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PROFESSOR: So by now you have
seen pretty much every

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possible trick there is in
basic probability theory,

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about how to calculate
distributions, and so on.

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00:00:33,100 --> 00:00:37,230
You have the basic tools to
do pretty much anything.

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00:00:37,230 --> 00:00:40,610
So what's coming after this?

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Well, probability is useful for
developing the science of

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inference, and this is a subject
to which we're going

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00:00:48,120 --> 00:00:51,280
to come back at the end
of the semester.

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00:00:51,280 --> 00:00:55,240
Another chapter, which is what
we will be doing over the next

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00:00:55,240 --> 00:00:59,210
few weeks, is to deal with
phenomena that evolve in time.

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So so-called random processes
or stochastic processes.

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00:01:03,300 --> 00:01:05,069
So what is this about?

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So in the real world, you don't
just throw two random

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variables and go home.

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Rather the world goes on.

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00:01:11,410 --> 00:01:14,880
So you generate the random
variable, then you get more

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random variables, and things
evolve in time.

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And random processes are
supposed to be models that

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capture the evolution of random
phenomena over time.

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So that's what we
will be doing.

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00:01:27,620 --> 00:01:31,230
Now when we have evolution in
time, mathematically speaking,

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you can use discrete time
or continuous time.

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Of course, discrete
time is easier.

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And that's where we're
going to start from.

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And we're going to start from
the easiest, simplest random

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process, which is the so-called
Bernoulli process,

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which is nothing but just a
sequence of coin flips.

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00:01:50,250 --> 00:01:54,650
You keep flipping a coin
and keep going forever.

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00:01:54,650 --> 00:01:56,380
That's what the Bernoulli
process is.

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00:01:56,380 --> 00:01:58,290
So in some sense it's something
that you have

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00:01:58,290 --> 00:01:59,160
already seen.

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00:01:59,160 --> 00:02:03,180
But we're going to introduce a
few additional ideas here that

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will be useful and relevant as
we go along and we move on to

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00:02:07,460 --> 00:02:09,949
continuous time processes.

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00:02:09,949 --> 00:02:12,640
So we're going to define the
Bernoulli process, talk about

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00:02:12,640 --> 00:02:17,800
some basic properties that the
process has, and derive a few

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00:02:17,800 --> 00:02:21,440
formulas, and exploit the
special structure that it has

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00:02:21,440 --> 00:02:24,930
to do a few quite interesting
things.

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00:02:24,930 --> 00:02:29,550
By the way, where does the
word Bernoulli come from?

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00:02:29,550 --> 00:02:33,010
Well the Bernoulli's were a
family of mathematicians,

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00:02:33,010 --> 00:02:37,200
Swiss mathematicians and
scientists around the 1700s.

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00:02:37,200 --> 00:02:39,680
There were so many of
them that actually--

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and some of them had the
same first name--

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historians even have difficulty
of figuring out who

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exactly did what.

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But in any case, you can imagine
that at the dinner

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table they were probably
flipping coins and doing

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Bernoulli trials.

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So maybe that was
their pass-time.

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OK.

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So what is the Bernoulli
process?

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The Bernoulli process is nothing
but a sequence of

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independent Bernoulli
trials that you can

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think of as coin flips.

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So you can think the result
of each trial

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being heads or tails.

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It's a little more convenient
maybe to talk about successes

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and failures instead
of heads or tails.

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Or if you wish numerical values,
to use a 1 for a

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success and 0 for a failure.

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So the model is that each one
of these trials has the same

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probability of success, p.

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And the other assumption is
that these trials are

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statistically independent
of each other.

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So what could be some examples
of Bernoulli trials?

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You buy a lottery ticket every
week and you win or lose.

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Presumably, these are
independent of each other.

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And if it's the same kind of
lottery, the probability of

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winning should be the same
during every week.

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Maybe you want to model
the financial markets.

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And a crude model could be that
on any given day the Dow

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Jones is going to go up
or down with a certain

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probability.

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Well that probability must be
somewhere around 0.5, or so.

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This is a crude model of
financial markets.

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You say, probably there
is more into them.

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Life is not that simple.

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But actually it's a pretty
reasonable model.

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It takes quite a bit of work
to come up with more

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sophisticated models that can
do better predictions than

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just pure heads and tails.

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Now more interesting, perhaps
to the examples we will be

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00:04:36,270 --> 00:04:37,975
dealing with in this class--

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a Bernoulli process is a good
model for streams of arrivals

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of any kind to a facility.

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So it could be a bank, and
you are sitting at

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the door of the bank.

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00:04:50,710 --> 00:04:54,180
And at every second, you check
whether a customer came in

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during that second or not.

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Or you can think about arrivals
of jobs to a server.

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00:05:00,670 --> 00:05:04,890
Or any other kind of requests
to a service system.

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So requests, or jobs, arrive
at random times.

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You split the time
into time slots.

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And during each time slot
something comes or something

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does not come.

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And for many applications, it's
a reasonable assumption

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to make that arrivals on any
given slot are independent of

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00:05:24,510 --> 00:05:27,530
arrivals in any other
time slot.

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00:05:27,530 --> 00:05:30,910
So each time slot can be viewed
as a trial, where

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either something comes
or doesn't come.

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00:05:32,810 --> 00:05:35,830
And different trials are
independent of each other.

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Now there's two assumptions
that we're making here.

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One is the independence
assumption.

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The other is that this number,
p, probability

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00:05:42,190 --> 00:05:44,520
of success, is constant.

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00:05:44,520 --> 00:05:47,840
Now if you think about the bank
example, if you stand

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outside the bank at 9:30 in
the morning, you'll see

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00:05:53,330 --> 00:05:55,830
arrivals happening at
a certain rate.

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If you stand outside the bank
at 12:00 noon, probably

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arrivals are more frequent.

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Which means that the given
time slot has a higher

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00:06:03,680 --> 00:06:08,030
probability of seeing an arrival
around noon time.

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00:06:08,030 --> 00:06:11,780
This means that the assumption
of a constant p is probably

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00:06:11,780 --> 00:06:15,080
not correct in that setting,
if you're talking

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about the whole day.

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So the probability of successes
or arrivals in the

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00:06:19,670 --> 00:06:24,340
morning is going to be smaller
than what it would be at noon.

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But if you're talking about a
time period, let's say 10:00

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00:06:27,775 --> 00:06:31,980
to 10:15, probably all slots
have the same probability of

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00:06:31,980 --> 00:06:34,880
seeing an arrival and it's
a good approximation.

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00:06:34,880 --> 00:06:37,450
So we're going to stick with
the assumption that p is

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constant, doesn't change
with time.

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Now that we have our model
what do we do with it?

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Well, we start talking about
the statistical properties

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that it has.

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And here there's two slightly
different perspectives of

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thinking about what a
random process is.

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The simplest version is to think
about the random process

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00:06:59,060 --> 00:07:01,605
as being just a sequence
of random variables.

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00:07:01,605 --> 00:07:04,210

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We know what random
variables are.

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We know what multiple random
variables are.

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So it's just an experiment that
has associated with it a

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00:07:12,760 --> 00:07:14,730
bunch of random variables.

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00:07:14,730 --> 00:07:17,410
So once you have random
variables, what do you do

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00:07:17,410 --> 00:07:18,480
instinctively?

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00:07:18,480 --> 00:07:20,140
You talk about the
distribution of

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00:07:20,140 --> 00:07:21,610
these random variables.

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00:07:21,610 --> 00:07:25,690
We already specified for the
Bernoulli process that each Xi

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is a Bernoulli random variable,
with probability of

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00:07:27,830 --> 00:07:29,480
success equal to p.

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That specifies the distribution
of the random

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00:07:31,640 --> 00:07:35,500
variable X, or Xt, for
general time t.

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Then you can calculate
expected values and

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00:07:37,780 --> 00:07:39,420
variances, and so on.

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So the expected value is, with
probability p, you get a 1.

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And with probability
1 - p, you get a 0.

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00:07:46,650 --> 00:07:49,510
So the expected value
is equal to p.

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And then we have seen before a
formula for the variance of

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the Bernoulli random variable,
which is p times 1-p.

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So this way we basically now
have all the statistical

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00:08:00,350 --> 00:08:04,680
properties of the random
variable Xt, and we have those

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00:08:04,680 --> 00:08:06,670
properties for every t.

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00:08:06,670 --> 00:08:10,140
Is this enough of a
probabilistic description of a

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random process?

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Well, no.

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00:08:12,240 --> 00:08:15,070
You need to know how the
different random variables

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relate to each other.

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If you're talking about a
general random process, you

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00:08:20,870 --> 00:08:23,490
would like to know things.

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00:08:23,490 --> 00:08:26,370
For example, the joint
distribution of X2,

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00:08:26,370 --> 00:08:29,220
with X5, and X7.

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For example, that might be
something that you're

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interested in.

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And the way you specify it is
by giving the joint PMF of

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these random variables.

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And you have to do that for
every collection, or any

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00:08:44,440 --> 00:08:46,150
subset, of the random
variables you

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00:08:46,150 --> 00:08:47,300
are interested in.

