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PROFESSOR: OK, so let's
get started.

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00:00:26,680 --> 00:00:30,530
We're going to essentially
finish up on

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00:00:30,530 --> 00:00:33,600
Poisson processes today.

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00:00:33,600 --> 00:00:38,960
And today, we have the part of
it that's really the fun part.

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00:00:38,960 --> 00:00:44,870
What we had until now was the
dry stuff, defining everything

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00:00:44,870 --> 00:00:46,150
and so forth.

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As I said before, Poisson
processes are these perfect

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processes where everything that
could be true is true.

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00:00:54,210 --> 00:00:58,760
And you have so many different
ways of looking at problems

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00:00:58,760 --> 00:01:02,950
that you can solve problems
in a wide variety of ways.

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00:01:02,950 --> 00:01:08,900
What you want try to do in the
problem set this week is to

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00:01:08,900 --> 00:01:13,650
get out of the mode of starting
to write equations

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00:01:13,650 --> 00:01:15,770
before you think about it.

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00:01:15,770 --> 00:01:20,060
I mean, write equations, yes,
but while you're doing it,

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00:01:20,060 --> 00:01:24,910
think about what the right way
of approaching the problem is.

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00:01:24,910 --> 00:01:29,730
As far as I know, every part of
every problem there can be

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00:01:29,730 --> 00:01:31,860
solved in one or two lines.

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00:01:31,860 --> 00:01:32,800
There's nothing long.

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00:01:32,800 --> 00:01:34,360
There's nothing complicated.

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00:01:34,360 --> 00:01:38,740
But you have to find exactly
the right way of doing it.

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00:01:38,740 --> 00:01:43,010
And that's what you're supposed
to learn because you

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00:01:43,010 --> 00:01:48,880
find all sorts of processes in
the world, which are poorly

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00:01:48,880 --> 00:01:52,400
modelled as Poisson processes.

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00:01:52,400 --> 00:02:00,370
You get a lot of insight about
the real process by looking at

32
00:02:00,370 --> 00:02:04,370
Poisson processes, but you don't
get the whole story.

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00:02:04,370 --> 00:02:08,530
And if you don't understand how
these relationships are

34
00:02:08,530 --> 00:02:13,000
connected to each other then
you have no hope of getting

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00:02:13,000 --> 00:02:18,000
some sense of when Poisson
process theory is telling you

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00:02:18,000 --> 00:02:21,580
something and when it isn't.

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00:02:21,580 --> 00:02:24,910
That's why engineers are
different than mathematicians.

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00:02:24,910 --> 00:02:28,570
Mathematicians live in
this beautiful world.

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00:02:28,570 --> 00:02:29,380
And I love it.

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00:02:29,380 --> 00:02:30,880
I love to live there.

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00:02:30,880 --> 00:02:34,570
Love to go there for vacations
and so on because everything

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00:02:34,570 --> 00:02:36,250
is perfectly clean.

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00:02:36,250 --> 00:02:39,560
Everything has a
right solution.

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00:02:39,560 --> 00:02:41,100
And if it's not a
right solution,

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00:02:41,100 --> 00:02:42,545
it's a wrong solution.

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00:02:42,545 --> 00:02:47,850
In engineering, everything
is kind of hazy.

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00:02:47,850 --> 00:02:50,830
And you get insights about
things when you put a lot of

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00:02:50,830 --> 00:02:52,300
insights together.

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00:02:52,300 --> 00:02:54,850
You finally make judgments
about things.

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00:02:54,850 --> 00:02:57,780
You use all sorts of
models to do this.

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00:02:57,780 --> 00:03:03,050
And what you use a course like
this for is to understand what

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00:03:03,050 --> 00:03:04,990
all these models are saying.

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00:03:04,990 --> 00:03:07,090
And then be able to use them.

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00:03:07,090 --> 00:03:11,380
So, the stuff that we'll be
talking about today is stuff

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00:03:11,380 --> 00:03:15,090
you will use all the
rest of the term.

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00:03:15,090 --> 00:03:18,770
Because everything we do,
surprisingly enough, from now

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00:03:18,770 --> 00:03:24,800
on, has some relationships
with Poisson processes.

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00:03:24,800 --> 00:03:29,710
It doesn't sound like they do,
but, in fact, they do.

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00:03:29,710 --> 00:03:33,110
And this has a lot of the things
that look like tricks,

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00:03:33,110 --> 00:03:34,690
but which are more
than tricks.

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00:03:34,690 --> 00:03:39,540
They're really what comes from
a basic understanding of

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00:03:39,540 --> 00:03:40,850
Poisson processes.

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00:03:40,850 --> 00:03:42,660
So first, I'm going to
review a little bit

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what we did last time.

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Poisson process is an
arrival process.

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Remember what an arrival
process is.

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00:03:51,800 --> 00:03:57,030
It's just a bunch of arrival
epochs, which have some

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00:03:57,030 --> 00:03:59,660
statistics associated
with them.

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00:03:59,660 --> 00:04:02,980
And it has IID
exponentially-distributed

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00:04:02,980 --> 00:04:04,240
interarrival times.

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00:04:04,240 --> 00:04:09,190
So the time between successive
arrivals is independent from

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00:04:09,190 --> 00:04:11,700
one arrival to the next.

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00:04:11,700 --> 00:04:14,670
And it has this exponential
distribution, which is what

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00:04:14,670 --> 00:04:18,512
gives the Poisson process it's
very special characteristic.

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00:04:21,620 --> 00:04:24,930
It can be represented by
its arrival epochs.

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00:04:24,930 --> 00:04:31,360
These are the things you see
here, S1, S2, S3, and so

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00:04:31,360 --> 00:04:32,960
forth, are the arrival epochs.

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00:04:32,960 --> 00:04:38,450
If you can specify what the
probability relationship is

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for this set of joint random
variables then you know

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00:04:42,100 --> 00:04:45,230
everything there is to know
about a Poisson process.

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00:04:45,230 --> 00:04:48,810
If you specify the joint
distribution of these

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00:04:48,810 --> 00:04:52,700
interarrival times, and that's
trivial because they're IID

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00:04:52,700 --> 00:04:56,045
exponentially-distributed random
variables, then you

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00:04:56,045 --> 00:04:58,790
know everything there is to
know about the process.

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00:04:58,790 --> 00:05:04,280
And if you can specify N of
t, which is the number of

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00:05:04,280 --> 00:05:10,380
arrivals up until time t for
every t then that specifies

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00:05:10,380 --> 00:05:12,730
the process completely also.

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00:05:12,730 --> 00:05:17,360
So we take the viewpoint here,
I mean, usually we view a

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00:05:17,360 --> 00:05:22,130
stochastic process as either a
sequence of random variables

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00:05:22,130 --> 00:05:24,940
or a continuum of random
variables.

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00:05:24,940 --> 00:05:28,490
Here, we're viewing as this
three ways of looking at the

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00:05:28,490 --> 00:05:30,200
same thing.

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So a Poisson process is then
either the sequence of

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00:05:36,620 --> 00:05:40,590
interarrival times, the sequence
of arrival epochs,

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00:05:40,590 --> 00:05:44,920
or, what we call the counting
process, N of t at each t

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00:05:44,920 --> 00:05:46,360
greater than zero.

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00:05:46,360 --> 00:05:50,030
The arrival epochs in N of
t are related either

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00:05:50,030 --> 00:05:53,180
this way or this way.

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00:05:53,180 --> 00:05:55,850
And we talked about that.

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The interarrival times
are memoryless.

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00:06:01,980 --> 00:06:05,650
In other words, they satisfy
this relationship here.

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00:06:05,650 --> 00:06:08,680
The probability that an
interarrival time Xi is

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00:06:08,680 --> 00:06:12,970
greater than t plus x, for any
t and any x, which are

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00:06:12,970 --> 00:06:15,850
positive, given that it's
greater than t.

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00:06:15,850 --> 00:06:18,730
In other words, given that
you've already wasted time t

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00:06:18,730 --> 00:06:23,400
waiting, the probability that
the time from here until the

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00:06:23,400 --> 00:06:27,876
actual occurrence occurs
is again exponential.

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00:06:32,870 --> 00:06:35,340
This is stated in conditional
form.

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00:06:35,340 --> 00:06:41,330
We stated it before in just
joint probability form.

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00:06:41,330 --> 00:06:45,380
OK, which says, that if you wait
for a while and nothing

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00:06:45,380 --> 00:06:49,180
has happened, you just
keep on waiting.

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00:06:49,180 --> 00:06:51,080
You're right where
you started.

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00:06:51,080 --> 00:06:52,240
You haven't lost anything.

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00:06:52,240 --> 00:06:54,990
You haven't gained anything.

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00:06:54,990 --> 00:06:59,100
And we said that other renewal
processes, which are IID

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00:06:59,100 --> 00:07:05,790
interarrival random variables,
you can have these heavy

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00:07:05,790 --> 00:07:11,090
tailed distributions where if
nothing happens after while

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then you start to really feel
badly because you know

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00:07:14,700 --> 00:07:18,290
nothing's going to happen
for an awful lot longer.

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00:07:18,290 --> 00:07:21,420
Heavy tailed distribution's best
example is when you're

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00:07:21,420 --> 00:07:26,130
trying to catch an airplane and
they say, it's going to be

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00:07:26,130 --> 00:07:28,180
10 minutes late.

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00:07:28,180 --> 00:07:31,780
That's the worst heavy tailed
distribution there is.

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00:07:31,780 --> 00:07:33,300
And it drives you crazy.

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00:07:33,300 --> 00:07:36,680
Because I've never caught a
plane that was supposed to be

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00:07:36,680 --> 00:07:41,380
10 minutes late that wasn't
at least an hour late.

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00:07:41,380 --> 00:07:46,280
And often, it got canceled,
which makes it not a random

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00:07:46,280 --> 00:07:47,530
variable at all.

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00:07:54,770 --> 00:07:57,330
One of the things we're
interested in now, and we

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00:07:57,330 --> 00:08:00,800
talked about it a lot last
time, is you pick some

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00:08:00,800 --> 00:08:05,170
arbitrary time t, that can
be any time at all.

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00:08:05,170 --> 00:08:10,550
And you ask, how long is it from
time t-- t might be when

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00:08:10,550 --> 00:08:13,400
you arrive to wait for
a bus or something--

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00:08:13,400 --> 00:08:16,780
how long is it until
the next bus comes?

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00:08:16,780 --> 00:08:21,410
So Z is the random variable
that goes.

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00:08:21,410 --> 00:08:25,370
And you really should put
some indices on this.

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00:08:25,370 --> 00:08:32,360
But what it is is the random
variable from t until this

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00:08:32,360 --> 00:08:38,070
slow arrival here that's poking
along finally comes in.

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00:08:38,070 --> 00:08:44,200
Now, what we found is that the
interval Z, which is the time

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00:08:44,200 --> 00:08:51,760
from this arrival back
to t, is exponential.

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00:08:51,760 --> 00:08:58,860
And the way we showed that is
to say, let's condition Z on

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00:08:58,860 --> 00:09:00,940
anything we want to
condition it on.

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00:09:00,940 --> 00:09:03,780
And the things that's important
too condition it on

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00:09:03,780 --> 00:09:07,870
is the value of N of t here,
which is N. And once we

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00:09:07,870 --> 00:09:12,520
condition it on the fact that N
of t is n, we then condition

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00:09:12,520 --> 00:09:18,100
on S sub n, which is the time
that this last arrival came.

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00:09:18,100 --> 00:09:29,750
So if we condition Z on N of t
at time t and the time that

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00:09:29,750 --> 00:09:33,570
this last arrival came in, which
is S sub two, in this

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00:09:33,570 --> 00:09:39,080
case, then Z turns out to be,
again, just the time left to

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00:09:39,080 --> 00:09:40,990
wait after we've already
waited for this

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00:09:40,990 --> 00:09:41,990
given amount of time.

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00:09:41,990 --> 00:09:45,250
We then find out that Z,
conditional on these two

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00:09:45,250 --> 00:09:48,370
things that we don't understand
at all, it's just

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00:09:48,370 --> 00:09:50,960
exponential, no matter
what they are.

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00:09:50,960 --> 00:09:55,210
And since it's exponential no
matter what they are, we don't

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00:09:55,210 --> 00:09:57,720
go to the trouble of trying to
figure out what they are.

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00:09:57,720 --> 00:10:02,920
We just say, whatever
distribution they have, Z

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00:10:02,920 --> 00:10:07,440
itself, unconditionally, is
just the same exponential

159
00:10:07,440 --> 00:10:08,900
random variable.

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00:10:08,900 --> 00:10:12,930
So that was one of the main
things we did last time.

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00:10:15,710 --> 00:10:20,860
Next thing we did was we started
to look at any set of

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00:10:20,860 --> 00:10:26,310
time, say t1, t2, up to t sub
k, and then we looked at the

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00:10:26,310 --> 00:10:27,720
increments.

