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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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00:00:30,273 --> 00:00:33,100
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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00:00:45,370 --> 00:00:48,120
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.
18
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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00:00:59,210 --> 00:01:03,300
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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00:01:05,069 --> 00:01:08,100
So in the real world, you don't
just throw two random
23
00:01:08,100 --> 00:01:09,520
variables and go home.
24
00:01:09,520 --> 00:01:11,410
Rather the world goes on.
25
00:01:11,410 --> 00:01:14,880
So you generate the random
variable, then you get more
26
00:01:14,880 --> 00:01:18,100
random variables, and things
evolve in time.
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00:01:18,100 --> 00:01:21,560
And random processes are
supposed to be models that
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00:01:21,560 --> 00:01:25,680
capture the evolution of random
phenomena over time.
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00:01:25,680 --> 00:01:27,620
So that's what we
will be doing.
30
00:01:27,620 --> 00:01:31,230
Now when we have evolution in
time, mathematically speaking,
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00:01:31,230 --> 00:01:34,840
you can use discrete time
or continuous time.
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00:01:34,840 --> 00:01:36,800
Of course, discrete
time is easier.
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00:01:36,800 --> 00:01:39,280
And that's where we're
going to start from.
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00:01:39,280 --> 00:01:43,000
And we're going to start from
the easiest, simplest random
35
00:01:43,000 --> 00:01:46,740
process, which is the so-called
Bernoulli process,
36
00:01:46,740 --> 00:01:50,250
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.
38
00:01:54,650 --> 00:01:56,380
That's what the Bernoulli
process is.
39
00:01:56,380 --> 00:01:58,290
So in some sense it's something
that you have
40
00:01:58,290 --> 00:01:59,160
already seen.
41
00:01:59,160 --> 00:02:03,180
But we're going to introduce a
few additional ideas here that
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00:02:03,180 --> 00:02:07,460
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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00:02:46,920 --> 00:02:48,660
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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00:02:53,710 --> 00:02:55,570
Bernoulli trials.
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00:02:55,570 --> 00:02:58,290
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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00:03:05,750 --> 00:03:08,700
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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00:03:37,050 --> 00:03:40,880
statistically independent
of each other.
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00:03:40,880 --> 00:03:43,520
So what could be some examples
of Bernoulli trials?
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00:03:43,520 --> 00:03:48,010
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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00:04:02,810 --> 00:04:05,860
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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00:04:19,670 --> 00:04:23,770
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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00:04:26,260 --> 00:04:29,590
sophisticated models that can
do better predictions than
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00:04:29,590 --> 00:04:32,150
just pure heads and tails.
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00:04:32,150 --> 00:04:36,270
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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00:04:43,760 --> 00:04:45,940
of any kind to a facility.
95
00:04:45,940 --> 00:04:49,790
So it could be a bank, and
you are sitting at
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00:04:49,790 --> 00:04:50,710
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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00:04:54,180 --> 00:04:56,210
during that second or not.
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00:04:56,210 --> 00:05:00,670
Or you can think about arrivals
of jobs to a server.
100
00:05:00,670 --> 00:05:04,890
Or any other kind of requests
to a service system.
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00:05:04,890 --> 00:05:08,560
So requests, or jobs, arrive
at random times.
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You split the time
into time slots.
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00:05:12,360 --> 00:05:15,180
And during each time slot
something comes or something
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00:05:15,180 --> 00:05:16,610
does not come.
105
00:05:16,610 --> 00:05:20,110
And for many applications, it's
a reasonable assumption
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00:05:20,110 --> 00:05:24,510
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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00:05:30,910 --> 00:05:32,810
either something comes
or doesn't come.
110
00:05:32,810 --> 00:05:35,830
And different trials are
independent of each other.
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00:05:35,830 --> 00:05:38,000
Now there's two assumptions
that we're making here.
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00:05:38,000 --> 00:05:40,000
One is the independence
assumption.
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00:05:40,000 --> 00:05:42,190
The other is that this number,
p, probability
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00:05:42,190 --> 00:05:44,520
of success, is constant.
115
00:05:44,520 --> 00:05:47,840
Now if you think about the bank
example, if you stand
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00:05:47,840 --> 00:05:53,330
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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00:05:55,830 --> 00:05:59,370
If you stand outside the bank
at 12:00 noon, probably
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00:05:59,370 --> 00:06:01,240
arrivals are more frequent.