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00:08:47,300 --> 00:08:49,860
So to have a complete
description of a random

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00:08:49,860 --> 00:08:54,770
processes, you need to specify
for me all the possible joint

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00:08:54,770 --> 00:08:55,960
distributions.

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And once you have all the
possible joint distributions,

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00:08:58,780 --> 00:09:01,580
then you can answer, in
principle, any questions you

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00:09:01,580 --> 00:09:03,280
might be interested in.

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00:09:03,280 --> 00:09:05,670
How did we get around
this issue for

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00:09:05,670 --> 00:09:06,690
the Bernoulli process?

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00:09:06,690 --> 00:09:10,030
I didn't give you the joint
distributions explicitly.

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00:09:10,030 --> 00:09:12,160
But I gave them to
you implicitly.

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00:09:12,160 --> 00:09:15,250
And this is because I told you
that the different random

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00:09:15,250 --> 00:09:18,390
variables are independent
of each other.

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00:09:18,390 --> 00:09:21,490
So at least for the Bernoulli
process, where we make the

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00:09:21,490 --> 00:09:24,320
independence assumption, we know
that this is going to be

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00:09:24,320 --> 00:09:25,970
the product of the PMFs.

194
00:09:25,970 --> 00:09:29,000

195
00:09:29,000 --> 00:09:33,850
And since I have told you what
the individual PMFs are, this

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00:09:33,850 --> 00:09:37,090
means that you automatically
know all the joint PMFs.

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00:09:37,090 --> 00:09:40,940
And we can go to business
based on that.

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00:09:40,940 --> 00:09:41,310
All right.

199
00:09:41,310 --> 00:09:45,160
So this is one view of what a
random process is, just a

200
00:09:45,160 --> 00:09:47,180
collection of random
variables.

201
00:09:47,180 --> 00:09:50,660
There's another view that's a
little more abstract, which is

202
00:09:50,660 --> 00:09:53,170
the following.

203
00:09:53,170 --> 00:09:57,100
The entire process is to be
thought of as one long

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00:09:57,100 --> 00:09:58,520
experiment.

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00:09:58,520 --> 00:10:01,970
So we go back to the
chapter one view of

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00:10:01,970 --> 00:10:03,550
probabilistic models.

207
00:10:03,550 --> 00:10:06,240
So there must be a sample
space involved.

208
00:10:06,240 --> 00:10:07,660
What is the sample space?

209
00:10:07,660 --> 00:10:11,760
If I do my infinite, long
experiment of flipping an

210
00:10:11,760 --> 00:10:15,330
infinite number of coins,
a typical outcome of the

211
00:10:15,330 --> 00:10:21,140
experiment would be a sequence
of 0's and 1's.

212
00:10:21,140 --> 00:10:25,640
So this could be one possible
outcome of the experiment,

213
00:10:25,640 --> 00:10:28,630
just an infinite sequence
of 0's and 1's.

214
00:10:28,630 --> 00:10:33,020
My sample space is the
set of all possible

215
00:10:33,020 --> 00:10:35,050
outcomes of this kind.

216
00:10:35,050 --> 00:10:40,660
Here's another possible
outcome, and so on.

217
00:10:40,660 --> 00:10:44,060
And essentially we're dealing
with a sample space, which is

218
00:10:44,060 --> 00:10:46,980
the space of all sequences
of 0's and 1's.

219
00:10:46,980 --> 00:10:50,470
And we're making some sort of
probabilistic assumption about

220
00:10:50,470 --> 00:10:53,330
what may happen in
that experiment.

221
00:10:53,330 --> 00:10:56,350
So one particular sequence that
we may be interested in

222
00:10:56,350 --> 00:10:59,660
is the sequence of obtaining
all 1's.

223
00:10:59,660 --> 00:11:05,510
So this is the sequence that
gives you 1's forever.

224
00:11:05,510 --> 00:11:08,330
Once you take the point of view
that this is our sample

225
00:11:08,330 --> 00:11:10,760
space-- its the space of all
infinite sequences--

226
00:11:10,760 --> 00:11:13,770
you can start asking questions
that have to do

227
00:11:13,770 --> 00:11:15,470
with infinite sequences.

228
00:11:15,470 --> 00:11:19,120
Such as the question, what's the
probability of obtaining

229
00:11:19,120 --> 00:11:23,180
the infinite sequence that
consists of all 1's?

230
00:11:23,180 --> 00:11:24,690
So what is this probability?

231
00:11:24,690 --> 00:11:27,240
Let's see how we could
calculate it.

232
00:11:27,240 --> 00:11:34,000
So the probability of obtaining
all 1's is certainly

233
00:11:34,000 --> 00:11:39,890
less than or equal to the
probability of obtaining 1's,

234
00:11:39,890 --> 00:11:42,335
just in the first 10 tosses.

235
00:11:42,335 --> 00:11:45,075

236
00:11:45,075 --> 00:11:47,030
OK.

237
00:11:47,030 --> 00:11:50,810
This is asking for more things
to happen than this.

238
00:11:50,810 --> 00:11:55,780
If this event is true, then
this is also true.

239
00:11:55,780 --> 00:11:58,660
Therefore the probability of
this is smaller than the

240
00:11:58,660 --> 00:11:59,540
probability of that.

241
00:11:59,540 --> 00:12:03,410
This event is contained
in that event.

242
00:12:03,410 --> 00:12:04,950
This implies this.

243
00:12:04,950 --> 00:12:06,880
So we have this inequality.

244
00:12:06,880 --> 00:12:12,360
Now what's the probability of
obtaining 1's in 10 trials?

245
00:12:12,360 --> 00:12:15,980
This is just p to the 10th
because the trials are

246
00:12:15,980 --> 00:12:18,530
independent.

247
00:12:18,530 --> 00:12:22,780
Now of course there's no reason
why I chose 10 here.

248
00:12:22,780 --> 00:12:26,160
The same argument goes
through if I use an

249
00:12:26,160 --> 00:12:29,850
arbitrary number, k.

250
00:12:29,850 --> 00:12:34,250
And this has to be
true for all k.

251
00:12:34,250 --> 00:12:38,690
So this probability is less
than p to the k, no matter

252
00:12:38,690 --> 00:12:41,670
what k I choose.

253
00:12:41,670 --> 00:12:46,350
Therefore, this must be less
than or equal to the limit of

254
00:12:46,350 --> 00:12:48,660
this, as k goes to infinity.

255
00:12:48,660 --> 00:12:51,210
This is smaller than
that for all k's.

256
00:12:51,210 --> 00:12:55,860
Let k go to infinity, take k
arbitrarily large, this number

257
00:12:55,860 --> 00:12:57,770
is going to become arbitrarily
small.

258
00:12:57,770 --> 00:12:59,190
It goes to 0.

259
00:12:59,190 --> 00:13:02,480
And that proves that the
probability of an infinite

260
00:13:02,480 --> 00:13:06,080
sequence of 1's is equal to 0.

261
00:13:06,080 --> 00:13:09,800
So take limits of both sides.

262
00:13:09,800 --> 00:13:13,220

263
00:13:13,220 --> 00:13:16,217
It's going to be less than
or equal to the limit--

264
00:13:16,217 --> 00:13:18,380
I shouldn't take a limit here.

265
00:13:18,380 --> 00:13:21,745
The probability is less than or
equal to the limit of p to

266
00:13:21,745 --> 00:13:26,000
the k, as k goes to infinity,
which is 0.

267
00:13:26,000 --> 00:13:30,880
So this proves in a formal way
that the sequence of all 1's

268
00:13:30,880 --> 00:13:32,650
has 0 probability.

269
00:13:32,650 --> 00:13:35,770
If you have an infinite number
of coin flips, what's the

270
00:13:35,770 --> 00:13:40,610
probability that all of the coin
flips result in heads?

271
00:13:40,610 --> 00:13:43,690
The probability of this
happening is equal to zero.

272
00:13:43,690 --> 00:13:48,310
So this particular sequence
has 0 probability.

273
00:13:48,310 --> 00:13:51,480
Of course, I'm assuming here
that p is less than 1,

274
00:13:51,480 --> 00:13:53,420
strictly less than 1.

275
00:13:53,420 --> 00:13:56,280
Now the interesting thing is
that if you look at any other

276
00:13:56,280 --> 00:13:59,690
infinite sequence, and you try
to calculate the probability

277
00:13:59,690 --> 00:14:03,526
of that infinite sequence, you
would get a product of (1-p)

278
00:14:03,526 --> 00:14:07,600
times 1, 1-p times 1, 1-p,
times p times p,

279
00:14:07,600 --> 00:14:09,570
times 1-p and so on.

280
00:14:09,570 --> 00:14:13,560
You keep multiplying numbers
that are less than 1.

281
00:14:13,560 --> 00:14:16,760
Again, I'm making the
assumption that p is

282
00:14:16,760 --> 00:14:17,940
between 0 and 1.

283
00:14:17,940 --> 00:14:21,140
So 1-p is less than 1,
p is less than 1.

284
00:14:21,140 --> 00:14:23,600
You keep multiplying numbers
less than 1.

285
00:14:23,600 --> 00:14:26,190
If you multiply infinitely
many such numbers, the

286
00:14:26,190 --> 00:14:28,470
infinite product becomes 0.