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00:10:27,720 --> 00:10:31,690
How many arrivals occurred
between zero and t1?

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00:10:31,690 --> 00:10:35,060
How many arrivals occurred
between t1 and t2?

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00:10:35,060 --> 00:10:37,850
How many between t2 and t3?

167
00:10:37,850 --> 00:10:44,030
So we looked at these Poisson
process increments, a whole

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00:10:44,030 --> 00:10:49,780
bunch of random variables, and
we said, these are stationary

169
00:10:49,780 --> 00:10:54,080
and independent Poisson counting
processes, over their

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00:10:54,080 --> 00:10:54,950
given intervals.

171
00:10:54,950 --> 00:11:07,410
In other words, if you stop the
first process, you stop

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00:11:07,410 --> 00:11:10,370
looking at this at time t1.

173
00:11:10,370 --> 00:11:19,620
Then you look at this
from time t1 to t2.

174
00:11:19,620 --> 00:11:23,480
You look at this one from
tk minus 1 to tk.

175
00:11:23,480 --> 00:11:26,500
And you look at the last one,
the next last one, all

176
00:11:26,500 --> 00:11:27,940
the way up to tk.

177
00:11:27,940 --> 00:11:30,150
And there should be one
where you look at it

178
00:11:30,150 --> 00:11:34,530
from tk on to t.

179
00:11:34,530 --> 00:11:37,810
No, you're only looking
at k of them.

180
00:11:37,810 --> 00:11:40,180
So these are the things that
you're looking at.

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00:11:40,180 --> 00:11:42,260
The statement is that these are

182
00:11:42,260 --> 00:11:44,430
independent random variables.

183
00:11:44,430 --> 00:11:48,150
The other statement is they're
stationary, which means, if

184
00:11:48,150 --> 00:11:51,550
you look at the number of
arrivals in this interval,

185
00:11:51,550 --> 00:11:56,500
here, it's a function
of t2 minus t1.

186
00:11:56,500 --> 00:11:59,700
But it's not a function
of t1 alone.

187
00:11:59,700 --> 00:12:03,260
It's only a function of the
length of the interval.

188
00:12:03,260 --> 00:12:09,540
The number of arrivals in any
interval of length tk minus tk

189
00:12:09,540 --> 00:12:16,780
minus 1 is a function only of
the length of that interval

190
00:12:16,780 --> 00:12:19,020
and not of where the
interval is.

191
00:12:19,020 --> 00:12:23,540
That's a reasonable way to look
at stationarity, I think.

192
00:12:23,540 --> 00:12:28,240
How many arrivals come in in a
given area is independent of

193
00:12:28,240 --> 00:12:29,460
where the area is.

194
00:12:29,460 --> 00:12:34,020
It depends only on how
long the interval is.

195
00:12:34,020 --> 00:12:38,970
OK, then we found that the
probability mass function for

196
00:12:38,970 --> 00:12:43,430
N of t, and now we're just
looking from zero to t because

197
00:12:43,430 --> 00:12:47,440
we know we get the same thing
over any interval, this

198
00:12:47,440 --> 00:12:52,670
probability mass function is
this nice function, which

199
00:12:52,670 --> 00:12:54,950
depends only and the
product lambda t.

200
00:12:54,950 --> 00:12:59,610
It never depends on lambda
alone or t alone.

201
00:12:59,610 --> 00:13:04,820
Lambda t to n, e to the minus
lambda t over n factorial.

202
00:13:04,820 --> 00:13:07,210
What the n factorial
is doing there.

203
00:13:07,210 --> 00:13:10,760
Well, it came out in
the derivation.

204
00:13:10,760 --> 00:13:15,740
By the time we finish today you
should have more ideas of

205
00:13:15,740 --> 00:13:18,120
where that factorial
came from.

206
00:13:18,120 --> 00:13:21,580
And we'll try to understand
that.

207
00:13:21,580 --> 00:13:23,800
By the stationary and
independent increment

208
00:13:23,800 --> 00:13:27,390
property, we know that these two
things are independent, N

209
00:13:27,390 --> 00:13:32,450
of t1 and the number of
arrivals in t1 to t.

210
00:13:32,450 --> 00:13:36,160
This is a Poisson
random variable.

211
00:13:36,160 --> 00:13:38,580
This is a Poisson
random variable.

212
00:13:38,580 --> 00:13:44,190
We know that the number of
arrivals between zero and t is

213
00:13:44,190 --> 00:13:46,140
also a Poisson random
variable.

214
00:13:46,140 --> 00:13:48,300
And what does that tell you?

215
00:13:48,300 --> 00:13:50,900
It tells you, you don't have
to go through all of this

216
00:13:50,900 --> 00:13:54,190
discrete convolution stuff.

217
00:13:54,190 --> 00:13:56,690
You probably should go through
it once just for your own

218
00:13:56,690 --> 00:13:59,410
edification to see that
this all works.

219
00:13:59,410 --> 00:14:05,180
But for a very lazy person
like me, who likes using

220
00:14:05,180 --> 00:14:10,000
arguments like this, I say,
well, these two things are

221
00:14:10,000 --> 00:14:11,620
independent.

222
00:14:11,620 --> 00:14:14,100
They are Poisson random
variables.

223
00:14:14,100 --> 00:14:16,040
Their sum is Poisson.

224
00:14:16,040 --> 00:14:20,400
And therefore, whenever you have
two independent Poisson

225
00:14:20,400 --> 00:14:23,390
random variables and you add
them together, what you get is

226
00:14:23,390 --> 00:14:29,930
a Poisson random variable whose
mean is the sum of the

227
00:14:29,930 --> 00:14:36,070
means of the two individual
random variables.

228
00:14:36,070 --> 00:14:38,390
In general, sums of independent
Poisson random

229
00:14:38,390 --> 00:14:40,880
variables are Poisson with
the means adding.

230
00:14:44,440 --> 00:14:47,500
Them we went through a couple of
alternate definitions of a

231
00:14:47,500 --> 00:14:48,750
Poisson process.

232
00:14:51,130 --> 00:14:55,980
And at this point, just from
what I've said so far, and

233
00:14:55,980 --> 00:14:59,520
from reading the notes and
understanding what I've said

234
00:14:59,520 --> 00:15:04,060
so far, it ought to be almost
clear that these alternate

235
00:15:04,060 --> 00:15:06,190
definitions, which we
talked about last

236
00:15:06,190 --> 00:15:09,030
time have to be valid.

237
00:15:09,030 --> 00:15:13,020
If an arrival process has the
stationary and independent

238
00:15:13,020 --> 00:15:17,590
increment properties and if N
of t has the Poisson PMF for

239
00:15:17,590 --> 00:15:23,520
given lambda, and all t, then
the process is Poisson.

240
00:15:23,520 --> 00:15:26,640
Now what is that saying?

241
00:15:26,640 --> 00:15:30,970
I mean, we've said that if we
can specify all of the random

242
00:15:30,970 --> 00:15:34,030
variables N of t for
all t, then we've

243
00:15:34,030 --> 00:15:36,170
specified the process.

244
00:15:36,170 --> 00:15:38,970
What does it mean to
specify a whole

245
00:15:38,970 --> 00:15:41,650
bunch of random variables?

246
00:15:41,650 --> 00:15:44,770
It is not sufficient to find the
distribution function of

247
00:15:44,770 --> 00:15:48,270
all those random variables.

248
00:15:48,270 --> 00:15:53,190
And one of the problems in the
homeworks at this time is to

249
00:15:53,190 --> 00:15:57,000
explicitly show that for
a simpler process,

250
00:15:57,000 --> 00:15:59,240
the Bernoulli process.

251
00:15:59,240 --> 00:16:07,080
And to actually construct an
example of where N of t is

252
00:16:07,080 --> 00:16:10,380
specified everywhere.

253
00:16:10,380 --> 00:16:13,110
But you don't have the
independence between different

254
00:16:13,110 --> 00:16:16,500
intervals, and therefore,
you don't

255
00:16:16,500 --> 00:16:18,310
have a Bernoulli process.

256
00:16:18,310 --> 00:16:22,980
You just have this nice binomial
formula everywhere.

257
00:16:22,980 --> 00:16:26,400
But it doesn't really
tell you much.

258
00:16:26,400 --> 00:16:31,080
OK, so, but here we're adding
on the independent increment

259
00:16:31,080 --> 00:16:36,650
properties, which says over any
set of intervals the joint

260
00:16:36,650 --> 00:16:39,840
distribution of how many
arrivals there are here, how

261
00:16:39,840 --> 00:16:54,080
many here, how many here, how
many here, those joint random

262
00:16:54,080 --> 00:16:59,600
variables are independent of
each other, which is what the

263
00:16:59,600 --> 00:17:02,210
independent increment
property says.

264
00:17:02,210 --> 00:17:07,140
So in fact, this tells you
everything you want to know

265
00:17:07,140 --> 00:17:11,380
because you now know the
relationship between each one

266
00:17:11,380 --> 00:17:12,630
of these intervals.

267
00:17:18,970 --> 00:17:21,420
So we see why the process
is Poisson.

268
00:17:21,420 --> 00:17:23,230
This one's a little trickier.

269
00:17:23,230 --> 00:17:26,220
If an arrival process has the
stationary and independent

270
00:17:26,220 --> 00:17:30,190
increment properties and it
satisfies this incremental

271
00:17:30,190 --> 00:17:33,670
condition, then the
processes Poisson.

272
00:17:33,670 --> 00:17:36,870
And the incremental condition
says that if you're looking at

273
00:17:36,870 --> 00:17:46,040
the number of arrivals in some
interval of size delta, the

274
00:17:46,040 --> 00:17:51,830
probability that this is equal
to N has the form 1 minus

275
00:17:51,830 --> 00:17:57,030
lambda delta plus o of delta,
where n equals zero.

276
00:17:57,030 --> 00:18:00,480
Lambda delta plus o of delta,
for n equals 1.

277
00:18:00,480 --> 00:18:04,480
o of delta for n greater
than or equal to 2.

278
00:18:04,480 --> 00:18:08,620
This intuitively is only
supposed to talk about very,

279
00:18:08,620 --> 00:18:10,170
very small delta.

280
00:18:10,170 --> 00:18:14,442
So if you take the Poisson
distribution lambda t to the

281
00:18:14,442 --> 00:18:18,660
N, e to the minus lambda t over
n factorial, and you look

282
00:18:18,660 --> 00:18:22,380
at what happens when t is
very small, this is

283
00:18:22,380 --> 00:18:24,330
what it turns into.

284
00:18:24,330 --> 00:18:28,420
When t is very small the
probability that there are no

285
00:18:28,420 --> 00:18:32,700
arrivals in this interval of
size delta is very large.

286
00:18:32,700 --> 00:18:42,490
It's 1 minus lambda delta plus
this extra term that says--

287
00:18:42,490 --> 00:18:45,130
first point of view whenever
you see an o of delta is to

288
00:18:45,130 --> 00:18:48,320
say, oh, that's not important.

289
00:18:48,320 --> 00:18:53,890
And for N equals 1, there's
going to be one arrival with

290
00:18:53,890 --> 00:18:56,680
some fudge factor, which
is not important.

291
00:18:56,680 --> 00:18:59,770
And there's going to be two or
more arrivals with some fudge

292
00:18:59,770 --> 00:19:02,400
factor, which is
not important.

293
00:19:02,400 --> 00:19:07,830
The next thing we talked about
is that o of delta really is

294
00:19:07,830 --> 00:19:15,650
defined as any function which
is a function of delta where

295
00:19:15,650 --> 00:19:18,800
the limit of o of
delta divided by

296
00:19:18,800 --> 00:19:20,960
delta is equal to 0.

297
00:19:20,960 --> 00:19:23,660
In other words, it's some
function that goes to zero

298
00:19:23,660 --> 00:19:25,840
faster than delta does.

299
00:19:25,840 --> 00:19:29,090
So it's something which
is insignificant with

300
00:19:29,090 --> 00:19:34,950
relationship to this as
delta gets very small.

301
00:19:34,950 --> 00:19:38,440
Now, how do you use this kind
of statement to make some

302
00:19:38,440 --> 00:19:41,430
statement about larger
intervals?

303
00:19:41,430 --> 00:19:44,650
Well, you're clearly
stuck looking at

304
00:19:44,650 --> 00:19:46,660
differential equations.

305
00:19:46,660 --> 00:19:48,720
And the text does that.

306
00:19:48,720 --> 00:19:52,130
I refuse to talk about
differential equations in

307
00:19:52,130 --> 00:19:53,780
lecture or anyplace else.

308
00:19:53,780 --> 00:19:57,000
When I retired I said, I will
no longer talk about

309
00:19:57,000 --> 00:19:58,250
differential equations
anymore.

