yes.

1322
01:10:31,740 --> 01:10:35,470
Because we insisted
that it did.

1323
01:10:35,470 --> 01:10:40,520
Namely, we insisted that this
event actually was an event

1324
01:10:40,520 --> 01:10:42,990
for all little x.

1325
01:10:42,990 --> 01:10:46,410
That's part of the definition.

1326
01:10:46,410 --> 01:10:51,560
So, in fact, when you do these
things a little more carefully

1327
01:10:51,560 --> 01:10:56,810
than you might be used to, the
definition implies that the

1328
01:10:56,810 --> 01:11:01,790
distribution function
always exists.

1329
01:11:01,790 --> 01:11:06,110
As a more real-world kind of
argument, we now have a way of

1330
01:11:06,110 --> 01:11:09,680
dealing with things that are
discrete, and continuous, and

1331
01:11:09,680 --> 01:11:14,530
mixed continuous and discrete,
and anything else that you

1332
01:11:14,530 --> 01:11:19,095
might think of because the
definition restricts it.

1333
01:11:19,095 --> 01:11:23,310
Now, one other thing.

1334
01:11:23,310 --> 01:11:25,185
How do I know that
this starts at 0?

1335
01:11:29,650 --> 01:11:32,680
That's a more complicated
thing.

1336
01:11:32,680 --> 01:11:37,820
And I'm not even going
to do it in detail.

1337
01:11:37,820 --> 01:11:43,370
But since every omega maps
into a finite number, you

1338
01:11:43,370 --> 01:11:47,300
can't have a jump down here
at minus infinity.

1339
01:11:47,300 --> 01:11:51,280
And you can't have a jump
here at plus infinity.

1340
01:11:51,280 --> 01:11:54,400
Because omegas don't
map into plus

1341
01:11:54,400 --> 01:11:55,910
infinity or minus infinity.

1342
01:11:55,910 --> 01:11:58,760
So you have to start
down here at 0.

1343
01:11:58,760 --> 01:12:00,510
You have to climb
up here to 1.

1344
01:12:00,510 --> 01:12:01,870
You might never reach 1.

1345
01:12:01,870 --> 01:12:05,610
You might reach it only as a
limit, but you have to reach

1346
01:12:05,610 --> 01:12:07,940
it as a limit.

1347
01:12:07,940 --> 01:12:09,002
Yes?

1348
01:12:09,002 --> 01:12:11,312
AUDIENCE: In the first
paragraph, [INAUDIBLE].

1349
01:12:19,760 --> 01:12:21,790
PROFESSOR: If we have a sequence
of [INAUDIBLE], yeah.

1350
01:12:21,790 --> 01:12:25,230
AUDIENCE: It's [INAUDIBLE].

1351
01:12:25,230 --> 01:12:27,180
PROFESSOR: You are
probably right.

1352
01:12:27,180 --> 01:12:29,820
Yes.

1353
01:12:29,820 --> 01:12:33,030
Well, I don't know I don't
think about that one.

1354
01:12:33,030 --> 01:12:38,320
I don't think you're right,
but we can argue about it.

1355
01:12:38,320 --> 01:12:41,130
But anyway, this has
to start at 0.

1356
01:12:41,130 --> 01:12:42,380
It has to go up to 1.

1357
01:12:44,860 --> 01:12:50,400
OK, we did this.

1358
01:12:50,400 --> 01:12:54,150
Now, I'm going to go through a
theoretical nitpick for the

1359
01:12:54,150 --> 01:12:56,960
last five minutes
of the class.

1360
01:12:56,960 --> 01:12:59,340
Anyone who doesn't like
theoretical nitpicks, you're

1361
01:12:59,340 --> 01:13:03,380
welcome to either go to sleep
for five minutes, or you're

1362
01:13:03,380 --> 01:13:06,030
welcome to go out and get a cup
of coffee, or whatever you

1363
01:13:06,030 --> 01:13:07,090
want to do.

1364
01:13:07,090 --> 01:13:10,230
I will do this to you
occasionally.

1365
01:13:10,230 --> 01:13:15,260
And I realize it's almost
torture for some of you,

1366
01:13:15,260 --> 01:13:20,890
because I want to get you used
to thinking about how

1367
01:13:20,890 --> 01:13:26,590
relatively obvious things
actually get proven.

1368
01:13:26,590 --> 01:13:29,650
I want to increase your ability
to prove things.

1369
01:13:29,650 --> 01:13:34,010
The general statement about
proving things, or at least

1370
01:13:34,010 --> 01:13:38,000
the way I prove things, is not
the way most mathematicians

1371
01:13:38,000 --> 01:13:38,750
prove things.