287
00:14:28,470 --> 00:14:33,310
So any individual sequence in
this sample space actually has

288
00:14:33,310 --> 00:14:35,230
0 probability.

289
00:14:35,230 --> 00:14:39,560
And that is a little bit
counter-intuitive perhaps.

290
00:14:39,560 --> 00:14:42,670
But the situation is more like
the situation where we deal

291
00:14:42,670 --> 00:14:44,680
with continuous random
variables.

292
00:14:44,680 --> 00:14:47,430
So if you could draw a
continuous random variable,

293
00:14:47,430 --> 00:14:50,810
every possible outcome
has 0 probability.

294
00:14:50,810 --> 00:14:52,320
And that's fine.

295
00:14:52,320 --> 00:14:54,860
But all of the outcomes
collectively still have

296
00:14:54,860 --> 00:14:56,280
positive probability.

297
00:14:56,280 --> 00:14:59,580
So the situation here is
very much similar.

298
00:14:59,580 --> 00:15:03,940
So the space of infinite
sequences of 0's and 1's, that

299
00:15:03,940 --> 00:15:07,340
sample space is very much
like a continuous space.

300
00:15:07,340 --> 00:15:10,340
If you want to push that analogy
further, you could

301
00:15:10,340 --> 00:15:15,830
think of this as the expansion
of a real number.

302
00:15:15,830 --> 00:15:18,950
Or the representation of a
real number in binary.

303
00:15:18,950 --> 00:15:22,540
Take a real number, write it
down in binary, you are going

304
00:15:22,540 --> 00:15:25,580
to get an infinite sequence
of 0's and 1's.

305
00:15:25,580 --> 00:15:28,780
So you can think of each
possible outcome here

306
00:15:28,780 --> 00:15:30,920
essentially as a real number.

307
00:15:30,920 --> 00:15:36,060
So the experiment of doing an
infinite number of coin flips

308
00:15:36,060 --> 00:15:39,670
is sort of similar to the
experiment of picking a real

309
00:15:39,670 --> 00:15:41,060
number at random.

310
00:15:41,060 --> 00:15:44,990
When you pick real numbers at
random, any particular real

311
00:15:44,990 --> 00:15:46,500
number has 0 probability.

312
00:15:46,500 --> 00:15:49,780
So similarly here, any
particular infinite sequence

313
00:15:49,780 --> 00:15:52,440
has 0 probability.

314
00:15:52,440 --> 00:15:55,260
So if we were to push that
analogy further, there would

315
00:15:55,260 --> 00:15:57,290
be a few interesting
things we could do.

316
00:15:57,290 --> 00:15:59,880
But we will not push
it further.

317
00:15:59,880 --> 00:16:05,400
This is just to give you an
indication that things can get

318
00:16:05,400 --> 00:16:08,710
pretty subtle and interesting
once you start talking about

319
00:16:08,710 --> 00:16:12,640
random processes that involve
forever, over the infinite

320
00:16:12,640 --> 00:16:13,900
time horizon.

321
00:16:13,900 --> 00:16:17,960
So things get interesting even
in this context of the simple

322
00:16:17,960 --> 00:16:19,750
Bernoulli process.

323
00:16:19,750 --> 00:16:23,130
Just to give you a preview of
what's coming further, today

324
00:16:23,130 --> 00:16:26,170
we're going to talk just about
the Bernoulli process.

325
00:16:26,170 --> 00:16:30,810
And you should make sure before
the next lecture--

326
00:16:30,810 --> 00:16:34,590
I guess between the exam
and the next lecture--

327
00:16:34,590 --> 00:16:36,740
to understand everything
we do today.

328
00:16:36,740 --> 00:16:39,930
Because next time we're going
to do everything once more,

329
00:16:39,930 --> 00:16:41,640
but in continuous time.

330
00:16:41,640 --> 00:16:46,360
And in continuous time, things
become more subtle and a

331
00:16:46,360 --> 00:16:47,700
little more difficult.

332
00:16:47,700 --> 00:16:50,580
But we are going to build on
what we understand for the

333
00:16:50,580 --> 00:16:52,030
discrete time case.

334
00:16:52,030 --> 00:16:55,370
Now both the Bernoulli process
and its continuous time analog

335
00:16:55,370 --> 00:16:58,470
has a property that we call
memorylessness, whatever

336
00:16:58,470 --> 00:17:01,590
happened in the past does
not affect the future.

337
00:17:01,590 --> 00:17:03,920
Later on in this class we're
going to talk about more

338
00:17:03,920 --> 00:17:07,310
general random processes,
so-called Markov chains, in

339
00:17:07,310 --> 00:17:10,890
which there are certain
dependences across time.

340
00:17:10,890 --> 00:17:15,349
That is, what has happened in
the past will have some

341
00:17:15,349 --> 00:17:18,390
bearing on what may happen
in the future.

342
00:17:18,390 --> 00:17:22,440
So it's like having coin flips
where the outcome of the next

343
00:17:22,440 --> 00:17:25,720
coin flip has some dependence
on the previous coin flip.

344
00:17:25,720 --> 00:17:28,400
And that gives us a richer
class of models.

345
00:17:28,400 --> 00:17:31,670
And once we get there,
essentially we will have

346
00:17:31,670 --> 00:17:34,400
covered all possible models.

347
00:17:34,400 --> 00:17:38,070
So for random processes that
are practically useful and

348
00:17:38,070 --> 00:17:41,480
which you can manipulate, Markov
chains are a pretty

349
00:17:41,480 --> 00:17:43,250
general class of models.

350
00:17:43,250 --> 00:17:47,260
And almost any real world
phenomenon that evolves in

351
00:17:47,260 --> 00:17:52,190
time can be approximately
modeled using Markov chains.

352
00:17:52,190 --> 00:17:55,580
So even though this is a first
class in probability, we will

353
00:17:55,580 --> 00:17:59,560
get pretty far in
that direction.

354
00:17:59,560 --> 00:17:59,950
All right.

355
00:17:59,950 --> 00:18:04,300
So now let's start doing a few
calculations and answer some

356
00:18:04,300 --> 00:18:06,690
questions about the
Bernoulli process.

357
00:18:06,690 --> 00:18:11,010
So again, the best way to think
in terms of models that

358
00:18:11,010 --> 00:18:13,350
correspond to the Bernoulli
process is in terms of

359
00:18:13,350 --> 00:18:15,870
arrivals of jobs
to a facility.

360
00:18:15,870 --> 00:18:18,230
And there's two types of
questions that you can ask.

361
00:18:18,230 --> 00:18:21,990
In a given amount of time,
how many jobs arrived?

362
00:18:21,990 --> 00:18:26,250
Or conversely, for a given
number of jobs, how much time

363
00:18:26,250 --> 00:18:28,720
did it take for them
to arrive?

364
00:18:28,720 --> 00:18:31,420
So we're going to deal with
these two questions, starting

365
00:18:31,420 --> 00:18:32,450
with the first.

366
00:18:32,450 --> 00:18:34,370
For a given amount of time--

367
00:18:34,370 --> 00:18:37,870
that is, for a given number
of time periods--

368
00:18:37,870 --> 00:18:40,150
how many arrivals have we had?

369
00:18:40,150 --> 00:18:44,180
How many of those Xi's
happen to be 1's?

370
00:18:44,180 --> 00:18:46,270
We fix the number
of time slots--

371
00:18:46,270 --> 00:18:47,970
let's say n time slots--

372
00:18:47,970 --> 00:18:50,920
and you measure the number
of successes.

373
00:18:50,920 --> 00:18:54,320
Well this is a very familiar
random variable.

374
00:18:54,320 --> 00:18:58,990
The number of successes in n
independent coin flips--

375
00:18:58,990 --> 00:19:01,430
or in n independent trials--

376
00:19:01,430 --> 00:19:03,660
is a binomial random variable.

377
00:19:03,660 --> 00:19:10,380
So we know its distribution is
given by the binomial PMF, and

378
00:19:10,380 --> 00:19:15,820
it's just this, for k going
from 0 up to n.

379
00:19:15,820 --> 00:19:18,940
And we know everything by now
about this random variable.

380
00:19:18,940 --> 00:19:21,990
We know its expected
value is n times p.

381
00:19:21,990 --> 00:19:27,980
And we know the variance, which
is n times p, times 1-p.

382
00:19:27,980 --> 00:19:31,740
So there's nothing new here.

383
00:19:31,740 --> 00:19:34,290
That's the easy part.

384
00:19:34,290 --> 00:19:37,590
So now let's look at the
opposite kind of question.

385
00:19:37,590 --> 00:19:42,210
Instead of fixing the time and
asking how many arrivals, now

386
00:19:42,210 --> 00:19:46,160
let us fix the number of
arrivals and ask how much time

387
00:19:46,160 --> 00:19:47,810
did it take.

388
00:19:47,810 --> 00:19:52,780
And let's start with the time
until the first arrival.

389
00:19:52,780 --> 00:19:59,470
So the process starts.

390
00:19:59,470 --> 00:20:00,720
We got our slots.