310
00:20:01,310 --> 00:20:04,510
And you know, you don't need to
because you can see what's

311
00:20:04,510 --> 00:20:07,870
happening here.

312
00:20:07,870 --> 00:20:11,870
And what you see is happening
is, in fact, what's happening.

313
00:20:11,870 --> 00:20:15,790
OK, so, that's where we
finished up last time.

314
00:20:15,790 --> 00:20:20,210
And now, we come to the really
fun stuff where we want to

315
00:20:20,210 --> 00:20:24,000
combine independent
Poisson processes.

316
00:20:24,000 --> 00:20:26,970
And then, we want to split
Poisson processes.

317
00:20:26,970 --> 00:20:29,900
And we want to play all sorts of
games with multiple Poisson

318
00:20:29,900 --> 00:20:33,690
processes, which looks very
hard, and because of this,

319
00:20:33,690 --> 00:20:34,780
it's very easy.

320
00:20:34,780 --> 00:20:35,996
Yes?

321
00:20:35,996 --> 00:20:38,550
AUDIENCE: The previous
definitions, they are if and

322
00:20:38,550 --> 00:20:39,960
only statements, right?

323
00:20:39,960 --> 00:20:40,910
PROFESSOR: They are what?

324
00:20:40,910 --> 00:20:42,480
AUDIENCE: If and only,
all Poisson

325
00:20:42,480 --> 00:20:45,070
processes satisfy tho--

326
00:20:45,070 --> 00:20:46,450
PROFESSOR: Yes.

327
00:20:46,450 --> 00:20:49,590
OK, in other words, what you're
saying is, if you can

328
00:20:49,590 --> 00:20:53,510
satisfy those properties, this
is a Poisson process.

329
00:20:53,510 --> 00:20:56,190
I mean, we've already shown
that a Poisson process

330
00:20:56,190 --> 00:20:57,440
satisfies those properties.

331
00:21:03,170 --> 00:21:06,510
As a matter of fact, the way I
put it in the notes is these

332
00:21:06,510 --> 00:21:09,140
are three alternate definitions
where you could

333
00:21:09,140 --> 00:21:13,940
start out with any one of them
and derive the whole thing.

334
00:21:13,940 --> 00:21:16,660
Many people like to start out
with this incremental

335
00:21:16,660 --> 00:21:21,030
definition because it's
very physical.

336
00:21:21,030 --> 00:21:24,770
But it makes all the mathematics
much, much harder.

337
00:21:24,770 --> 00:21:28,930
And so, it's just a question
of what you prefer.

338
00:21:28,930 --> 00:21:32,280
I like to start out with
something clean, then derive

339
00:21:32,280 --> 00:21:34,530
things, and then say, does
it make any sense for the

340
00:21:34,530 --> 00:21:36,920
physical situation?

341
00:21:36,920 --> 00:21:38,970
And that's what we usually do.

342
00:21:38,970 --> 00:21:42,340
We don't usually start out with
a physical situation and

343
00:21:42,340 --> 00:21:48,120
analyze the hell out of it and
say, aha, this is a Poisson

344
00:21:48,120 --> 00:21:51,100
process because it satisfies
all these properties.

345
00:21:51,100 --> 00:21:54,470
It never satisfies all
those properties.

346
00:21:54,470 --> 00:21:57,810
I mean, you say it's a Poisson
process because a Poisson

347
00:21:57,810 --> 00:22:02,340
process is simple and you can
get some insight from it, not

348
00:22:02,340 --> 00:22:06,760
because it really is
a Poisson process.

349
00:22:06,760 --> 00:22:12,890
Let's talk about taking two
independent Poisson processes.

350
00:22:12,890 --> 00:22:17,340
Just to be a little more
precise, two counting

351
00:22:17,340 --> 00:22:24,030
processes, N1 of t and N2 of t
are independent if for all t1

352
00:22:24,030 --> 00:22:29,840
up to t sub n the random
variables N1 of t1 to N1 of tn

353
00:22:29,840 --> 00:22:35,280
are independent of N1 of
t1 up to N2 of tn.

354
00:22:35,280 --> 00:22:38,590
Why don't I just say that for
all t they're independent?

355
00:22:38,590 --> 00:22:40,930
Because I don't even know
what that means.

356
00:22:40,930 --> 00:22:45,010
I mean, we've never defined
independence for an infinite

357
00:22:45,010 --> 00:22:47,100
number of things.

358
00:22:47,100 --> 00:22:51,760
So all we can do is say, for all
finite sets, we have this

359
00:22:51,760 --> 00:22:53,460
independence.

360
00:22:53,460 --> 00:22:58,210
Now, give you a short
pop quiz.

361
00:22:58,210 --> 00:23:03,375
Suppose that instead of doing it
this way, I say, for all t1

362
00:23:03,375 --> 00:23:09,390
to tn the random variables
N1 of t1 to N1 of tn are

363
00:23:09,390 --> 00:23:17,325
independent of for all tao1,
tao2, tao3, tao4, N2 of tao1

364
00:23:17,325 --> 00:23:20,300
up to N2 of tao sub n?

365
00:23:20,300 --> 00:23:22,250
That sounds much more
general, doesn't it?

366
00:23:22,250 --> 00:23:26,600
Because it's saying that I can
count one process at one set

367
00:23:26,600 --> 00:23:31,610
of times and the other process
at another set of times.

368
00:23:31,610 --> 00:23:34,490
Now, why isn't it any more
general to do it that way?

369
00:23:41,700 --> 00:23:45,470
Well, it's an unfair pop quiz
because if you can answer that

370
00:23:45,470 --> 00:23:49,520
question in the short one
sentence answer that you'd be

371
00:23:49,520 --> 00:23:54,570
willing to give in a class like
this, I would just give

372
00:23:54,570 --> 00:23:59,250
you A plus and tell
you go away.

373
00:23:59,250 --> 00:24:02,800
And you already know all the
things you should know.

374
00:24:02,800 --> 00:24:06,870
The argument is the following,
if you order these different

375
00:24:06,870 --> 00:24:14,760
times, first tao1 of tao1 is
less than t1 then t1, if

376
00:24:14,760 --> 00:24:18,560
that's the next one, and you
order them all along.

377
00:24:18,560 --> 00:24:22,440
And then you apply this
definition to

378
00:24:22,440 --> 00:24:25,060
that ordered set.

379
00:24:25,060 --> 00:24:33,830
t1, tao1, t2, t3, t4, tao2, t5,
tao3, and so forth, and

380
00:24:33,830 --> 00:24:37,200
you apply this definition, and
then you get the other

381
00:24:37,200 --> 00:24:38,320
definition.

382
00:24:38,320 --> 00:24:40,690
So one is not more general
than the other.

383
00:24:44,940 --> 00:24:49,980
The theorem then, is that if
N1 of t and N2 of t are

384
00:24:49,980 --> 00:24:53,140
independent Poisson processes,
one of them has

385
00:24:53,140 --> 00:24:54,740
a rate lambda 1.

386
00:24:54,740 --> 00:24:57,120
One of them has a
rate lambda 2.

387
00:24:57,120 --> 00:25:02,530
And if N of t is equal to N1 of
t plus N2 of t, that just

388
00:25:02,530 --> 00:25:08,230
means for every t this random
variable N of t is the sum of

389
00:25:08,230 --> 00:25:11,190
the random variable N1
of t plus the random

390
00:25:11,190 --> 00:25:12,960
variable N2 of 2.

391
00:25:12,960 --> 00:25:14,530
This is true for all.

392
00:25:14,530 --> 00:25:18,775
This is, definition, for all t
greater than 0, then the sum N

393
00:25:18,775 --> 00:25:24,920
of t is a Poisson process of
rate lambda equals lambda 1

394
00:25:24,920 --> 00:25:26,600
plus lambda 2.

395
00:25:26,600 --> 00:25:30,680
Looks almost obvious,
doesn't it?

396
00:25:30,680 --> 00:25:33,380
I said that today there
was lots of fun stuff.

397
00:25:33,380 --> 00:25:37,360
There's also a little bit of
ugly stuff and this is one of

398
00:25:37,360 --> 00:25:42,160
those half obvious things
that's ugly.

399
00:25:42,160 --> 00:25:44,370
And I'm not going to waste
a lot of time on it.

400
00:25:44,370 --> 00:25:47,290
You can read all about the
details in the notes.

401
00:25:47,290 --> 00:25:49,610
But I will spend a little
bit of time on it

402
00:25:49,610 --> 00:25:52,850
because it's important.

403
00:25:52,850 --> 00:25:57,620
The idea is that if you look at
any small increment, t to t

404
00:25:57,620 --> 00:26:02,757
plus delta, the number of
arrivals in the interval t to

405
00:26:02,757 --> 00:26:07,280
t plus delta is equal to the
number of arrivals in the

406
00:26:07,280 --> 00:26:10,970
interval t to t plus delta from
the first process plus

407
00:26:10,970 --> 00:26:12,870
that in the second process.

408
00:26:12,870 --> 00:26:17,230
So the probability that there's
one arrival in this

409
00:26:17,230 --> 00:26:21,600
combined process is the
probability that there's one

410
00:26:21,600 --> 00:26:27,940
arrival in the first process and
no arrivals in the second

411
00:26:27,940 --> 00:26:31,890
process, or that there's no
arrivals in the first process

412
00:26:31,890 --> 00:26:34,270
and one arrival in the
second process.

413
00:26:34,270 --> 00:26:37,315
That's just a very simple
case of convolution.

414
00:26:39,870 --> 00:26:42,020
Those are the only ways
you can get one in

415
00:26:42,020 --> 00:26:44,270
the combined process.

416
00:26:44,270 --> 00:26:51,310
This term here is delta
times lambda 1--

417
00:26:51,310 --> 00:26:52,420
too early in the morning.

418
00:26:52,420 --> 00:26:55,586
I'm confusing my deltas
and lambdas.

419
00:26:55,586 --> 00:26:57,820
There are too many
of each of them.

420
00:26:57,820 --> 00:26:59,310
--plus o of delta.

421
00:26:59,310 --> 00:27:02,680
And this term here, probability
that there's zero

422
00:27:02,680 --> 00:27:05,620
for the second process
is 1 minus delta

423
00:27:05,620 --> 00:27:08,300
lambda 2 plus o of delta.

424
00:27:08,300 --> 00:27:11,370
And then this term is just
the opposite term

425
00:27:11,370 --> 00:27:13,010
corresponding to this.

426
00:27:13,010 --> 00:27:15,670
Now I multiply these
terms out.

427
00:27:15,670 --> 00:27:17,560
And what do I get?

428
00:27:17,560 --> 00:27:22,750
Well, this 1 here combines
with a delta lambda 1.

429
00:27:22,750 --> 00:27:28,660
Then there's a delta lambda 2
times delta lambda 1, which is

430
00:27:28,660 --> 00:27:29,860
delta squared.

431
00:27:29,860 --> 00:27:32,800
It's a delta squared term,
so that's really

432
00:27:32,800 --> 00:27:34,980
an o of delta term.

433
00:27:34,980 --> 00:27:39,540
It's negligible as delta
goes to zero.

434
00:27:39,540 --> 00:27:41,490
So we forget about that.

435
00:27:41,490 --> 00:27:45,710
There's an o of delta times 1,
that's still an o of delta.

436
00:27:45,710 --> 00:27:48,380
There's an o of delta times
a delta lambda.

437
00:27:48,380 --> 00:27:53,610
And that's sort of an o of delta
squared if you wish.

438
00:27:53,610 --> 00:27:54,950
But it's still an o of delta.

439
00:27:54,950 --> 00:27:58,140
It goes to zero as
delta gets large.

440
00:27:58,140 --> 00:28:00,160
And goes to zero faster
than delta.

441
00:28:00,160 --> 00:28:02,770
What we're trying to do is to
find the terms that are

442
00:28:02,770 --> 00:28:06,480
significant in terms of delta.

443
00:28:06,480 --> 00:28:10,130
Namely, when delta gets very
small, I want to find things

444
00:28:10,130 --> 00:28:13,990
that are at least proportional
to delta and not of lower

445
00:28:13,990 --> 00:28:15,450
order than delta.

446
00:28:15,450 --> 00:28:18,790
So when I get done with all of
that, this is delta times

447
00:28:18,790 --> 00:28:22,500
lambda 1 plus lambda
2 plus o of delta.

448
00:28:22,500 --> 00:28:27,040
That's the incremental property
that we want a

449
00:28:27,040 --> 00:28:28,370
Poisson process to have.

450
00:28:28,370 --> 00:28:30,895
So it has that incremental
property.

451
00:28:35,190 --> 00:28:42,150
And those are the same sort of
argument if you want to for

452
00:28:42,150 --> 00:28:47,160
the number of arrivals in
t, that t plus delta.

453
00:28:47,160 --> 00:28:48,820
Maybe a picture of
this would help.