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00:06:01,240 --> 00:06:03,680
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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00:06:15,080 --> 00:06:16,630
about the whole day.
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00:06:16,630 --> 00:06:19,670
So the probability of successes
or arrivals in the
126
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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00:06:24,340 --> 00:06:27,775
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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00:06:37,450 --> 00:06:41,110
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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00:06:47,200 --> 00:06:48,510
that it has.
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00:06:48,510 --> 00:06:52,400
And here there's two slightly
different perspectives of
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00:06:52,400 --> 00:06:55,680
thinking about what a
random process is.
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00:06:55,680 --> 00:06:59,060
The simplest version is to think
about the random process
138
00:06:59,060 --> 00:07:01,605
as being just a sequence
of random variables.
139
00:07:01,605 --> 00:07:04,210
140
00:07:04,210 --> 00:07:06,900
We know what random
variables are.
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00:07:06,900 --> 00:07:09,440
We know what multiple random
variables are.
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00:07:09,440 --> 00:07:12,760
So it's just an experiment that
has associated with it a
143
00:07:12,760 --> 00:07:14,730
bunch of random variables.
144
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?
146
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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00:07:25,690 --> 00:07:27,830
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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00:07:29,480 --> 00:07:31,640
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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00:07:35,500 --> 00:07:37,780
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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00:07:39,420 --> 00:07:44,050
So the expected value is, with
probability p, you get a 1.
156
00:07:44,050 --> 00:07:46,650
And with probability
1 - p, you get a 0.
157
00:07:46,650 --> 00:07:49,510
So the expected value
is equal to p.
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00:07:49,510 --> 00:07:52,890
And then we have seen before a
formula for the variance of
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00:07:52,890 --> 00:07:57,040
the Bernoulli random variable,
which is p times 1-p.
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00:07:57,040 --> 00:08:00,350
So this way we basically now
have all the statistical
161
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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00:08:10,140 --> 00:08:11,580
random process?
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00:08:11,580 --> 00:08:12,240
Well, no.
166
00:08:12,240 --> 00:08:15,070
You need to know how the
different random variables
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00:08:15,070 --> 00:08:16,820
relate to each other.
168
00:08:16,820 --> 00:08:20,870
If you're talking about a
general random process, you
169
00:08:20,870 --> 00:08:23,490
would like to know things.
170
00:08:23,490 --> 00:08:26,370
For example, the joint
distribution of X2,
171
00:08:26,370 --> 00:08:29,220
with X5, and X7.
172
00:08:29,220 --> 00:08:31,680
For example, that might be
something that you're
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00:08:31,680 --> 00:08:32,809
interested in.
174
00:08:32,809 --> 00:08:38,820
And the way you specify it is
by giving the joint PMF of
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00:08:38,820 --> 00:08:40,530
these random variables.
176
00:08:40,530 --> 00:08:44,440
And you have to do that for
every collection, or any
177
00:08:44,440 --> 00:08:46,150
subset, of the random
variables you
178
00:08:46,150 --> 00:08:47,300
are interested in.
179
00:08:47,300 --> 00:08:49,860
So to have a complete
description of a random
180
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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00:08:55,960 --> 00:08:58,780
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
184
00:09:01,580 --> 00:09:03,280
might be interested in.
185
00:09:03,280 --> 00:09:05,670
How did we get around
this issue for
186
00:09:05,670 --> 00:09:06,690
the Bernoulli process?
187
00:09:06,690 --> 00:09:10,030
I didn't give you the joint
distributions explicitly.
188
00:09:10,030 --> 00:09:12,160
But I gave them to
you implicitly.
189
00:09:12,160 --> 00:09:15,250
And this is because I told you
that the different random
190
00:09:15,250 --> 00:09:18,390
variables are independent
of each other.
191
00:09:18,390 --> 00:09:21,490
So at least for the Bernoulli
process, where we make the
192
00:09:21,490 --> 00:09:24,320
independence assumption, we know
that this is going to be
193
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
196
00:09:33,850 --> 00:09:37,090
means that you automatically
know all the joint PMFs.
197
00:09:37,090 --> 00:09:40,940
And we can go to business
based on that.
198
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
204
00:09:57,100 --> 00:09:58,520
experiment.
205
00:09:58,520 --> 00:10:01,970
So we go back to the
chapter one view of
206
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