391
00:20:00,720 --> 00:20:04,070

392
00:20:04,070 --> 00:20:08,250
And we see, perhaps, a sequence
of 0's and then at

393
00:20:08,250 --> 00:20:10,660
some point we get a 1.

394
00:20:10,660 --> 00:20:14,810
The number of trials it took
until we get a 1, we're going

395
00:20:14,810 --> 00:20:16,500
to call it T1.

396
00:20:16,500 --> 00:20:19,020
And it's the time of
the first arrival.

397
00:20:19,020 --> 00:20:23,020

398
00:20:23,020 --> 00:20:23,680
OK.

399
00:20:23,680 --> 00:20:27,130
What is the probability
distribution of T1?

400
00:20:27,130 --> 00:20:30,430
What kind of random
variable is it?

401
00:20:30,430 --> 00:20:31,835
We've gone through
this before.

402
00:20:31,835 --> 00:20:34,360

403
00:20:34,360 --> 00:20:40,560
The event that the first arrival
happens at time little

404
00:20:40,560 --> 00:20:48,400
t is the event that the first
t-1 trials were failures, and

405
00:20:48,400 --> 00:20:52,850
the trial number t happens
to be a success.

406
00:20:52,850 --> 00:20:57,960
So for the first success to
happen at time slot number 5,

407
00:20:57,960 --> 00:21:02,390
it means that the first 4 slots
had failures and the 5th

408
00:21:02,390 --> 00:21:04,800
slot had a success.

409
00:21:04,800 --> 00:21:08,440
So the probability of this
happening is the probability

410
00:21:08,440 --> 00:21:13,580
of having failures in the first
t -1 trials, and having

411
00:21:13,580 --> 00:21:16,300
a success at trial number 1.

412
00:21:16,300 --> 00:21:20,060
And this is the formula for
t equal 1,2, and so on.

413
00:21:20,060 --> 00:21:22,460
So we know what this
distribution is.

414
00:21:22,460 --> 00:21:25,200
It's the so-called geometric
distribution.

415
00:21:25,200 --> 00:21:30,900

416
00:21:30,900 --> 00:21:35,010
Let me jump this through
this for a minute.

417
00:21:35,010 --> 00:21:38,360
In the past, we did calculate
the expected value of the

418
00:21:38,360 --> 00:21:42,240
geometric distribution,
and it's 1/p.

419
00:21:42,240 --> 00:21:46,010
Which means that if p is small,
you expect to take a

420
00:21:46,010 --> 00:21:48,810
long time until the
first success.

421
00:21:48,810 --> 00:21:52,540
And then there's a formula also
for the variance of T1,

422
00:21:52,540 --> 00:21:56,410
which we never formally derived
in class, but it was

423
00:21:56,410 --> 00:22:01,600
in your textbook and it just
happens to be this.

424
00:22:01,600 --> 00:22:02,310
All right.

425
00:22:02,310 --> 00:22:05,920
So nothing new until
this point.

426
00:22:05,920 --> 00:22:08,700
Now, let's talk about
this property, the

427
00:22:08,700 --> 00:22:10,210
memorylessness property.

428
00:22:10,210 --> 00:22:13,240
We kind of touched on this
property when we discussed--

429
00:22:13,240 --> 00:22:15,760
when we did the derivation
in class of the

430
00:22:15,760 --> 00:22:18,010
expected value of T1.

431
00:22:18,010 --> 00:22:20,040
Now what is the memoryless
property?

432
00:22:20,040 --> 00:22:22,980
It's essentially a consequence
of independence.

433
00:22:22,980 --> 00:22:26,740
If I tell you the results of my
coin flips up to a certain

434
00:22:26,740 --> 00:22:30,550
time, this, because of
independence, doesn't give you

435
00:22:30,550 --> 00:22:34,410
any information about the coin
flips after that time.

436
00:22:34,410 --> 00:22:37,180

437
00:22:37,180 --> 00:22:41,180
So knowing that we had lots of
0's here does not change what

438
00:22:41,180 --> 00:22:44,920
I believe about the future coin
flips, because the future

439
00:22:44,920 --> 00:22:47,580
coin flips are going to be just
independent coin flips

440
00:22:47,580 --> 00:22:53,170
with a given probability,
p, for obtaining tails.

441
00:22:53,170 --> 00:22:58,240
So this is a statement that I
made about a specific time.

442
00:22:58,240 --> 00:23:02,270
That is, you do coin flips
until 12 o'clock.

443
00:23:02,270 --> 00:23:05,210
And then at 12 o'clock,
you start watching.

444
00:23:05,210 --> 00:23:09,900
No matter what happens before 12
o'clock, after 12:00, what

445
00:23:09,900 --> 00:23:12,890
you're going to see is just
a sequence of independent

446
00:23:12,890 --> 00:23:15,610
Bernoulli trials with the
same probability, p.

447
00:23:15,610 --> 00:23:18,450
Whatever happened in the
past is irrelevant.

448
00:23:18,450 --> 00:23:21,590
Now instead of talking about
the fixed time at which you

449
00:23:21,590 --> 00:23:26,940
start watching, let's think
about a situation where your

450
00:23:26,940 --> 00:23:31,240
sister sits in the next room,
flips the coins until she

451
00:23:31,240 --> 00:23:35,760
observes the first success,
and then calls you inside.

452
00:23:35,760 --> 00:23:38,700
And you start watching
after this time.

453
00:23:38,700 --> 00:23:40,970
What are you're going to see?

454
00:23:40,970 --> 00:23:45,160
Well, you're going to see a coin
flip with probability p

455
00:23:45,160 --> 00:23:46,850
of success.

456
00:23:46,850 --> 00:23:49,870
You're going to see another
trial that has probability p

457
00:23:49,870 --> 00:23:53,250
as a success, and these are all
independent of each other.

458
00:23:53,250 --> 00:23:56,800
So what you're going to see
starting at that time is going

459
00:23:56,800 --> 00:24:02,610
to be just a sequence of
independent Bernoulli trials,

460
00:24:02,610 --> 00:24:06,190
as if the process was starting
at this time.

461
00:24:06,190 --> 00:24:10,880
How long it took for the first
success to occur doesn't have

462
00:24:10,880 --> 00:24:15,850
any bearing on what is going
to happen afterwards.

463
00:24:15,850 --> 00:24:19,170
What happens afterwards
is still a sequence of

464
00:24:19,170 --> 00:24:21,230
independent coin flips.

465
00:24:21,230 --> 00:24:24,680
And this story is actually
even more general.

466
00:24:24,680 --> 00:24:28,980
So your sister watches the coin
flips and at some point

467
00:24:28,980 --> 00:24:31,690
tells you, oh, something
really interesting is

468
00:24:31,690 --> 00:24:32,440
happening here.

469
00:24:32,440 --> 00:24:35,250
I got this string of a
hundred 1's in a row.

470
00:24:35,250 --> 00:24:37,250
Come and watch.

471
00:24:37,250 --> 00:24:40,260
Now when you go in there and
you start watching, do you

472
00:24:40,260 --> 00:24:43,890
expect to see something
unusual?

473
00:24:43,890 --> 00:24:46,830
There were unusual things
that happened before

474
00:24:46,830 --> 00:24:48,180
you were called in.

475
00:24:48,180 --> 00:24:50,620
Does this means that you're
going to see unusual things

476
00:24:50,620 --> 00:24:51,780
afterwards?

477
00:24:51,780 --> 00:24:52,180
No.

478
00:24:52,180 --> 00:24:55,060
Afterwards, what you're going
to see is, again, just a

479
00:24:55,060 --> 00:24:57,700
sequence of independent
coin flips.

480
00:24:57,700 --> 00:25:00,640
The fact that some strange
things happened before doesn't

481
00:25:00,640 --> 00:25:03,940
have any bearing as to what is
going to happen in the future.

482
00:25:03,940 --> 00:25:08,560
So if the roulettes in the
casino are properly made, the

483
00:25:08,560 --> 00:25:12,610
fact that there were 3 reds in
a row doesn't affect the odds

484
00:25:12,610 --> 00:25:16,430
of whether in the next
roll it's going to

485
00:25:16,430 --> 00:25:19,570
be a red or a black.

486
00:25:19,570 --> 00:25:22,850
So whatever happens in
the past-- no matter

487
00:25:22,850 --> 00:25:25,010
how unusual it is--

488
00:25:25,010 --> 00:25:28,910
at the time when you're called
in, what's going to happen in

489
00:25:28,910 --> 00:25:32,510
the future is going to be just
independent Bernoulli trials,

490
00:25:32,510 --> 00:25:34,170
with the same probability, p.

491
00:25:34,170 --> 00:25:36,730

492
00:25:36,730 --> 00:25:41,900
The only case where this story
changes is if your sister has

493
00:25:41,900 --> 00:25:43,850
a little bit of foresight.

494
00:25:43,850 --> 00:25:48,500
So your sister can look ahead
into the future and knows that

495
00:25:48,500 --> 00:25:54,230
the next 10 coin flips will be
heads, and calls you before

496
00:25:54,230 --> 00:25:56,430
those 10 flips will happen.

497
00:25:56,430 --> 00:25:59,660
If she calls you in, then what
are you going to see?