454
00:28:52,320 --> 00:28:56,030
Pictures always help
in the morning.

455
00:28:56,030 --> 00:28:57,280
Here we have two processes.

456
00:29:00,220 --> 00:29:06,815
We're looking at some interval
of time, t to t plus delta.

457
00:29:09,530 --> 00:29:14,160
t to plus delta.

458
00:29:14,160 --> 00:29:15,790
And we might have
an arrival here.

459
00:29:18,580 --> 00:29:22,120
We might have an arrival
here, an arrival

460
00:29:22,120 --> 00:29:25,460
here, an arrival here.

461
00:29:25,460 --> 00:29:29,040
Well, the probability of an
arrival here and an arrival

462
00:29:29,040 --> 00:29:32,990
here is something of order
delta squared.

463
00:29:32,990 --> 00:29:36,050
So that's something we ignore.

464
00:29:36,050 --> 00:29:38,760
So it says, we might have
an arrival here

465
00:29:38,760 --> 00:29:41,810
and no arrival here.

466
00:29:41,810 --> 00:29:43,910
This is a lambda 1.

467
00:29:43,910 --> 00:29:45,960
This is lambda 2 here.

468
00:29:48,840 --> 00:29:51,700
We might have an arrival here
and none here, or we might

469
00:29:51,700 --> 00:29:54,470
have an arrival here
and none there.

470
00:29:54,470 --> 00:29:58,080
Two arrivals is just too
unlikely to worry about, so we

471
00:29:58,080 --> 00:30:00,784
forget about it at least
for the time being.

472
00:30:05,160 --> 00:30:09,190
Now, after going through this
incremental argument, if you

473
00:30:09,190 --> 00:30:13,050
go back and say, let's forget
about all these o of deltas

474
00:30:13,050 --> 00:30:15,670
because they're very confusing,
let's just do the

475
00:30:15,670 --> 00:30:18,370
convolution knowing
what N of t is in

476
00:30:18,370 --> 00:30:20,110
both of these intervals.

477
00:30:20,110 --> 00:30:24,150
If you do that, it's much easier
to find out that the

478
00:30:24,150 --> 00:30:29,940
number of arrivals in this
sum of intervals.

479
00:30:29,940 --> 00:30:33,760
Number arrivals here, plus the
number of arrivals here, this

480
00:30:33,760 --> 00:30:34,810
is Poisson.

481
00:30:34,810 --> 00:30:35,870
This is Poisson.

482
00:30:35,870 --> 00:30:39,040
You add two Poisson,
you get Poisson.

483
00:30:39,040 --> 00:30:44,294
The rate of the sum of the
two Poisson is Poisson.

484
00:30:44,294 --> 00:30:49,290
So it's Poisson with lambda
1 plus lambda 2.

485
00:30:49,290 --> 00:30:52,870
Now how many of you saw that
and says, why is this idiot

486
00:30:52,870 --> 00:30:55,650
going through this incremental
argument?

487
00:30:55,650 --> 00:30:56,930
Anyone?

488
00:30:56,930 --> 00:30:58,880
I won't be embarrassed.

489
00:30:58,880 --> 00:30:59,740
I knew it anyway.

490
00:30:59,740 --> 00:31:00,990
And I did it for a reason.

491
00:31:04,120 --> 00:31:07,060
But I must confess, when I wrote
the first edition of

492
00:31:07,060 --> 00:31:09,980
this book I didn't
recognize that.

493
00:31:09,980 --> 00:31:13,710
So I went through this terribly
tedious argument.

494
00:31:13,710 --> 00:31:22,630
Anyway, the more important issue
is if you have the sum

495
00:31:22,630 --> 00:31:25,870
of many, many small independent
arrival

496
00:31:25,870 --> 00:31:27,810
processes--

497
00:31:27,810 --> 00:31:32,110
OK, in other words, you
have the internet.

498
00:31:32,110 --> 00:31:37,680
And a node in the internet is
getting jobs, or messages,

499
00:31:37,680 --> 00:31:41,920
from all sorts of people, and
all sorts of processes, and

500
00:31:41,920 --> 00:31:45,400
all sorts of nonsense going to
all sorts of people, all sorts

501
00:31:45,400 --> 00:31:49,540
of processes, and all sorts
of nonsense and those are

502
00:31:49,540 --> 00:31:51,360
independent of each
other, not really

503
00:31:51,360 --> 00:31:53,240
independent of each other.

504
00:31:53,240 --> 00:32:00,750
But relative to the data rate
that's travelling over this

505
00:32:00,750 --> 00:32:03,760
internet, each of
those processes

506
00:32:03,760 --> 00:32:06,190
are very, very small.

507
00:32:06,190 --> 00:32:11,330
And what happens is the sum of
many, many small independent

508
00:32:11,330 --> 00:32:15,820
arrival processes tend to be
Poisson even if the small

509
00:32:15,820 --> 00:32:18,230
processes are not.

510
00:32:18,230 --> 00:32:21,690
In a sense, the independence
between the processes

511
00:32:21,690 --> 00:32:25,860
overcomes the dependence between
successive arrivals in

512
00:32:25,860 --> 00:32:27,660
each process.

513
00:32:27,660 --> 00:32:29,930
Now, I look at that and I say,
well, it's sort of a

514
00:32:29,930 --> 00:32:31,900
plausibility argument.

515
00:32:31,900 --> 00:32:35,230
You look at the argument in
the text, and you say, ah,

516
00:32:35,230 --> 00:32:37,185
it's sort of a plausibility
argument.

517
00:32:39,990 --> 00:32:42,290
I mean, proving this statement,
you need to put a

518
00:32:42,290 --> 00:32:44,080
lot of conditions on it.

519
00:32:44,080 --> 00:32:46,450
And you need to really go
through an awful lot of work.

520
00:32:49,060 --> 00:32:53,020
It's like proving the central
limit theorem, but it's

521
00:32:53,020 --> 00:32:55,510
probably harder than that.

522
00:32:55,510 --> 00:33:00,150
So, if you read the text,
and say you don't really

523
00:33:00,150 --> 00:33:04,120
understand the argument there,
I don't understand it either

524
00:33:04,120 --> 00:33:06,980
because I don't think
it's exactly right.

525
00:33:06,980 --> 00:33:10,520
And I was just trying to say
something to give some idea of

526
00:33:10,520 --> 00:33:11,770
why this is plausible.

527
00:33:14,980 --> 00:33:16,230
It should probably be changed.

528
00:33:18,830 --> 00:33:23,760
Next we want to talk about
splitting a Poisson process.

529
00:33:23,760 --> 00:33:26,440
So we start out with a
Poisson process here

530
00:33:26,440 --> 00:33:29,350
and a t of rate lambda.

531
00:33:29,350 --> 00:33:30,390
And what happens?

532
00:33:30,390 --> 00:33:33,690
These arrivals come
to some point.

533
00:33:33,690 --> 00:33:36,110
And some character is
standing there.

534
00:33:38,670 --> 00:33:41,720
It's like when you're having
your passport checked when you

535
00:33:41,720 --> 00:33:44,100
come back to Boston
after being away.

536
00:33:47,390 --> 00:33:52,630
In some places you press a
button, and if the button come

537
00:33:52,630 --> 00:33:56,790
up one way, you're sent off to
one line to be interrogated

538
00:33:56,790 --> 00:33:57,840
and all sorts of junk.

539
00:33:57,840 --> 00:33:59,490
And it's supposed
to be random.

540
00:33:59,490 --> 00:34:02,440
I have no idea whether it's
random or not, but it's

541
00:34:02,440 --> 00:34:04,820
supposed to be random.

542
00:34:04,820 --> 00:34:06,890
And otherwise, you go
through and you get

543
00:34:06,890 --> 00:34:08,790
through very quickly.

544
00:34:08,790 --> 00:34:11,110
So it's the same sort
of thing here.

545
00:34:11,110 --> 00:34:13,530
You have a bunch of arrivals.

546
00:34:13,530 --> 00:34:20,310
And each arrival is effectively
randomly shoveled

547
00:34:20,310 --> 00:34:23,050
this way or shoveled this way.

548
00:34:23,050 --> 00:34:26,330
With probability p, it's
shoveled this way.

549
00:34:26,330 --> 00:34:30,360
With probability 1 minus p,
it's shoveled this way.

550
00:34:30,360 --> 00:34:38,889
So you can characterize this
switch as a Bernoulli process.

551
00:34:38,889 --> 00:34:42,330
It's a Bernoulli process
because it's random and

552
00:34:42,330 --> 00:34:48,030
independent from, not
time to time now,

553
00:34:48,030 --> 00:34:50,150
but arrival to arrival.

554
00:34:50,150 --> 00:34:53,540
When we first looked at a
Poisson process, we said it's

555
00:34:53,540 --> 00:34:56,110
a sequence of random
variables.

556
00:34:56,110 --> 00:34:58,820
Sometimes we look at
in terms of time.

557
00:34:58,820 --> 00:35:01,680
Time doesn't really make
any difference there.

558
00:35:01,680 --> 00:35:05,940
It's just a sequence of
IID random variables.

559
00:35:05,940 --> 00:35:09,290
So you have a sequence of IID
random variables doing this

560
00:35:09,290 --> 00:35:10,450
switching here.

561
00:35:10,450 --> 00:35:13,520
You have a Poisson process
coming in.

562
00:35:13,520 --> 00:35:17,130
And when you look at the
combination of the Poisson

563
00:35:17,130 --> 00:35:21,990
process and the Bernoulli
process, you get some kind of

564
00:35:21,990 --> 00:35:24,870
process of things
coming out here.

565
00:35:24,870 --> 00:35:28,840
And another kind of process
of things coming out here.

566
00:35:28,840 --> 00:35:35,120
And the theorem says, that when
you combine this Poisson

567
00:35:35,120 --> 00:35:39,500
process with this independent
Bernoulli process, what you

568
00:35:39,500 --> 00:35:45,240
get is a Poisson process here
and an independent Poisson

569
00:35:45,240 --> 00:35:48,040
process here.

570
00:35:48,040 --> 00:35:51,600
And, of course, you need to know
what the probability is

571
00:35:51,600 --> 00:35:55,970
of being switched this way and
being switched that way.

572
00:35:55,970 --> 00:35:59,790
Each new process clearly has a
stationary and independent

573
00:35:59,790 --> 00:36:01,360
increment property.

574
00:36:01,360 --> 00:36:04,450
Why is that?

575
00:36:04,450 --> 00:36:10,010
Well, you look at some increment
of time and this

576
00:36:10,010 --> 00:36:15,680
process here is independent
from one period of time to

577
00:36:15,680 --> 00:36:18,650
another period of time.

578
00:36:18,650 --> 00:36:23,930
The Bernoulli process is just
switching the terms within

579
00:36:23,930 --> 00:36:27,220
that interval of time, which
is independent of all other

580
00:36:27,220 --> 00:36:29,060
intervals of time.

581
00:36:29,060 --> 00:36:31,680
So that when you look at the
combination of the Bernoulli

582
00:36:31,680 --> 00:36:35,580
process and the Poisson
process, you have the

583
00:36:35,580 --> 00:36:40,680
stationary and independent
increment property.

584
00:36:40,680 --> 00:36:44,890
And each satisfies the small
increment property.

585
00:36:44,890 --> 00:36:48,410
If you look at very
small delta here,

586
00:36:48,410 --> 00:36:49,960
unless each is Poisson.

587
00:36:49,960 --> 00:36:52,930
There's a more careful argument
in the notes.

588
00:36:52,930 --> 00:36:56,150
What I'm trying to do in the
lecture is not to give you

589
00:36:56,150 --> 00:36:59,580
careful proofs of things, but to
give you some insight into

590
00:36:59,580 --> 00:37:01,520
why they're true.

591
00:37:01,520 --> 00:37:04,570
So that you can read
the proof.

592
00:37:04,570 --> 00:37:07,580
And instead of going through
each line and saying, yes, I

593
00:37:07,580 --> 00:37:10,410
agree with that, yes, I agree
with that and you finally come

594
00:37:10,410 --> 00:37:13,780
to the end of the proof
and you think what?

595
00:37:13,780 --> 00:37:17,020
Which I've done all the time
because I don't really know

596
00:37:17,020 --> 00:37:19,960
what's going on.

597
00:37:19,960 --> 00:37:23,680
And you don't learn anything
from that kind of proof.

598
00:37:23,680 --> 00:37:28,200
If you're surprised when it's
done it means you ought to go

599
00:37:28,200 --> 00:37:30,270
back and look at it more
carefully because there's

600
00:37:30,270 --> 00:37:36,540
something there that you didn't
understand while it was

601
00:37:36,540 --> 00:37:37,790
wandering past you.

602
00:37:41,230 --> 00:37:44,250
The small increment property
really doesn't make it clear

603
00:37:44,250 --> 00:37:48,090
that the split processes
are independent.