1372
01:13:38,750 --> 01:13:41,440
Most mathematicians prove things
by starting out with

1373
01:13:41,440 --> 01:13:46,190
axioms and going step by step
until they get to what they're

1374
01:13:46,190 --> 01:13:48,340
trying to prove.

1375
01:13:48,340 --> 01:13:53,020
Now, every time I talk to a good
mathematician, I find out

1376
01:13:53,020 --> 01:13:56,660
that's what they write down when
they prove something, but

1377
01:13:56,660 --> 01:14:00,140
that's not the way they
think about it at all.

1378
01:14:00,140 --> 01:14:04,540
All of us-- engineers,
businesspeople, everyone--

1379
01:14:04,540 --> 01:14:06,470
thinks about problems
in a different way.

1380
01:14:06,470 --> 01:14:11,080
If we're trying to prove
something, we first give a

1381
01:14:11,080 --> 01:14:13,770
really half-assed proof of it.

1382
01:14:13,770 --> 01:14:16,580
And after we do that, we look
at it and we say, well, I

1383
01:14:16,580 --> 01:14:19,570
don't see why this is true and
I don't see why that's true.

1384
01:14:19,570 --> 01:14:22,390
And then you go back and you
patch these things up.

1385
01:14:22,390 --> 01:14:24,020
And then after you patch
things up, it

1386
01:14:24,020 --> 01:14:25,290
starts to look ugly.

1387
01:14:25,290 --> 01:14:27,830
So you go back and do
it a nicer way.

1388
01:14:27,830 --> 01:14:30,190
And you go back and forth and
back and forth and back and

1389
01:14:30,190 --> 01:14:37,070
forth, using both contradiction
and implication.

1390
01:14:37,070 --> 01:14:39,270
You use both of them.

1391
01:14:39,270 --> 01:14:42,750
Now, when you're proving things
in this class, I don't

1392
01:14:42,750 --> 01:14:46,620
care whether you make it look
like you're a formal

1393
01:14:46,620 --> 01:14:48,040
mathematician or not.

1394
01:14:48,040 --> 01:14:51,230
I would just assume you didn't
pretend you were a formal

1395
01:14:51,230 --> 01:14:52,560
mathematician.

1396
01:14:52,560 --> 01:14:56,870
I would like to see you prove
things in such a way that it

1397
01:14:56,870 --> 01:15:00,890
is at least difficult to poke
a hole in your argument.

1398
01:15:00,890 --> 01:15:03,620
In other words, I would like you
to give an argument which

1399
01:15:03,620 --> 01:15:06,590
you've thought about enough
that there aren't obvious

1400
01:15:06,590 --> 01:15:08,380
counter examples to it.

1401
01:15:08,380 --> 01:15:10,940
And if you learn to do that,
you're well on your way to

1402
01:15:10,940 --> 01:15:16,080
learning to use this theory in
a way where you can actually

1403
01:15:16,080 --> 01:15:20,020
come up with correct answer.

1404
01:15:20,020 --> 01:15:23,090
And in fact, I'm not going to go
through this proof at all.

1405
01:15:23,090 --> 01:15:24,830
And I don't think I
really wanted to.

1406
01:15:24,830 --> 01:15:26,440
I just did it because--

1407
01:15:26,440 --> 01:15:27,960
well, I think it's something
you ought to read.

1408
01:15:31,180 --> 01:15:34,410
It is important to learn to
prove things because when you

1409
01:15:34,410 --> 01:15:38,800
get to complicated systems,
you cannot see your way

1410
01:15:38,800 --> 01:15:41,000
through them intuitively.

1411
01:15:41,000 --> 01:15:44,010
And if you can't see your way
through it intuitively, you

1412
01:15:44,010 --> 01:15:47,170
need to understand something
about how to prove things, and

1413
01:15:47,170 --> 01:15:50,320
you need to put all the
techniques of proving things

1414
01:15:50,320 --> 01:15:53,920
that you learned together with
all the techniques that you've

1415
01:15:53,920 --> 01:15:56,390
learned for doing things
intuitively.

1416
01:15:56,390 --> 01:15:59,410
And you need to know how
to put them together.

1417
01:15:59,410 --> 01:16:02,170
If you're stuck dealing only
with things that are

1418
01:16:02,170 --> 01:16:05,100
intuitive, or things that you
learned in high school like

1419
01:16:05,100 --> 01:16:11,660
calculus, then you really can't
deal with complicated

1420
01:16:11,660 --> 01:16:12,960
systems very well.

1421
01:16:12,960 --> 01:16:16,270
OK, I'm going to end
at that point.

1422
01:16:16,270 --> 01:16:19,040
You can read this theoretical
nitpick if you want,

1423
01:16:19,040 --> 01:16:21,100
and play with it.

1424
01:16:21,100 --> 01:16:22,350
And we'll go on next time.