498
00:25:59,660 --> 00:26:02,310
You're not going to see
independent Bernoulli trials,

499
00:26:02,310 --> 00:26:05,510
since she has psychic powers
and she knows that the next

500
00:26:05,510 --> 00:26:06,910
ones would be 1's.

501
00:26:06,910 --> 00:26:12,770
She called you in and you will
see a sequence of 1's.

502
00:26:12,770 --> 00:26:15,470
So it's no more independent
Bernoulli trials.

503
00:26:15,470 --> 00:26:19,420
So what's the subtle
difference here?

504
00:26:19,420 --> 00:26:24,010
The future is independent from
the past, provided that the

505
00:26:24,010 --> 00:26:28,310
time that you are called and
asked to start watching is

506
00:26:28,310 --> 00:26:31,460
determined by someone who
doesn't have any foresight,

507
00:26:31,460 --> 00:26:33,470
who cannot see the future.

508
00:26:33,470 --> 00:26:36,960
If you are called in, just on
the basis of what has happened

509
00:26:36,960 --> 00:26:39,630
so far, then you don't have any

510
00:26:39,630 --> 00:26:41,430
information about the future.

511
00:26:41,430 --> 00:26:44,870
And one special case is
the picture here.

512
00:26:44,870 --> 00:26:47,240
You have your coin flips.

513
00:26:47,240 --> 00:26:51,060
Once you see a one that happens,
once you see a

514
00:26:51,060 --> 00:26:53,310
success, you are called in.

515
00:26:53,310 --> 00:26:57,690
You are called in on the basis
of what happened in the past,

516
00:26:57,690 --> 00:26:59,070
but without any foresight.

517
00:26:59,070 --> 00:27:02,208

518
00:27:02,208 --> 00:27:03,140
OK.

519
00:27:03,140 --> 00:27:07,020
And this subtle distinction is
what's going to make our next

520
00:27:07,020 --> 00:27:10,420
example interesting
and subtle.

521
00:27:10,420 --> 00:27:13,380
So here's the question.

522
00:27:13,380 --> 00:27:17,790
You buy a lottery ticket every
day, so we have a Bernoulli

523
00:27:17,790 --> 00:27:21,880
process that's running
in time.

524
00:27:21,880 --> 00:27:26,050
And you're interested in the
length of the first string of

525
00:27:26,050 --> 00:27:26,860
losing days.

526
00:27:26,860 --> 00:27:28,410
What does that mean?

527
00:27:28,410 --> 00:27:33,700
So suppose that a typical
sequence of events

528
00:27:33,700 --> 00:27:35,040
could be this one.

529
00:27:35,040 --> 00:27:40,970

530
00:27:40,970 --> 00:27:43,470
So what are we discussing
here?

531
00:27:43,470 --> 00:27:47,180
We're looking at the first
string of losing days, where

532
00:27:47,180 --> 00:27:49,140
losing days means 0's.

533
00:27:49,140 --> 00:27:51,940

534
00:27:51,940 --> 00:27:56,910
So the string of losing days
is this string here.

535
00:27:56,910 --> 00:28:02,030
Let's call the length of that
string, L. We're interested in

536
00:28:02,030 --> 00:28:06,520
the random variable, which is
the length of this interval.

537
00:28:06,520 --> 00:28:08,670
What kind of random
variable is it?

538
00:28:08,670 --> 00:28:11,230

539
00:28:11,230 --> 00:28:11,600
OK.

540
00:28:11,600 --> 00:28:16,190
Here's one possible way you
might think about the problem.

541
00:28:16,190 --> 00:28:16,550
OK.

542
00:28:16,550 --> 00:28:24,140
Starting from this time, and
looking until this time here,

543
00:28:24,140 --> 00:28:27,420
what are we looking at?

544
00:28:27,420 --> 00:28:31,640
We're looking at the time,
starting from here, until the

545
00:28:31,640 --> 00:28:35,040
first success.

546
00:28:35,040 --> 00:28:40,530
So the past doesn't matter.

547
00:28:40,530 --> 00:28:43,550
Starting from here we
have coin flips

548
00:28:43,550 --> 00:28:45,870
until the first success.

549
00:28:45,870 --> 00:28:48,160
The time until the
first success

550
00:28:48,160 --> 00:28:50,280
in a Bernoulli process--

551
00:28:50,280 --> 00:28:54,700
we just discussed that it's a
geometric random variable.

552
00:28:54,700 --> 00:28:58,090
So your first conjecture would
be that this random variable

553
00:28:58,090 --> 00:29:02,960
here, which is 1 longer than the
one we are interested in,

554
00:29:02,960 --> 00:29:06,310
that perhaps is a geometric
random variable.

555
00:29:06,310 --> 00:29:14,040

556
00:29:14,040 --> 00:29:18,460
And if this were so, then you
could say that the random

557
00:29:18,460 --> 00:29:23,300
variable, L, is a geometric,
minus 1.

558
00:29:23,300 --> 00:29:26,160
Can that be the correct
answer?

559
00:29:26,160 --> 00:29:29,590
A geometric random variable,
what values does it take?

560
00:29:29,590 --> 00:29:33,160
It takes values 1,
2, 3, and so on.

561
00:29:33,160 --> 00:29:37,576
1 minus a geometric would
take values from 0,

562
00:29:37,576 --> 00:29:40,940
1, 2, and so on.

563
00:29:40,940 --> 00:29:45,200
Can the random variable
L be 0?

564
00:29:45,200 --> 00:29:45,920
No.

565
00:29:45,920 --> 00:29:48,500
The random variable L
is the length of a

566
00:29:48,500 --> 00:29:50,390
string of losing days.

567
00:29:50,390 --> 00:29:56,190
So the shortest that L could
be, would be just 1.

568
00:29:56,190 --> 00:29:59,770
If you get just one losing day
and then you start winning, L

569
00:29:59,770 --> 00:30:01,520
would be equal to 1.

570
00:30:01,520 --> 00:30:05,338
So L cannot be 0 by definition,
which means that L

571
00:30:05,338 --> 00:30:09,820
+ 1 cannot be 1,
by definition.

572
00:30:09,820 --> 00:30:14,310
But if L +1 were geometric,
it could be equal to 1.

573
00:30:14,310 --> 00:30:16,226
Therefore this random
variable, L

574
00:30:16,226 --> 00:30:18,830
+ 1, is not a geometric.

575
00:30:18,830 --> 00:30:23,550

576
00:30:23,550 --> 00:30:23,920
OK.

577
00:30:23,920 --> 00:30:26,720
Why is it not geometric?

578
00:30:26,720 --> 00:30:29,100
I started watching
at this time.

579
00:30:29,100 --> 00:30:33,810
From this time until the first
success, that should be a

580
00:30:33,810 --> 00:30:35,690
geometric random variable.

581
00:30:35,690 --> 00:30:38,180
Where's the catch?

582
00:30:38,180 --> 00:30:42,670
If I'm asked to start watching
at this time, it's because my

583
00:30:42,670 --> 00:30:48,260
sister knows that the next
one was a failure.

584
00:30:48,260 --> 00:30:52,360
This is the time where the
string of failures starts.

585
00:30:52,360 --> 00:30:56,050
In order to know that they
should start watching here,

586
00:30:56,050 --> 00:30:58,770
it's the same as if
I'm told that the

587
00:30:58,770 --> 00:31:01,240
next one is a failure.

588
00:31:01,240 --> 00:31:05,180
So to be asked to start watching
at this time requires

589
00:31:05,180 --> 00:31:08,050
that someone looked
in the future.

590
00:31:08,050 --> 00:31:13,210
And in that case, it's no longer
true that these will be

591
00:31:13,210 --> 00:31:14,850
independent Bernoulli trials.

592
00:31:14,850 --> 00:31:16,000
In fact, they're not.

593
00:31:16,000 --> 00:31:18,990
If you start watching here,
you're certain that the next

594
00:31:18,990 --> 00:31:20,210
one is a failure.

595
00:31:20,210 --> 00:31:23,510
The next one is not an
independent Bernoulli trial.

596
00:31:23,510 --> 00:31:26,860
That's why the argument that
would claim that this L + 1 is

597
00:31:26,860 --> 00:31:30,400
geometric would be incorrect.

598
00:31:30,400 --> 00:31:33,680
So if this is not the correct
answer, which

599
00:31:33,680 --> 00:31:35,150
is the correct answer?

600
00:31:35,150 --> 00:31:37,700
The correct answer
goes as follows.

601
00:31:37,700 --> 00:31:39,400
Your sister is watching.

602
00:31:39,400 --> 00:31:44,080
Your sister sees the first
failure, and then tells you,

603
00:31:44,080 --> 00:31:45,530
OK, the failures--

604
00:31:45,530 --> 00:31:46,670
or losing days--

605
00:31:46,670 --> 00:31:47,790
have started.

606
00:31:47,790 --> 00:31:49,260
Come in and watch.

607
00:31:49,260 --> 00:31:51,550
So you start to watching
at this time.

608
00:31:51,550 --> 00:31:56,060
And you start watching until
the first success comes.

609
00:31:56,060 --> 00:31:59,290
This will be a geometric
random variable.

610
00:31:59,290 --> 00:32:05,005
So from here to here, this
will be geometric.