604
00:37:48,090 --> 00:37:54,000
And for independence both
processes must sometimes have

605
00:37:54,000 --> 00:37:56,560
arrivals in the same
small increment.

606
00:37:56,560 --> 00:37:58,470
And this independence
is hidden in

607
00:37:58,470 --> 00:38:00,350
those o of delta terms.

608
00:38:00,350 --> 00:38:03,220
And if you want to resolve that
for yourself, you really

609
00:38:03,220 --> 00:38:06,440
have to look at the text.

610
00:38:06,440 --> 00:38:09,380
And you have to do
some work too.

611
00:38:09,380 --> 00:38:13,030
The nice thing about combining
and splitting is that you

612
00:38:13,030 --> 00:38:15,350
typically do them together.

613
00:38:15,350 --> 00:38:19,600
Most of the places where you
use combining and splitting

614
00:38:19,600 --> 00:38:22,690
you use both of them
repeatedly.

615
00:38:22,690 --> 00:38:26,400
The typical thing you do is
first, you look at separate

616
00:38:26,400 --> 00:38:29,000
independent Poisson processes.

617
00:38:29,000 --> 00:38:32,610
And you take those separate
independent Poisson processes,

618
00:38:32,610 --> 00:38:37,440
and you say, I want to look at
those as a combined process.

619
00:38:37,440 --> 00:38:40,760
And after you look at them as
a combined process, you then

620
00:38:40,760 --> 00:38:42,880
split them again.

621
00:38:42,880 --> 00:38:46,540
And what you're doing, when you
then split them again, is

622
00:38:46,540 --> 00:38:50,740
you're saying these two
independent Poisson processes

623
00:38:50,740 --> 00:38:54,720
that I started with, I can
view them as one Poisson

624
00:38:54,720 --> 00:38:58,310
process plus a Bernoulli
process.

625
00:38:58,310 --> 00:39:01,630
And you'd be amazed at how many
problems you can solve by

626
00:39:01,630 --> 00:39:03,130
doing that.

627
00:39:03,130 --> 00:39:06,560
You will be amazed when you
do the homework this time.

628
00:39:06,560 --> 00:39:12,220
If you don't use that property
at least 10 times, you haven't

629
00:39:12,220 --> 00:39:18,005
really understood what's being
asked for in those problems.

630
00:39:21,370 --> 00:39:25,060
Let's look at a simple
example.

631
00:39:25,060 --> 00:39:28,030
Let's look at a last come,
first served queue.

632
00:39:28,030 --> 00:39:31,980
Last come, first served queues
are rather peculiar.

633
00:39:31,980 --> 00:39:38,310
They are queues where arrivals
come in, and for some reason

634
00:39:38,310 --> 00:39:42,580
or other, when a new arrival
comes in that arrival goes

635
00:39:42,580 --> 00:39:44,930
into the server immediately.

636
00:39:44,930 --> 00:39:47,010
And the queue gets backed up.

637
00:39:47,010 --> 00:39:49,795
Whatever was being served
gets backed up.

638
00:39:49,795 --> 00:39:53,680
And the new arrival gets served,
which is fine for the

639
00:39:53,680 --> 00:39:57,260
new arrival, unless another
arrival comes

640
00:39:57,260 --> 00:40:00,410
in before it finishes.

641
00:40:00,410 --> 00:40:02,500
And then it gets
backed up too.

642
00:40:02,500 --> 00:40:05,518
So things get backed
up in the queue.

643
00:40:05,518 --> 00:40:07,380
And those that are lucky
enough to get

644
00:40:07,380 --> 00:40:08,880
through, get through.

645
00:40:08,880 --> 00:40:11,400
And those that aren't, don't.

646
00:40:11,400 --> 00:40:13,950
Anybody have any
idea why that--

647
00:40:13,950 --> 00:40:17,470
I mean, it sounds like a very
unfair thing to do.

648
00:40:17,470 --> 00:40:22,570
But, in fact, it's not unfair
because you're not singling

649
00:40:22,570 --> 00:40:25,270
out any particular group
or anything.

650
00:40:25,270 --> 00:40:29,120
You're just that's
the rule you use.

651
00:40:29,120 --> 00:40:32,540
And it applies to everyone
equally, so it's not unfair.

652
00:40:32,540 --> 00:40:35,880
But why might it make
sense sometimes?

653
00:40:35,880 --> 00:40:37,122
Yeah?

654
00:40:37,122 --> 00:40:40,098
AUDIENCE: If you're doing a
first come, first served

655
00:40:40,098 --> 00:40:43,580
you're going to have a
certain distribution

656
00:40:43,580 --> 00:40:47,250
for your serve time.

657
00:40:47,250 --> 00:40:50,948
Whereas, if you do last
come, first served--

658
00:40:50,948 --> 00:40:51,894
I don't know.

659
00:40:51,894 --> 00:40:54,505
I'm just trying to think that
there might some situations

660
00:40:54,505 --> 00:40:57,110
where the distribution of
service times in that is

661
00:40:57,110 --> 00:41:00,427
favorable overall, even though
some people, every once in a

662
00:41:00,427 --> 00:41:02,850
while, are feeling screwed.

663
00:41:02,850 --> 00:41:06,080
PROFESSOR: It's a good try.

664
00:41:06,080 --> 00:41:07,790
But it's not quite right.

665
00:41:07,790 --> 00:41:11,250
And, in fact, it's not right for
a Poisson process because

666
00:41:11,250 --> 00:41:16,700
for a Poisson process the server
is just chunking things

667
00:41:16,700 --> 00:41:21,490
out at rate mu, no
matter what.

668
00:41:21,490 --> 00:41:23,980
So it doesn't really
help anyone.

669
00:41:23,980 --> 00:41:27,450
If you have a heavy tail
distribution, if somebody who

670
00:41:27,450 --> 00:41:31,180
comes in and requires an
enormous amount of service,

671
00:41:31,180 --> 00:41:35,340
then you get everybody else
done first because that

672
00:41:35,340 --> 00:41:40,430
customer with a huge service
requirement keeps getting

673
00:41:40,430 --> 00:41:44,210
pushed back every time the queue
is empty he gets some

674
00:41:44,210 --> 00:41:45,220
service again.

675
00:41:45,220 --> 00:41:47,100
And then other people come in.

676
00:41:47,100 --> 00:41:48,440
And people with small service

677
00:41:48,440 --> 00:41:50,200
requirements get served quickly.

678
00:41:53,700 --> 00:41:55,200
And what that means
is it's not quite

679
00:41:55,200 --> 00:41:57,550
as crazy as it sounds.

680
00:41:57,550 --> 00:42:02,450
But the reason we want to look
at it here is because it's a

681
00:42:02,450 --> 00:42:04,470
nice example of this
combining and

682
00:42:04,470 --> 00:42:07,430
splitting of Poisson processes.

683
00:42:07,430 --> 00:42:09,970
So how does that happen?

684
00:42:09,970 --> 00:42:14,530
Well, you view services
as a Poisson process.

685
00:42:14,530 --> 00:42:18,640
Namely, we have an exponential
server here, where the time

686
00:42:18,640 --> 00:42:22,040
for each service is
exponentially distributed.

687
00:42:22,040 --> 00:42:24,660
Now, if you're awake, you point
out, well, what happens

688
00:42:24,660 --> 00:42:27,850
when the server has
nothing to do?

689
00:42:27,850 --> 00:42:32,710
Well, just suppose that there's
some very low priority

690
00:42:32,710 --> 00:42:36,870
set of jobs that the server
is doing, which also are

691
00:42:36,870 --> 00:42:40,200
exponentially distributed that
the server goes to when it has

692
00:42:40,200 --> 00:42:41,510
nothing to do.

693
00:42:41,510 --> 00:42:47,520
So the server is still doing
things exponentially at rate

694
00:42:47,520 --> 00:42:51,770
lambda, but if there's nothing
real to do then

695
00:42:51,770 --> 00:42:53,570
the output is wasted.

696
00:42:53,570 --> 00:42:55,910
So that's a Poisson process.

697
00:42:55,910 --> 00:42:59,940
The income is Poisson also.

698
00:42:59,940 --> 00:43:04,880
So the arrival process, plus the
service process, they're

699
00:43:04,880 --> 00:43:06,030
independent.

700
00:43:06,030 --> 00:43:06,980
So they're Poisson.

701
00:43:06,980 --> 00:43:10,900
And the rate is lambda
plus mu.

702
00:43:10,900 --> 00:43:16,020
Now, interesting question,
what's the probability that an

703
00:43:16,020 --> 00:43:20,030
arrival completes service before
being interrupted?

704
00:43:20,030 --> 00:43:22,120
I'm lucky, I get into
the server.

705
00:43:22,120 --> 00:43:23,830
I start getting served.

706
00:43:23,830 --> 00:43:26,950
What's the probability that
I finish before I get

707
00:43:26,950 --> 00:43:28,200
interrupted?

708
00:43:33,380 --> 00:43:37,200
Well, I get into service
at a particular time t.

709
00:43:37,200 --> 00:43:41,640
I look at this combined Poisson
process of arrivals

710
00:43:41,640 --> 00:43:47,630
and services and if the first
arrival in this combined

711
00:43:47,630 --> 00:43:53,740
process gets switched to
service, I'm done.

712
00:43:53,740 --> 00:43:58,460
If gets switched to arrival,
I'm interrupted.

713
00:43:58,460 --> 00:44:02,180
So the question is, what's the
probability that I get

714
00:44:02,180 --> 00:44:04,380
switched to--

715
00:44:08,360 --> 00:44:10,940
I want to find the probability
that I'm interrupted, so

716
00:44:10,940 --> 00:44:14,240
what's the probability that a
new arrival in the combined

717
00:44:14,240 --> 00:44:20,480
process gets switched to
arrivals because that's the

718
00:44:20,480 --> 00:44:22,270
case where I get interrupted.

719
00:44:22,270 --> 00:44:23,520
And it's just lambda.

720
00:44:29,470 --> 00:44:32,730
The probability that I get
interrupted is lambda divided

721
00:44:32,730 --> 00:44:34,290
by lambda plus mu.

722
00:44:34,290 --> 00:44:39,180
And the probability that I
complete my service is mu over

723
00:44:39,180 --> 00:44:40,440
lambda plus mu.

724
00:44:40,440 --> 00:44:44,450
When mu is very large, when the
server is going very, very

725
00:44:44,450 --> 00:44:47,640
fast, I have a good chance
of finishing before being

726
00:44:47,640 --> 00:44:48,600
interrupted.

727
00:44:48,600 --> 00:44:51,120
When it's the other
way, I have a much

728
00:44:51,120 --> 00:44:54,120
smaller chance of finishing.

729
00:44:54,120 --> 00:44:57,340
More interesting, and more
difficult case, given that

730
00:44:57,340 --> 00:45:01,410
you're interrupted, what is the
probability that you have

731
00:45:01,410 --> 00:45:03,730
no further interruptions?

732
00:45:03,730 --> 00:45:07,360
In other words, I'm being
served, I get interrupted, so

733
00:45:07,360 --> 00:45:10,760
now I'm sitting at the
front of the queue.

734
00:45:10,760 --> 00:45:14,260
Everybody else that came in
before me is in back of me.

735
00:45:14,260 --> 00:45:16,590
I'm sitting there at the
front of the queue.

736
00:45:16,590 --> 00:45:21,360
This interrupting customer
is being served.

737
00:45:21,360 --> 00:45:33,700
What's the probability that I'm
going to finish my service

738
00:45:33,700 --> 00:45:36,110
before any further interruption
occurs?

739
00:45:36,110 --> 00:45:39,430
Have to be careful about
how to state this.

740
00:45:39,430 --> 00:45:42,900
The probability that there is
no further interruption is

741
00:45:42,900 --> 00:45:48,700
that two services occur before
the next arrival.

742
00:45:52,580 --> 00:45:56,710
And the probability of that is
mu over lambda plus mu the

743
00:45:56,710 --> 00:45:57,960
quantity squared.

744
00:46:01,990 --> 00:46:04,380
Now, whether you agree with
that number or not is

745
00:46:04,380 --> 00:46:05,050
immaterial.

746
00:46:05,050 --> 00:46:08,670
The thing I want you to
understand is that this is a

747
00:46:08,670 --> 00:46:14,140
method which you can use
in quite complicated

748
00:46:14,140 --> 00:46:15,390
circumstances.

749
00:46:17,100 --> 00:46:23,380
And it's something that applies
in so many places that

750
00:46:23,380 --> 00:46:25,820
it's amazing.

751
00:46:25,820 --> 00:46:28,060
OK, let's talk a little
bit about

752
00:46:28,060 --> 00:46:31,325
non-homogeneous Poisson processes.

753
00:46:35,990 --> 00:46:38,960
Maybe the most important
application of this is optical

754
00:46:38,960 --> 00:46:40,260
transmission.

755
00:46:40,260 --> 00:46:44,410
There's an optical stream of
photons that's modulated by

756
00:46:44,410 --> 00:46:46,230
variable power.

757
00:46:46,230 --> 00:46:49,590
Photon stream is reasonably
modeled as a Poisson process,

758
00:46:49,590 --> 00:46:51,770
not perfectly modeled
that way.

759
00:46:51,770 --> 00:46:55,380
But again, what we're doing here
is saying, let's look at

760
00:46:55,380 --> 00:46:56,540
this model.

761
00:46:56,540 --> 00:46:59,850
And then, see whether the
consequences of the model

762
00:46:59,850 --> 00:47:02,010
apply to the physical
situation.

763
00:47:02,010 --> 00:47:05,720
The modulation converts the
steady photon rate into a

764
00:47:05,720 --> 00:47:07,370
variable rate.

765
00:47:07,370 --> 00:47:13,210
So the arrivals are being
served, namely, they're being

766
00:47:13,210 --> 00:47:16,190
transmitted at some
rate, lambda of

767
00:47:16,190 --> 00:47:19,570
t, where t is varying.

768
00:47:19,570 --> 00:47:22,800
I mean, the photons have
nothing to do the

769
00:47:22,800 --> 00:47:25,230
information at all.

770
00:47:25,230 --> 00:47:27,350
They're just random photons.

771
00:47:27,350 --> 00:47:30,640
And the information is
represented by lambda t.

772
00:47:30,640 --> 00:47:32,500
The question is can
you actually send

773
00:47:32,500 --> 00:47:34,400
anything this way?

774
00:47:34,400 --> 00:47:38,710
We model the number of photons
in any interval t and t plus

775
00:47:38,710 --> 00:47:43,660
delta as a Poisson random
variable whose rate parameter

776
00:47:43,660 --> 00:47:48,030
over t and t plus delta, or
very small delta, is the

777
00:47:48,030 --> 00:47:52,210
average photon rate over t and
t plus delta times delta.

778
00:47:52,210 --> 00:47:55,780
So we go through this small
increment model again.

779
00:47:55,780 --> 00:47:59,140
But in the small increment
model, we have a lambda t

780
00:47:59,140 --> 00:48:04,100
instead of a lambda and
everything else is the same.

781
00:48:04,100 --> 00:48:10,510
And when you carry this
out, what you find--

782
00:48:10,510 --> 00:48:13,740
and I'm not going to show
this or anything.

783
00:48:13,740 --> 00:48:16,550
I'm just going to tell it to you
because it's in the notes

784
00:48:16,550 --> 00:48:19,280
if you want to read
more about it.

785
00:48:19,280 --> 00:48:24,530
The probability that you have a
given number of arrivals in

786
00:48:24,530 --> 00:48:30,320
some given interval is a Poisson
random variable again.

787
00:48:30,320 --> 00:48:33,010
Namely, whether the process
is homogeneous or not

788
00:48:33,010 --> 00:48:42,320
homogeneous, if the original
photons are Poisson then what

789
00:48:42,320 --> 00:48:47,340
you wind up with is
this parameter

790
00:48:47,340 --> 00:48:49,680
of the Poisson process.

791
00:48:49,680 --> 00:48:56,370
m hat to the N e the minus m
hat divided by n factorial.

792
00:48:56,370 --> 00:48:58,820
And this parameter
here is what?

793
00:48:58,820 --> 00:49:02,080
It's the overall arrival
rate in that interval.

794
00:49:02,080 --> 00:49:05,770
So it's exactly what you'd
expect it to be.

795
00:49:05,770 --> 00:49:08,810
So what this is saying is
combining and splitting

796
00:49:08,810 --> 00:49:12,980
non-homogeneous processes
still works like in the

797
00:49:12,980 --> 00:49:14,680
homogeneous case.

798
00:49:14,680 --> 00:49:17,630
The independent exponential
in arrivals don't work.

799
00:49:24,280 --> 00:49:27,280
OK, let me say that
another way.

800
00:49:27,280 --> 00:49:30,490
When you're looking at
non-homogeneous Poisson

801
00:49:30,490 --> 00:49:35,310
processes, looking at them in
terms of a Poisson counting

802
00:49:35,310 --> 00:49:38,240
process is really a
neat thing to do.

803
00:49:38,240 --> 00:49:41,930
Because all of the Poisson
counting process formulas

804
00:49:41,930 --> 00:49:43,100
still work.

805
00:49:43,100 --> 00:49:45,040
And all you have to do when
you want to look at the

806
00:49:45,040 --> 00:49:48,430
distribution of the number
arrivals in an interval is

807
00:49:48,430 --> 00:49:50,950
look at the average arrival
rate over that interval.

808
00:49:50,950 --> 00:49:53,390
And that tells you everything.

809
00:49:53,390 --> 00:49:56,460
When you look at the
interarrival times and things

810
00:49:56,460 --> 00:50:00,180
like that they aren't
exponential anymore.

811
00:50:00,180 --> 00:50:01,430
And none of that works.

812
00:50:04,790 --> 00:50:09,050
So you get half the pie, but
not the whole thing.

813
00:50:09,050 --> 00:50:15,100
Now, the other thing, which is
really fun, and which will

814
00:50:15,100 --> 00:50:22,690
take us a bit of time, is to
study, now it's not a small

815
00:50:22,690 --> 00:50:25,380
increment but a bigger
increment, of a Poisson

816
00:50:25,380 --> 00:50:29,900
process when you condition
on N of t.

817
00:50:29,900 --> 00:50:32,970
In other words, if somebody
tells you how many arrivals

818
00:50:32,970 --> 00:50:37,590
have occurred by time t, how do
you analyze where those are

819
00:50:37,590 --> 00:50:41,240
arrivals occur, where the
arrival epochs are,

820
00:50:41,240 --> 00:50:42,820
and things like that?

821
00:50:42,820 --> 00:50:47,340
Well, since we're conditioning
on N of t, let's start out

822
00:50:47,340 --> 00:50:51,460
with N of t equals 23 because
that's obviously the simplest

823
00:50:51,460 --> 00:50:53,830
number to start out with.

824
00:50:53,830 --> 00:50:54,950
Obviously not.

825
00:50:54,950 --> 00:50:58,950
We start out with N of t equals
1 because that clearly

826
00:50:58,950 --> 00:51:01,130
is a simpler thing
to start with.

827
00:51:01,130 --> 00:51:03,500
So here's the situation
we have.

828
00:51:03,500 --> 00:51:06,200
We're assuming that there's
a one arrival

829
00:51:06,200 --> 00:51:07,670
up until this time.

830
00:51:07,670 --> 00:51:11,380
That one arrival has to
occur somewhere in

831
00:51:11,380 --> 00:51:12,660
this interval here.

832
00:51:12,660 --> 00:51:14,300
We don't know where it is.

833
00:51:14,300 --> 00:51:19,040
So we say, well, let's suppose
that it's in the interval s1

834
00:51:19,040 --> 00:51:24,460
to s1 plus delta and try to find
the probability that it

835
00:51:24,460 --> 00:51:26,840
actually is in that interval.

836
00:51:26,840 --> 00:51:30,070
So we start out with something
that looks ugly.

837
00:51:30,070 --> 00:51:32,770
But we'll see it isn't as
ugly as it looks like.

838
00:51:32,770 --> 00:51:38,297
We'll look at the conditional
density of s1 given that N of

839
00:51:38,297 --> 00:51:40,480
t is equal to 1.

840
00:51:40,480 --> 00:51:41,710
That's what this is.

841
00:51:41,710 --> 00:51:44,110
This is a conditional
density here.

842
00:51:44,110 --> 00:51:47,340
And we say, well, how do we find
the conditional density?

843
00:51:47,340 --> 00:51:49,710
This a one dimensional
density.

844
00:51:49,710 --> 00:51:51,680
So what we're going to do is
we're going to look at the

845
00:51:51,680 --> 00:51:56,540
probability that there's one
arrival in this tiny interval.

846
00:51:56,540 --> 00:51:58,840
And then we're going to divide
by delta and that gives us the

847
00:51:58,840 --> 00:52:01,700
density, if everything
is well defined.

848
00:52:01,700 --> 00:52:06,070
So the density is going to be
the limit as delta goes to

849
00:52:06,070 --> 00:52:11,200
zero, which is the way you
always define densities, of

850
00:52:11,200 --> 00:52:19,300
the probability that there
is no arrivals in here.

851
00:52:19,300 --> 00:52:21,560
That there's one arrival here.

852
00:52:21,560 --> 00:52:23,910
And that there's no
arrivals here.

853
00:52:23,910 --> 00:52:27,840
So probability zero
arrivals here.

854
00:52:27,840 --> 00:52:30,950
Probability of one
arrival here.

855
00:52:30,950 --> 00:52:33,560
And probability of zero
arrivals here.

856
00:52:33,560 --> 00:52:35,580
We want to find that
probability.

857
00:52:35,580 --> 00:52:38,740
We want to divide for the
conditioning by the

858
00:52:38,740 --> 00:52:44,240
probability that there was one
arrival in the whole interval.

859
00:52:44,240 --> 00:52:47,360
This delta here is because we
want to go to the limit and

860
00:52:47,360 --> 00:52:49,540
get a density.

861
00:52:49,540 --> 00:52:52,350
OK, when we look at this and
we calculate this, e to the

862
00:52:52,350 --> 00:52:56,100
minus lambda s1 is
what this is.

863
00:52:56,100 --> 00:52:59,780
Lambda delta times e to
the minus lambda delta

864
00:52:59,780 --> 00:53:02,200
is what this is.

865
00:53:02,200 --> 00:53:06,340
And e to the minus lambda
t minus s1 minus delta

866
00:53:06,340 --> 00:53:07,290
is what this is.

867
00:53:07,290 --> 00:53:14,665
Because this last interval is
t minus s1 minus delta.

868
00:53:18,070 --> 00:53:22,795
Now, the remarkable thing is
when I add, I multiply e to

869
00:53:22,795 --> 00:53:28,560
the minus lambda s1 times e to
the minus lambda delta times e

870
00:53:28,560 --> 00:53:31,940
to the minus lambda t minus
s1 minus delta.

871
00:53:31,940 --> 00:53:33,480
There's an s1 here.

872
00:53:33,480 --> 00:53:34,830
An s1 one here.

873
00:53:34,830 --> 00:53:36,190
There's a delta here.

874
00:53:36,190 --> 00:53:37,650
And a delta here.

875
00:53:37,650 --> 00:53:41,050
This whole thing is e to
the minus lambda delta.

876
00:53:41,050 --> 00:53:44,440
And it cancels out with this e
to the minus lambda delta.

877
00:53:44,440 --> 00:53:47,150
That's going to happen
no matter how many

878
00:53:47,150 --> 00:53:48,390
arrivals I put in here.

879
00:53:48,390 --> 00:53:51,740
So that's an interesting
thing to observe.

880
00:53:51,740 --> 00:53:57,150
So what I wind up with is lambda
delta up here, lambda

881
00:53:57,150 --> 00:54:00,330
delta t here, and
my probability

882
00:54:00,330 --> 00:54:04,050
density is 1 over t.

883
00:54:04,050 --> 00:54:07,530
And now we look at that and we
say, my God, how strange.

884
00:54:07,530 --> 00:54:08,930
And then we look at it
and we say, oh, of

885
00:54:08,930 --> 00:54:09,970
course, it's obvious.

886
00:54:09,970 --> 00:54:10,360
Yes?

887
00:54:10,360 --> 00:54:14,792
AUDIENCE: Why do you use N s1
and then the N tilde for the

888
00:54:14,792 --> 00:54:18,700
other two in the first
part in the limit?

889
00:54:18,700 --> 00:54:18,980
PROFESSOR: Here?

890
00:54:18,980 --> 00:54:21,590
AUDIENCE: You have p and s1 and
it's e N tilde on the top.

891
00:54:21,590 --> 00:54:23,920
Why do you use the N
and the N tilde?

892
00:54:27,615 --> 00:54:30,090
PROFESSOR: N of t is not
shown in the figure.

893
00:54:30,090 --> 00:54:35,220
N of t just says there is one
arrival up until time t, which

894
00:54:35,220 --> 00:54:36,980
is what lets me draw
the figure.

895
00:54:36,980 --> 00:54:38,852
AUDIENCE: I meant in
the limit as delta

896
00:54:38,852 --> 00:54:41,200
approaches zero, Pn, Professor.

897
00:54:41,200 --> 00:54:44,943
You used Pn and then Pn tilde
for the other two.

898
00:54:44,943 --> 00:54:46,899
Does that signify anything?

899
00:54:51,300 --> 00:54:56,560
PROFESSOR: OK, this term is the
probability that we have

900
00:54:56,560 --> 00:54:59,110
no arrivals between
zero and s1.