611
00:32:05,005 --> 00:32:09,430

612
00:32:09,430 --> 00:32:11,560
So things happen.

613
00:32:11,560 --> 00:32:14,270
You are asked to
start watching.

614
00:32:14,270 --> 00:32:18,660
After you start watching, the
future is just a sequence of

615
00:32:18,660 --> 00:32:20,540
independent Bernoulli trials.

616
00:32:20,540 --> 00:32:23,930
And the time until the first
failure occurs, this is going

617
00:32:23,930 --> 00:32:27,470
to be a geometric random
variable with parameter p.

618
00:32:27,470 --> 00:32:31,830
And then you notice that the
interval of interest is

619
00:32:31,830 --> 00:32:35,030
exactly the same as the length
of this interval.

620
00:32:35,030 --> 00:32:37,550
This starts one time step
later, and ends

621
00:32:37,550 --> 00:32:39,260
one time step later.

622
00:32:39,260 --> 00:32:43,370
So conclusion is that L is
actually geometric, with

623
00:32:43,370 --> 00:32:44,620
parameter p.

624
00:32:44,620 --> 00:33:33,090

625
00:33:33,090 --> 00:33:36,160
OK, it looks like I'm
missing one slide.

626
00:33:36,160 --> 00:33:38,540
Can I cheat a little
from here?

627
00:33:38,540 --> 00:33:46,830

628
00:33:46,830 --> 00:33:48,080
OK.

629
00:33:48,080 --> 00:33:52,550
So now that we dealt with the
time until the first arrival,

630
00:33:52,550 --> 00:33:56,110
we can start talking about
the time until the second

631
00:33:56,110 --> 00:33:57,190
arrival, and so on.

632
00:33:57,190 --> 00:33:59,550
How do we define these?

633
00:33:59,550 --> 00:34:02,920
After the first arrival happens,
we're going to have a

634
00:34:02,920 --> 00:34:06,100
sequence of time slots with no
arrivals, and then the next

635
00:34:06,100 --> 00:34:08,199
arrival is going to happen.

636
00:34:08,199 --> 00:34:11,080
So we call this time
that elapses--

637
00:34:11,080 --> 00:34:14,760
or number of time slots after
the first arrival

638
00:34:14,760 --> 00:34:16,730
until the next one--

639
00:34:16,730 --> 00:34:18,500
we call it T2.

640
00:34:18,500 --> 00:34:22,070
This is the second inter-arrival
time, that is,

641
00:34:22,070 --> 00:34:23,780
time between arrivals.

642
00:34:23,780 --> 00:34:28,260
Once this arrival has happened,
then we wait and see

643
00:34:28,260 --> 00:34:31,600
how many more it takes until
the third arrival.

644
00:34:31,600 --> 00:34:37,040
And we call this
time here, T3.

645
00:34:37,040 --> 00:34:42,429
We're interested in the time of
the k-th arrival, which is

646
00:34:42,429 --> 00:34:45,230
going to be just the
sum of the first k

647
00:34:45,230 --> 00:34:46,889
inter-arrival times.

648
00:34:46,889 --> 00:34:51,909
So for example, let's say Y3
is the time that the third

649
00:34:51,909 --> 00:34:53,510
arrival comes.

650
00:34:53,510 --> 00:34:58,940
Y3 is just the sum of T1,
plus T2, plus T3.

651
00:34:58,940 --> 00:35:01,840

652
00:35:01,840 --> 00:35:06,310
So we're interested in this
random variable, Y3, and it's

653
00:35:06,310 --> 00:35:08,765
the sum of inter-arrival
times.

654
00:35:08,765 --> 00:35:12,190
To understand what kind of
random variable it is, I guess

655
00:35:12,190 --> 00:35:16,970
we should understand what kind
of random variables these are

656
00:35:16,970 --> 00:35:18,230
going to be.

657
00:35:18,230 --> 00:35:22,040
So what kind of random
variable is T2?

658
00:35:22,040 --> 00:35:27,440
Your sister is doing her coin
flips until a success is

659
00:35:27,440 --> 00:35:29,750
observed for the first time.

660
00:35:29,750 --> 00:35:32,540
Based on that information about
what has happened so

661
00:35:32,540 --> 00:35:34,730
far, you are called
into the room.

662
00:35:34,730 --> 00:35:39,320
And you start watching until a
success is observed again.

663
00:35:39,320 --> 00:35:42,160
So after you start watching,
what you have is just a

664
00:35:42,160 --> 00:35:45,190
sequence of independent
Bernoulli trials.

665
00:35:45,190 --> 00:35:47,110
So each one of these
has probability

666
00:35:47,110 --> 00:35:49,400
p of being a success.

667
00:35:49,400 --> 00:35:52,180
The time it's going to take
until the first success, this

668
00:35:52,180 --> 00:35:57,600
number, T2, is going to be again
just another geometric

669
00:35:57,600 --> 00:35:58,840
random variable.

670
00:35:58,840 --> 00:36:01,860
It's as if the process
just started.

671
00:36:01,860 --> 00:36:06,280
After you are called into the
room, you have no foresight,

672
00:36:06,280 --> 00:36:09,540
you don't have any information
about the future, other than

673
00:36:09,540 --> 00:36:11,020
the fact that these
are going to be

674
00:36:11,020 --> 00:36:13,280
independent Bernoulli trials.

675
00:36:13,280 --> 00:36:18,480
So T2 itself is going to be
geometric with the same

676
00:36:18,480 --> 00:36:20,590
parameter p.

677
00:36:20,590 --> 00:36:24,700
And then you can continue the
arguments and argue that T3 is

678
00:36:24,700 --> 00:36:27,880
also geometric with the
same parameter p.

679
00:36:27,880 --> 00:36:30,850
Furthermore, whatever happened,
how long it took

680
00:36:30,850 --> 00:36:34,330
until you were called in, it
doesn't change the statistics

681
00:36:34,330 --> 00:36:36,810
about what's going to happen
in the future.

682
00:36:36,810 --> 00:36:39,130
So whatever happens
in the future is

683
00:36:39,130 --> 00:36:41,460
independent from the past.

684
00:36:41,460 --> 00:36:47,340
So T1, T2, and T3 are
independent random variables.

685
00:36:47,340 --> 00:36:54,220
So conclusion is that the time
until the third arrival is the

686
00:36:54,220 --> 00:37:00,510
sum of 3 independent geometric
random variables, with the

687
00:37:00,510 --> 00:37:02,150
same parameter.

688
00:37:02,150 --> 00:37:04,550
And this is true
more generally.

689
00:37:04,550 --> 00:37:08,820
The time until the k-th arrival
is going to be the sum

690
00:37:08,820 --> 00:37:14,900
of k independent random
variables.

691
00:37:14,900 --> 00:37:21,540
So in general, Yk is going to be
T1 plus Tk, where the Ti's

692
00:37:21,540 --> 00:37:26,850
are geometric, with the same
parameter p, and independent.

693
00:37:26,850 --> 00:37:30,440

694
00:37:30,440 --> 00:37:33,520
So now what's more natural
than trying to find the

695
00:37:33,520 --> 00:37:37,100
distribution of the random
variable Yk?

696
00:37:37,100 --> 00:37:38,350
How can we find it?

697
00:37:38,350 --> 00:37:40,260
So I fixed k for you.

698
00:37:40,260 --> 00:37:41,680
Let's say k is 100.

699
00:37:41,680 --> 00:37:43,580
I'm interested in how
long it takes until

700
00:37:43,580 --> 00:37:46,180
100 customers arrive.

701
00:37:46,180 --> 00:37:48,850
How can we find the distribution
of Yk?

702
00:37:48,850 --> 00:37:51,980
Well one way of doing
it is to use this

703
00:37:51,980 --> 00:37:54,200
lovely convolution formula.

704
00:37:54,200 --> 00:37:57,950
Take a geometric, convolve it
with another geometric, you

705
00:37:57,950 --> 00:37:59,430
get something.

706
00:37:59,430 --> 00:38:02,040
Take that something that you
got, convolve it with a

707
00:38:02,040 --> 00:38:07,090
geometric once more, do this 99
times, and this gives you

708
00:38:07,090 --> 00:38:09,450
the distribution of Yk.

709
00:38:09,450 --> 00:38:14,300
So that's definitely doable,
and it's extremely tedious.

710
00:38:14,300 --> 00:38:16,900
Let's try to find
the distribution

711
00:38:16,900 --> 00:38:22,520
of Yk using a shortcut.

712
00:38:22,520 --> 00:38:28,660
So the probability that
Yk is equal to t.

713
00:38:28,660 --> 00:38:31,620
So we're trying to find
the PMF of Yk.

714
00:38:31,620 --> 00:38:34,030
k has been fixed for us.

715
00:38:34,030 --> 00:38:36,810
And we want to calculate this
probability for the various

716
00:38:36,810 --> 00:38:39,580
values of t, because this
is going to give

717
00:38:39,580 --> 00:38:43,642
us the PMF of Yk.

718
00:38:43,642 --> 00:38:45,850
OK.

719
00:38:45,850 --> 00:38:47,540
What is this event?

720
00:38:47,540 --> 00:38:53,210
What does it take for the k-th
arrival to be at time t?

721
00:38:53,210 --> 00:38:56,580
For that to happen, we
need two things.

722
00:38:56,580 --> 00:39:00,960
In the first t -1 slots,
how many arrivals

723
00:39:00,960 --> 00:39:03,130
should we have gotten?

724
00:39:03,130 --> 00:39:04,830
k - 1.

725
00:39:04,830 --> 00:39:09,400
And then in the last slot, we
get one more arrival, and

726
00:39:09,400 --> 00:39:11,430
that's the k-th one.

727
00:39:11,430 --> 00:39:20,340
So this is the probability that
we have k - 1 arrivals in

728
00:39:20,340 --> 00:39:24,250
the time interval
from 1 up to t.

729
00:39:24,250 --> 00:39:28,670

730
00:39:28,670 --> 00:39:34,120
And then, an arrival
at time t.

731
00:39:34,120 --> 00:39:39,860

732
00:39:39,860 --> 00:39:43,210
That's the only way that it
can happen, that the k-th

733
00:39:43,210 --> 00:39:45,450
arrival happens at time t.

734
00:39:45,450 --> 00:39:48,470
We need to have an arrival
at time t.

735
00:39:48,470 --> 00:39:50,150
And before that time,
we need to have

736
00:39:50,150 --> 00:39:53,380
exactly k - 1 arrivals.

737
00:39:53,380 --> 00:39:55,590
Now this is an event
that refers--

738
00:39:55,590 --> 00:39:58,710

739
00:39:58,710 --> 00:39:59,960
t-1.

740
00:39:59,960 --> 00:40:02,830

741
00:40:02,830 --> 00:40:07,460
In the previous time slots we
had exactly k -1 arrivals.

742
00:40:07,460 --> 00:40:10,680
And then at the last time slot
we get one more arrival.

743
00:40:10,680 --> 00:40:14,560
Now the interesting thing is
that this event here has to do

744
00:40:14,560 --> 00:40:18,620
with what happened from time
1 up to time t -1.

745
00:40:18,620 --> 00:40:22,350
This event has to do with
what happened at time t.

746
00:40:22,350 --> 00:40:25,510
Different time slots are
independent of each other.

747
00:40:25,510 --> 00:40:31,130
So this event and that event
are independent.

748
00:40:31,130 --> 00:40:34,930
So this means that we can
multiply their probabilities.

749
00:40:34,930 --> 00:40:37,280
So take the probability
of this.

750
00:40:37,280 --> 00:40:38,590
What is that?

751
00:40:38,590 --> 00:40:41,600
Well probability of having a
certain number of arrivals in

752
00:40:41,600 --> 00:40:44,650
a certain number of time slots,
these are just the

753
00:40:44,650 --> 00:40:46,530
binomial probabilities.

754
00:40:46,530 --> 00:40:51,250
So this is, out of t - 1 slots,
to get exactly k - 1

755
00:40:51,250 --> 00:41:02,670
arrivals, p to the k-1, (1-p)
to the t-1 - (k-1),

756
00:41:02,670 --> 00:41:05,480
this gives us t-k.

757
00:41:05,480 --> 00:41:07,960
And then we multiply with this
probability, the probability

758
00:41:07,960 --> 00:41:14,300
of an arrival, at time
t is equal to p.

759
00:41:14,300 --> 00:41:21,310
And so this is the formula for
the PMF of the number--

760
00:41:21,310 --> 00:41:27,410
of the time it takes until
the k-th arrival happens.

761
00:41:27,410 --> 00:41:32,760

762
00:41:32,760 --> 00:41:35,730
Does it agree with the formula
in your handout?

763
00:41:35,730 --> 00:41:37,850
Or its not there?

764
00:41:37,850 --> 00:41:38,710
It's not there.

765
00:41:38,710 --> 00:41:39,960
OK.

766
00:41:39,960 --> 00:41:48,018

767
00:41:48,018 --> 00:41:49,010
Yeah.

768
00:41:49,010 --> 00:41:50,020
OK.

769
00:41:50,020 --> 00:41:57,338
So that's the formula and it is
true for what values of t?

770
00:41:57,338 --> 00:41:58,588
[INAUDIBLE].

771
00:41:58,588 --> 00:42:03,182

772
00:42:03,182 --> 00:42:08,350
It takes at least k time slots
in order to get k arrivals, so

773
00:42:08,350 --> 00:42:12,370
this formula should be
true for k larger

774
00:42:12,370 --> 00:42:13,990
than or equal to t.

775
00:42:13,990 --> 00:42:20,330

776
00:42:20,330 --> 00:42:23,391
For t larger than
or equal to k.

777
00:42:23,391 --> 00:42:30,130

778
00:42:30,130 --> 00:42:31,040
All right.

779
00:42:31,040 --> 00:42:34,950
So this gives us the PMF of
the random variable Yk.

780
00:42:34,950 --> 00:42:37,430
Of course, we may also be
interested in the mean and

781
00:42:37,430 --> 00:42:39,260
variance of Yk.

782
00:42:39,260 --> 00:42:42,150
But this is a lot easier.

783
00:42:42,150 --> 00:42:46,350
Since Yk is the sum of
independent random variables,

784
00:42:46,350 --> 00:42:50,310
the expected value of Yk is
going to be just k times the

785
00:42:50,310 --> 00:42:52,980
expected value of
your typical t.

786
00:42:52,980 --> 00:43:03,080
So the expected value of Yk is
going to be just k times 1/p,

787
00:43:03,080 --> 00:43:06,600
which is the mean of
the geometric.

788
00:43:06,600 --> 00:43:09,470
And similarly for the variance,
it's going to be k

789
00:43:09,470 --> 00:43:12,960
times the variance
of a geometric.

790
00:43:12,960 --> 00:43:16,950
So we have everything there is
to know about the distribution

791
00:43:16,950 --> 00:43:19,965
of how long it takes until
the first arrival comes.

792
00:43:19,965 --> 00:43:23,760

793
00:43:23,760 --> 00:43:25,420
OK.

794
00:43:25,420 --> 00:43:27,810
Finally, let's do a few
more things about

795
00:43:27,810 --> 00:43:30,680
the Bernoulli process.

796
00:43:30,680 --> 00:43:34,970
It's interesting to talk about
several processes at the time.

797
00:43:34,970 --> 00:43:39,600
So in the situation here of
splitting a Bernoulli process

798
00:43:39,600 --> 00:43:43,730
is where you have arrivals
that come to a server.

799
00:43:43,730 --> 00:43:46,370
And that's a picture of which
slots get arrivals.

800
00:43:46,370 --> 00:43:48,900
But actually maybe you
have two servers.

801
00:43:48,900 --> 00:43:53,110
And whenever an arrival comes to
the system, you flip a coin

802
00:43:53,110 --> 00:43:56,620
and with some probability, q,
you send it to one server.

803
00:43:56,620 --> 00:44:00,390
And with probability 1-q, you
send it to another server.

804
00:44:00,390 --> 00:44:03,520
So there is a single
arrival stream, but

805
00:44:03,520 --> 00:44:05,040
two possible servers.

806
00:44:05,040 --> 00:44:07,280
And whenever there's an arrival,
you either send it

807
00:44:07,280 --> 00:44:09,270
here or you send it there.

808
00:44:09,270 --> 00:44:13,780
And each time you decide where
you send it by flipping an

809
00:44:13,780 --> 00:44:17,950
independent coin that
has its own bias q.

810
00:44:17,950 --> 00:44:22,450
The coin flips that decide
where do you send it are

811
00:44:22,450 --> 00:44:27,460
assumed to be independent from
the arrival process itself.

812
00:44:27,460 --> 00:44:30,480
So there's two coin flips
that are happening.

813
00:44:30,480 --> 00:44:33,850
At each time slot, there's a
coin flip that decides whether

814
00:44:33,850 --> 00:44:37,150
you have an arrival in this
process here, and that coin

815
00:44:37,150 --> 00:44:39,630
flip is with parameter p.

816
00:44:39,630 --> 00:44:43,000
And if you have something that
arrives, you flip another coin

817
00:44:43,000 --> 00:44:47,050
with probabilities q, and 1-q,
that decides whether you send

818
00:44:47,050 --> 00:44:49,770
it up there or you send
it down there.

819
00:44:49,770 --> 00:44:55,460
So what kind of arrival process
does this server see?

820
00:44:55,460 --> 00:44:59,510
At any given time slot, there's
probability p that

821
00:44:59,510 --> 00:45:01,480
there's an arrival here.

822
00:45:01,480 --> 00:45:04,300
And there's a further
probability q that this

823
00:45:04,300 --> 00:45:07,320
arrival gets sent up there.

824
00:45:07,320 --> 00:45:10,860
So the probability that this
server sees an arrival at any

825
00:45:10,860 --> 00:45:14,090
given time is p times q.

826
00:45:14,090 --> 00:45:18,900
So this process here is going to
be a Bernoulli process, but

827
00:45:18,900 --> 00:45:21,810
with a different parameter,
p times q.

828
00:45:21,810 --> 00:45:24,820
And this one down here, with the
same argument, is going to

829
00:45:24,820 --> 00:45:29,860
be Bernoulli with parameter
p times (1-q).