901
00:55:03,130 --> 00:55:08,440
This N tilde means number of
arrivals in an interval, which

902
00:55:08,440 --> 00:55:11,550
starts at s1 and goes
to s1 plus delta.

903
00:55:11,550 --> 00:55:15,240
So this is the probability we
have one arrival somewhere in

904
00:55:15,240 --> 00:55:16,590
this interval.

905
00:55:16,590 --> 00:55:20,420
And this term here is the
probability that we have no

906
00:55:20,420 --> 00:55:27,360
arrivals in s1 plus
delta up until t.

907
00:55:27,360 --> 00:55:31,130
And when we write all of those
things out everything cancels

908
00:55:31,130 --> 00:55:33,370
out except the 1 over t.

909
00:55:33,370 --> 00:55:35,320
And then we look at that
and we say, of course.

910
00:55:43,960 --> 00:55:47,940
If you look at the small
increment definition of a

911
00:55:47,940 --> 00:55:54,950
Poisson process, it says that
arrivals are equally likely at

912
00:55:54,950 --> 00:55:57,250
any point along here.

913
00:55:57,250 --> 00:56:00,290
If I condition on the fact that
there's been one arrival

914
00:56:00,290 --> 00:56:03,500
in this whole interval, and
they're equally likely to

915
00:56:03,500 --> 00:56:07,750
occur anywhere along here, the
only sensible thing that can

916
00:56:07,750 --> 00:56:12,320
happen is that the probability
density for where that arrival

917
00:56:12,320 --> 00:56:16,350
happens is uniform over
this whole interval.

918
00:56:20,966 --> 00:56:23,780
Now, the important point
is that this

919
00:56:23,780 --> 00:56:26,000
doesn't depend on s1.

920
00:56:26,000 --> 00:56:28,260
And it doesn't depend
on lambda.

921
00:56:28,260 --> 00:56:31,030
That's a little surprising
also.

922
00:56:31,030 --> 00:56:33,580
I mean, we have this Poisson
process where arrivals are

923
00:56:33,580 --> 00:56:38,340
dependent on lambda, but you
see, as soon as you say N of t

924
00:56:38,340 --> 00:56:43,410
equals 1, you sort
of wash that out.

925
00:56:43,410 --> 00:56:46,300
Because you're already
conditioning on the fact that

926
00:56:46,300 --> 00:56:49,280
there was only one arrival
by this time.

927
00:56:49,280 --> 00:56:52,710
So you have this nice
result here.

928
00:56:52,710 --> 00:56:58,200
But a result which looks
a little suspicious.

929
00:56:58,200 --> 00:57:00,560
Well, we were successful
with that.

930
00:57:00,560 --> 00:57:04,160
Let's go on to N
of t equals 2.

931
00:57:04,160 --> 00:57:07,370
We'll get the general
picture with this.

932
00:57:07,370 --> 00:57:17,290
We want to look at the
probability density of

933
00:57:17,290 --> 00:57:19,830
arrivals at s1.

934
00:57:19,830 --> 00:57:23,360
This s1 and an arrival here,
given that there were only two

935
00:57:23,360 --> 00:57:26,840
arrivals overall.

936
00:57:26,840 --> 00:57:32,930
And again, we take the limit
as delta goes to zero.

937
00:57:32,930 --> 00:57:36,420
And we take a little delta
interval here, a little delta

938
00:57:36,420 --> 00:57:37,240
interval here.

939
00:57:37,240 --> 00:57:41,600
So we're finding the probability
that there are two

940
00:57:41,600 --> 00:57:46,280
arrivals, one between s1 and
s1 plus delta and the other

941
00:57:46,280 --> 00:57:50,700
one between s2 and
s2 plus delta.

942
00:57:50,700 --> 00:57:55,025
So that probability should be
proportional to delta squared.

943
00:57:55,025 --> 00:57:58,080
And we're dividing by delta
squared to find the joint

944
00:57:58,080 --> 00:58:01,220
probability density.

945
00:58:01,220 --> 00:58:04,440
You don't believe that, go back
and look at how joint

946
00:58:04,440 --> 00:58:07,360
probability densities
are the defined.

947
00:58:07,360 --> 00:58:10,480
I mean, you have to define
them somehow.

948
00:58:10,480 --> 00:58:15,620
So you have to look at the
probability in a small area.

949
00:58:15,620 --> 00:58:18,380
And then shrink the
area to zero.

950
00:58:18,380 --> 00:58:22,920
And the area of the area
is delta squared.

951
00:58:22,920 --> 00:58:25,040
And you have to divide
by delta squared as

952
00:58:25,040 --> 00:58:26,490
you go to the limit.

953
00:58:26,490 --> 00:58:29,800
OK, so, we do the same thing
that we did before.

954
00:58:29,800 --> 00:58:34,090
The probability of no arrivals
in here is e to the

955
00:58:34,090 --> 00:58:35,430
minus lambda s1.

956
00:58:35,430 --> 00:58:40,710
In other words, we are taking
the unconditional probability

957
00:58:40,710 --> 00:58:44,450
and then we're dividing by the
conditioning probability.

958
00:58:44,450 --> 00:58:47,170
So the probability that there's
nothing arriving in

959
00:58:47,170 --> 00:58:50,170
here is that.

960
00:58:50,170 --> 00:58:53,890
The probability there's one
arrival in here is that.

961
00:58:53,890 --> 00:58:56,330
Probability that there's
nothing in

962
00:58:56,330 --> 00:58:58,720
this interval is this.

963
00:58:58,720 --> 00:59:02,600
Probability there's one in this
interval is lambda delta

964
00:59:02,600 --> 00:59:04,750
times e to the minus
lambda delta.

965
00:59:04,750 --> 00:59:09,350
And finally, this last term
here, we have to divide by

966
00:59:09,350 --> 00:59:11,470
delta squared to go
to a density.

967
00:59:11,470 --> 00:59:15,200
We have to divide by the
probability that there were

968
00:59:15,200 --> 00:59:18,620
actually just two
arrivals here.

969
00:59:18,620 --> 00:59:21,490
Well, again, the same
thing happens.

970
00:59:21,490 --> 00:59:26,100
If we take all the exponentials,
lambda s1,

971
00:59:26,100 --> 00:59:34,110
lambda delta, lambda 2
in here, this term

972
00:59:34,110 --> 00:59:36,350
cancels out this term.

973
00:59:36,350 --> 00:59:40,420
This term here cancels
out this term.

974
00:59:40,420 --> 00:59:47,750
And all we're left with is e to
the minus lambda t up here,

975
00:59:47,750 --> 00:59:50,620
e to the minus lambda
t down here.

976
00:59:50,620 --> 00:59:58,590
And because the PMF function for
N of t is e to the minus

977
00:59:58,590 --> 01:00:06,120
lambda t over n factorial, n
factorial for n equals 2 is 2.

978
01:00:06,120 --> 01:00:10,220
We wind up with 2
over t squared.

979
01:00:10,220 --> 01:00:12,670
Now that's a little
bit surprising.

980
01:00:12,670 --> 01:00:14,740
I'm going to show you a picture
in just a little bit,

981
01:00:14,740 --> 01:00:19,510
which makes it clear what
that 2 is doing there.

982
01:00:19,510 --> 01:00:22,470
But let's not worry about it too
much for the time being.

983
01:00:22,470 --> 01:00:25,880
The important thing
here is, again,

984
01:00:25,880 --> 01:00:28,580
it's independent lambda.

985
01:00:28,580 --> 01:00:34,020
And it's a independent
of s1 and s2.

986
01:00:34,020 --> 01:00:37,390
Namely, given that there are two
arrivals in here doesn't

987
01:00:37,390 --> 01:00:40,150
make any difference
where they are.

988
01:00:40,150 --> 01:00:45,060
This is, in some sense, uniform
over s1 and s2.

989
01:00:45,060 --> 01:00:49,250
And I have to be more careful
about what uniform means here.

990
01:00:49,250 --> 01:00:52,320
But we'll do that in
just a little bit.

991
01:00:52,320 --> 01:00:55,170
Now we can do the same thing
for N of t equals N, for

992
01:00:55,170 --> 01:00:59,690
arbitrary N. And when I put
in an arbitrary number of

993
01:00:59,690 --> 01:01:04,720
arrivals here I still get this
property that when I take e to

994
01:01:04,720 --> 01:01:08,420
the minus lambda of this region,
e to the minus lambda

995
01:01:08,420 --> 01:01:15,250
times this region, this, this,
this, this, this, this and so

996
01:01:15,250 --> 01:01:19,070
forth, when I get all done, it's
e to the minus lambda t.

997
01:01:19,070 --> 01:01:23,890
And it cancels out with
this term down here.

998
01:01:23,890 --> 01:01:25,440
We've done that down here.

999
01:01:29,150 --> 01:01:35,850
And what you come up with is
this joint density is equal to

1000
01:01:35,850 --> 01:01:42,740
N factorial over t to the N. So
again, it's the same story.

1001
01:01:42,740 --> 01:01:44,080
It's uniform.

1002
01:01:44,080 --> 01:01:48,230
It doesn't depend on where the
s's are, except for the fact

1003
01:01:48,230 --> 01:01:52,080
that s1 is less than
s2 is less than s3.

1004
01:01:52,080 --> 01:01:53,950
We have to sort that out.

1005
01:01:53,950 --> 01:01:58,520
It has this peculiar N factorial
here that doesn't

1006
01:01:58,520 --> 01:02:00,080
depend on lambda, as I said.

1007
01:02:00,080 --> 01:02:03,120
It does depend N in this way.

1008
01:02:03,120 --> 01:02:07,410
That's a uniform N dimensional
probability density over the

1009
01:02:07,410 --> 01:02:11,980
volume t to the N over N
factorial corresponding to the

1010
01:02:11,980 --> 01:02:17,060
constraint region zero less than
s1 less than blah, blah,

1011
01:02:17,060 --> 01:02:24,150
blah, less than s
sub N. Now that

1012
01:02:24,150 --> 01:02:27,410
region is a little peculiar.

1013
01:02:27,410 --> 01:02:30,310
This N factorial is
a little peculiar.

1014
01:02:30,310 --> 01:02:33,200
The t to the N is
pretty obvious.

1015
01:02:33,200 --> 01:02:38,220
Because when you have a joint
density over t things, each

1016
01:02:38,220 --> 01:02:41,470
bounded between zero and
t, you expect to see

1017
01:02:41,470 --> 01:02:42,720
a t to the N there.

1018
01:02:45,280 --> 01:02:49,930
So if you ask, what's
going on?

1019
01:02:49,930 --> 01:02:53,780
In a very simple minded way, the
question is how did this

1020
01:02:53,780 --> 01:02:59,050
derivation know that the volume
of s1 to sn over zero

1021
01:02:59,050 --> 01:03:04,360
less than s1, less than sn, less
than s2, is N factorial

1022
01:03:04,360 --> 01:03:05,930
over t to the N?

1023
01:03:05,930 --> 01:03:08,950
I have a uniform probability
density over

1024
01:03:08,950 --> 01:03:11,320
some strange volume.

1025
01:03:11,320 --> 01:03:13,790
And it's a strange volume
which satisfies this

1026
01:03:13,790 --> 01:03:15,360
constraint.

1027
01:03:15,360 --> 01:03:20,030
How do I know that this
is N factorial here?

1028
01:03:20,030 --> 01:03:21,750
Very confusing at first.

1029
01:03:21,750 --> 01:03:27,050
We saw the same question when
we derived the airline

1030
01:03:27,050 --> 01:03:28,810
density, remember?

1031
01:03:28,810 --> 01:03:31,600
There was an N factorial
there.

1032
01:03:31,600 --> 01:03:35,520
And that N factorial when
we derived the Poisson

1033
01:03:35,520 --> 01:03:39,850
distribution led to an
N factorial there.

1034
01:03:39,850 --> 01:03:43,370
And, in fact, the fact that
we were using the Poisson

1035
01:03:43,370 --> 01:03:48,540
distribution there is why that
N factorial appears here.

1036
01:03:48,540 --> 01:03:52,370
It was because we stuck that
in when we were doing the

1037
01:03:52,370 --> 01:03:54,790
derivation of the Poisson PMF.

1038
01:03:54,790 --> 01:03:56,900
But it still seems a
little mystifying.

1039
01:03:56,900 --> 01:03:58,800
So I want to try to resolve
that mystery.

1040
01:04:04,020 --> 01:04:07,090
And to resolve it
let's look at a

1041
01:04:07,090 --> 01:04:10,730
supposedly unrelated problem.

1042
01:04:10,730 --> 01:04:17,810
And the unrelated problem is the
following, suppose U1 up

1043
01:04:17,810 --> 01:04:22,790
to U sub n are n IID random
variables, independent and

1044
01:04:22,790 --> 01:04:26,900
identically distributed random
variables, each uniform

1045
01:04:26,900 --> 01:04:29,320
over zero to t.