830
00:45:29,860 --> 00:45:33,500
So by taking a Bernoulli
stream of arrivals and

831
00:45:33,500 --> 00:45:36,460
splitting it into
two, you get two

832
00:45:36,460 --> 00:45:39,050
separate Bernoulli processes.

833
00:45:39,050 --> 00:45:40,890
This is going to be a Bernoulli
process, that's

834
00:45:40,890 --> 00:45:42,980
going to be a Bernoulli
process.

835
00:45:42,980 --> 00:45:45,630
Well actually, I'm running
a little too fast.

836
00:45:45,630 --> 00:45:49,330
What does it take to verify that
it's a Bernoulli process?

837
00:45:49,330 --> 00:45:52,650
At each time slot,
it's a 0 or 1.

838
00:45:52,650 --> 00:45:55,330
And it's going to be a 1, you're
going to see an arrival

839
00:45:55,330 --> 00:45:57,450
with probability p times q.

840
00:45:57,450 --> 00:46:00,510
What else do we need to verify,
to be able to tell--

841
00:46:00,510 --> 00:46:02,820
to say that it's a Bernoulli
process?

842
00:46:02,820 --> 00:46:05,620
We need to make sure that
whatever happens in this

843
00:46:05,620 --> 00:46:09,340
process, in different time
slots, are statistically

844
00:46:09,340 --> 00:46:11,240
independent from each other.

845
00:46:11,240 --> 00:46:13,030
Is that property true?

846
00:46:13,030 --> 00:46:16,900
For example, what happens in
this time slot whether you got

847
00:46:16,900 --> 00:46:20,100
an arrival or not, is it
independent from what happened

848
00:46:20,100 --> 00:46:22,660
at that time slot?

849
00:46:22,660 --> 00:46:26,850
The answer is yes for the
following reason.

850
00:46:26,850 --> 00:46:30,760
What happens in this time slot
has to do with the coin flip

851
00:46:30,760 --> 00:46:34,840
associated with the original
process at this time, and the

852
00:46:34,840 --> 00:46:38,340
coin flip that decides
where to send things.

853
00:46:38,340 --> 00:46:41,370
What happens at that time slot
has to do with the coin flip

854
00:46:41,370 --> 00:46:45,010
here, and the additional coin
flip that decides where to

855
00:46:45,010 --> 00:46:47,130
send it if something came.

856
00:46:47,130 --> 00:46:50,570
Now all these coin flips are
independent of each other.

857
00:46:50,570 --> 00:46:53,460
The coin flips that determine
whether we have an arrival

858
00:46:53,460 --> 00:46:56,860
here is independent from the
coin flips that determined

859
00:46:56,860 --> 00:46:59,280
whether we had an
arrival there.

860
00:46:59,280 --> 00:47:02,770
And you can generalize this
argument and conclude that,

861
00:47:02,770 --> 00:47:07,390
indeed, every time slot here
is independent from

862
00:47:07,390 --> 00:47:09,030
any other time slot.

863
00:47:09,030 --> 00:47:12,020
And this does make it
a Bernoulli process.

864
00:47:12,020 --> 00:47:15,590
And the reason is that, in the
original process, every time

865
00:47:15,590 --> 00:47:18,390
slot is independent from
every other time slot.

866
00:47:18,390 --> 00:47:21,000
And the additional assumption
that the coin flips that we're

867
00:47:21,000 --> 00:47:24,370
using to decide where to send
things, these are also

868
00:47:24,370 --> 00:47:26,020
independent of each other.

869
00:47:26,020 --> 00:47:30,220
So we're using here the basic
property that functions of

870
00:47:30,220 --> 00:47:33,283
independent things remain
independent.

871
00:47:33,283 --> 00:47:36,390

872
00:47:36,390 --> 00:47:38,710
There's a converse
picture of this.

873
00:47:38,710 --> 00:47:41,970
Instead of taking one stream
and splitting it into two

874
00:47:41,970 --> 00:47:44,720
streams, you can do
the opposite.

875
00:47:44,720 --> 00:47:48,040
You could start from two
streams of arrivals.

876
00:47:48,040 --> 00:47:51,300
Let's say you have arrivals of
men and you have arrivals of

877
00:47:51,300 --> 00:47:54,000
women, but you don't
care about gender.

878
00:47:54,000 --> 00:47:57,430
And the only thing you record
is whether, in a given time

879
00:47:57,430 --> 00:48:00,450
slot, you had an
arrival or not.

880
00:48:00,450 --> 00:48:04,320
Notice that here we may have
an arrival of a man and the

881
00:48:04,320 --> 00:48:05,790
arrival of a woman.

882
00:48:05,790 --> 00:48:11,180
We just record it with a 1, by
saying there was an arrival.

883
00:48:11,180 --> 00:48:14,260
So in the merged process, we're
not keeping track of how

884
00:48:14,260 --> 00:48:16,880
many arrivals we had total.

885
00:48:16,880 --> 00:48:18,840
We just record whether
there was an

886
00:48:18,840 --> 00:48:21,400
arrival or not an arrival.

887
00:48:21,400 --> 00:48:25,830
So an arrival gets recorded here
if, and only if, one or

888
00:48:25,830 --> 00:48:28,800
both of these streams
had an arrival.

889
00:48:28,800 --> 00:48:31,780
So that we call a merging
of two Bernoull-- of two

890
00:48:31,780 --> 00:48:34,470
processes, of two arrival
processes.

891
00:48:34,470 --> 00:48:37,840
So let's make the assumption
that this arrival process is

892
00:48:37,840 --> 00:48:41,440
independent from that
arrival process.

893
00:48:41,440 --> 00:48:44,330
So what happens at the
typical slot here?

894
00:48:44,330 --> 00:48:49,680
I'm going to see an arrival,
unless none of

895
00:48:49,680 --> 00:48:51,950
these had an arrival.

896
00:48:51,950 --> 00:48:56,380
So the probability of an arrival
in a typical time slot

897
00:48:56,380 --> 00:49:02,650
is going to be 1 minus the
probability of no arrival.

898
00:49:02,650 --> 00:49:07,370
And the event of no arrival
corresponds to the first

899
00:49:07,370 --> 00:49:10,330
process having no arrival,
and the second

900
00:49:10,330 --> 00:49:14,350
process having no arrival.

901
00:49:14,350 --> 00:49:18,110
So there's no arrival in the
merged process if, and only

902
00:49:18,110 --> 00:49:21,270
if, there's no arrival in the
first process and no arrival

903
00:49:21,270 --> 00:49:22,710
in the second process.

904
00:49:22,710 --> 00:49:26,080
We're assuming that the two
processes are independent and

905
00:49:26,080 --> 00:49:29,160
that's why we can multiply
probabilities here.

906
00:49:29,160 --> 00:49:34,270
And then you can take this
formula and it simplifies to p

907
00:49:34,270 --> 00:49:38,120
+ q, minus p times q.

908
00:49:38,120 --> 00:49:41,360
So each time slot of the merged
process has a certain

909
00:49:41,360 --> 00:49:44,620
probability of seeing
an arrival.

910
00:49:44,620 --> 00:49:47,360
Is the merged process
a Bernoulli process?

911
00:49:47,360 --> 00:49:51,260
Yes, it is after you verify the
additional property that

912
00:49:51,260 --> 00:49:54,650
different slots are independent
of each other.

913
00:49:54,650 --> 00:49:56,560
Why are they independent?

914
00:49:56,560 --> 00:50:01,070
What happens in this slot has to
do with that slot, and that

915
00:50:01,070 --> 00:50:03,160
slot down here.

916
00:50:03,160 --> 00:50:05,790
These two slots--

917
00:50:05,790 --> 00:50:08,570
so what happens here,
has to do with what

918
00:50:08,570 --> 00:50:11,430
happens here and there.

919
00:50:11,430 --> 00:50:16,680
What happens in this slot has
to do with whatever happened

920
00:50:16,680 --> 00:50:19,200
here and there.

921
00:50:19,200 --> 00:50:23,190
Now, whatever happens here and
there is independent from

922
00:50:23,190 --> 00:50:25,330
whatever happens
here and there.

923
00:50:25,330 --> 00:50:29,180
Therefore, what happens here
is independent from what

924
00:50:29,180 --> 00:50:30,220
happens there.

925
00:50:30,220 --> 00:50:33,310
So the independence property
is preserved.

926
00:50:33,310 --> 00:50:36,640
The different slots of this
merged process are independent

927
00:50:36,640 --> 00:50:37,590
of each other.

928
00:50:37,590 --> 00:50:41,970
So the merged process is itself
a Bernoulli process.

929
00:50:41,970 --> 00:50:45,900
So please digest these two
pictures of merging and

930
00:50:45,900 --> 00:50:48,960
splitting, because we're going
to revisit them in continuous

931
00:50:48,960 --> 00:50:52,510
time where things are little
subtler than that.

932
00:50:52,510 --> 00:50:53,160
OK.

933
00:50:53,160 --> 00:50:56,240
Good luck on the exam and
see you in a week.

934
00:50:56,240 --> 00:50:57,490