1046
01:04:29,320 --> 01:04:32,430
Well, now, the probability
density for those n random

1047
01:04:32,430 --> 01:04:34,600
variables is very,
very simple.

1048
01:04:34,600 --> 01:04:38,380
Because each one of them is
uniform over one to t.

1049
01:04:38,380 --> 01:04:43,850
The joint density is
1 over t to the n.

1050
01:04:43,850 --> 01:04:45,440
OK, no problem there.

1051
01:04:45,440 --> 01:04:46,760
They're each independent.

1052
01:04:46,760 --> 01:04:49,440
So the probability densities
multiply.

1053
01:04:49,440 --> 01:04:53,400
Each one has a probability
density 1 over t.

1054
01:04:53,400 --> 01:04:56,230
No matter where it is because
it's uniform.

1055
01:04:56,230 --> 01:04:57,930
So you multiply them together.

1056
01:04:57,930 --> 01:05:02,050
And no matter where you are in
this n dimensional cube, the

1057
01:05:02,050 --> 01:05:07,090
probability density is
1 over t to the n.

1058
01:05:07,090 --> 01:05:11,380
The next thing I want
to do is define--

1059
01:05:11,380 --> 01:05:15,380
now these are supposedly not the
same random variables as

1060
01:05:15,380 --> 01:05:17,470
these arrival epochs
we had before.

1061
01:05:17,470 --> 01:05:21,340
These are just defined
in terms of these

1062
01:05:21,340 --> 01:05:22,805
uniform random variables.

1063
01:05:25,570 --> 01:05:30,610
You're seeing a good example
here of why in probability

1064
01:05:30,610 --> 01:05:35,450
theory we start out with axioms,
we derive properties.

1065
01:05:35,450 --> 01:05:39,620
and we don't lock ourselves in
completely to some physical

1066
01:05:39,620 --> 01:05:42,030
problem that we're
trying to solve.

1067
01:05:42,030 --> 01:05:44,280
Because if we had locked
ourselves completely into a

1068
01:05:44,280 --> 01:05:47,320
Poisson process, we wouldn't
even be able to look at this.

1069
01:05:47,320 --> 01:05:50,150
We'd say, ah, that's nothing
to do with our problem.

1070
01:05:50,150 --> 01:05:51,020
But it does.

1071
01:05:51,020 --> 01:05:52,950
So wait.

1072
01:05:52,950 --> 01:05:56,090
We're going to define
s1 as the minimum of

1073
01:05:56,090 --> 01:05:58,470
U1 up to U sub n.

1074
01:05:58,470 --> 01:06:01,770
In other words, these are
the order statistics of

1075
01:06:01,770 --> 01:06:04,300
U1 up to U sub n.

1076
01:06:04,300 --> 01:06:08,760
I choose U1 to be anything
between zero and t, U2 to be

1077
01:06:08,760 --> 01:06:11,170
anything between zero and t.

1078
01:06:11,170 --> 01:06:15,350
And then I choose s1 to be the
smaller of those two and s2 to

1079
01:06:15,350 --> 01:06:16,730
be the larger of those two.

1080
01:06:24,698 --> 01:06:28,070
This is zero to t.

1081
01:06:28,070 --> 01:06:30,770
And here is say, U1.

1082
01:06:35,295 --> 01:06:38,340
And here is U2.

1083
01:06:38,340 --> 01:06:45,650
And then I transfer and call
this s1 and this s2.

1084
01:06:45,650 --> 01:06:54,570
And if I have a U3 out here then
this becomes s2 and this

1085
01:06:54,570 --> 01:06:56,690
becomes s3.

1086
01:06:56,690 --> 01:07:00,220
So the order statistics are
just, you take these

1087
01:07:00,220 --> 01:07:02,145
arrivals and you--

1088
01:07:02,145 --> 01:07:03,670
well, they don't have
to be arrivals.

1089
01:07:03,670 --> 01:07:04,680
They're whatever they are.

1090
01:07:04,680 --> 01:07:06,780
They're just uniform
random variables.

1091
01:07:06,780 --> 01:07:09,650
And I just order them.

1092
01:07:09,650 --> 01:07:17,790
So now my question is, the
region of volume t to the n,

1093
01:07:17,790 --> 01:07:24,470
name I have a volume t to the n,
for the possible values of

1094
01:07:24,470 --> 01:07:26,195
U1 up to U sub n.

1095
01:07:29,940 --> 01:07:35,620
That volume where the density of
Un is non-zero, I can think

1096
01:07:35,620 --> 01:07:39,685
of it as being partitioned
into n factorial regions.

1097
01:07:42,370 --> 01:07:46,470
First region is U1 less
than U2, and so

1098
01:07:46,470 --> 01:07:48,930
forth up to U sub n.

1099
01:07:48,930 --> 01:07:53,410
The second one is U2 is less
than U1, less than U3

1100
01:07:53,410 --> 01:07:54,990
and so forth up.

1101
01:07:54,990 --> 01:08:03,380
And for every permutation of
U1 to U sub n, I get a

1102
01:08:03,380 --> 01:08:05,610
different ordering here.

1103
01:08:05,610 --> 01:08:07,880
I don't even care about
the ordering anymore.

1104
01:08:07,880 --> 01:08:15,110
All I care about is that the
number of permutations of U1

1105
01:08:15,110 --> 01:08:16,520
to U sub n--

1106
01:08:16,520 --> 01:08:18,890
or the integers one
to n, really is

1107
01:08:18,890 --> 01:08:19,840
what I'm talking about.

1108
01:08:19,840 --> 01:08:25,189
Number of permutations of
1 to n is n factorial.

1109
01:08:25,189 --> 01:08:31,720
And for each one of those
permutations, I can talk about

1110
01:08:31,720 --> 01:08:36,689
the region in which the first
permutation is less than the

1111
01:08:36,689 --> 01:08:39,920
second permutation less than
the third and so forth.

1112
01:08:39,920 --> 01:08:41,170
Those are disjoint--

1113
01:08:43,560 --> 01:08:47,075
they have to be the same
volume by symmetry.

1114
01:08:53,310 --> 01:08:56,729
Now how does that symmetry
argument work there?

1115
01:08:56,729 --> 01:09:00,439
I have n integers.

1116
01:09:00,439 --> 01:09:03,380
There's no way to tell the
difference between them,

1117
01:09:03,380 --> 01:09:07,380
except by giving them names.

1118
01:09:07,380 --> 01:09:14,390
And each of them corresponds
to a IID random variable.

1119
01:09:17,279 --> 01:09:25,140
And now, what I'm saying is that
the region which U1 is

1120
01:09:25,140 --> 01:09:31,490
less than U2 has the same area
as the region where U2

1121
01:09:31,490 --> 01:09:32,930
is less than U1.

1122
01:09:32,930 --> 01:09:38,029
And the argument is whatever you
want to do with U1 and U2

1123
01:09:38,029 --> 01:09:42,979
to find the area of that, I
can take your argument and

1124
01:09:42,979 --> 01:09:46,840
every time you've written a U1,
I'll substitute U2 for it.

1125
01:09:46,840 --> 01:09:48,740
And every time you've
written a U2 I'll

1126
01:09:48,740 --> 01:09:51,240
substitute U1 for it.

1127
01:09:51,240 --> 01:09:53,740
And it's the same argument.

1128
01:09:53,740 --> 01:09:58,900
So that if you find the area
using your permutation, I get

1129
01:09:58,900 --> 01:10:02,580
the same area just using
your argument again.

1130
01:10:02,580 --> 01:10:07,060
And I can do this for as many
terms as I want to.

1131
01:10:07,060 --> 01:10:12,790
OK, so this says that from
symmetry each volume is the

1132
01:10:12,790 --> 01:10:16,930
same and thus, each volume
is t to the N

1133
01:10:16,930 --> 01:10:19,860
divided by N factorial.

1134
01:10:19,860 --> 01:10:24,410
Now the region where s sub N is
non-zero, s sub N are these

1135
01:10:24,410 --> 01:10:25,950
ordering statistics.

1136
01:10:25,950 --> 01:10:29,140
These are the terms in which you
wanted less than U2 less

1137
01:10:29,140 --> 01:10:33,250
than U sub n after
I reordered them.

1138
01:10:35,810 --> 01:10:43,040
The volume of the region where
S1 is less than S2 less than

1139
01:10:43,040 --> 01:10:48,000
Sn less than or equal to t, the
region of that volume is

1140
01:10:48,000 --> 01:10:52,110
exactly t to the N divided
by N factorial.

1141
01:10:52,110 --> 01:10:55,630
Because it's one over
N factorial

1142
01:10:55,630 --> 01:10:58,750
of the entire region.

1143
01:10:58,750 --> 01:11:01,310
Now, you just have to think
about that symmetry argument

1144
01:11:01,310 --> 01:11:02,560
for a while.

1145
01:11:04,810 --> 01:11:07,940
The other thing you can do is
just integrate, which is what

1146
01:11:07,940 --> 01:11:13,370
we did when we were deriving
the airline distribution.

1147
01:11:13,370 --> 01:11:14,960
I mean, you just integrate
the thing out.

1148
01:11:14,960 --> 01:11:18,250
And you find out that
that N factorial

1149
01:11:18,250 --> 01:11:21,160
magically appears there.

1150
01:11:21,160 --> 01:11:24,000
I think you get a better idea
of what's going on here.

1151
01:11:31,540 --> 01:11:36,700
Let me go on to this slide,
which will explain a little

1152
01:11:36,700 --> 01:11:40,820
bit what's going on here.

1153
01:11:40,820 --> 01:11:43,740
Suppose I'm looking
at S1 and S2.

1154
01:11:43,740 --> 01:11:52,170
The area where S1 is less
than S2 is this cross

1155
01:11:52,170 --> 01:11:53,420
hatched area here.

1156
01:11:57,350 --> 01:12:00,110
All the points in here
are points where S2

1157
01:12:00,110 --> 01:12:01,830
is bigger than S1.

1158
01:12:01,830 --> 01:12:07,250
As advertised, the region where
S2 is bigger than S1 has

1159
01:12:07,250 --> 01:12:14,470
the same area as the region
where S2 is less than S1.

1160
01:12:14,470 --> 01:12:18,550
And all I'm saying is this same
property happens for N

1161
01:12:18,550 --> 01:12:22,850
equals 3, N equals
4, and so forth.

1162
01:12:22,850 --> 01:12:29,180
And the argument there really
is a symmetry argument.

1163
01:12:29,180 --> 01:12:30,720
OK, I don't want to worry
about what the

1164
01:12:30,720 --> 01:12:32,670
x's are at this point.

1165
01:12:32,670 --> 01:12:40,940
I'm just showing you that a
little tiny area volume there

1166
01:12:40,940 --> 01:12:43,100
has to be part of
this triangle.

1167
01:12:43,100 --> 01:12:46,740
And I get that factor
of 2 because as S2

1168
01:12:46,740 --> 01:12:47,990
is bigger than S1.

1169
01:13:06,540 --> 01:13:11,460
Let me give you an example of
using the ordering statistics,

1170
01:13:11,460 --> 01:13:12,880
which we just talked about.

1171
01:13:15,740 --> 01:13:19,800
Namely, looking at N IID random
variables, and then

1172
01:13:19,800 --> 01:13:22,560
choosing S1 to be the smallest,
S2 to be the next

1173
01:13:22,560 --> 01:13:24,380
smallest and so forth.

1174
01:13:24,380 --> 01:13:29,560
What we've seen is that this S1
to S sub N is identically

1175
01:13:29,560 --> 01:13:34,480
distributed to the problem of
the epochs of arrivals in a

1176
01:13:34,480 --> 01:13:39,420
Poisson process conditional on
N arrivals up until time t.

1177
01:13:39,420 --> 01:13:47,110
So we can use either the uniform
distribution and

1178
01:13:47,110 --> 01:13:50,740
ordering or we can use
Poisson results.

1179
01:13:50,740 --> 01:13:54,840
Either one can give us results
about either process.

1180
01:13:54,840 --> 01:13:58,430
Here what I'm going to do is use
order statistics to find

1181
01:13:58,430 --> 01:14:04,350
the distribution function of S1
conditional on N arrivals

1182
01:14:04,350 --> 01:14:08,410
until time N. Conditional in
the fact that I found 1,000

1183
01:14:08,410 --> 01:14:11,150
arrivals up until time 1,000.

1184
01:14:11,150 --> 01:14:14,380
I want to find the distribution
function of when

1185
01:14:14,380 --> 01:14:16,360
the first arrival occurs.

1186
01:14:16,360 --> 01:14:21,110
Now, if I know that 1,000
arrivals have occurred by time

1187
01:14:21,110 --> 01:14:25,490
1,000, the first arrival is
probably going to be pretty

1188
01:14:25,490 --> 01:14:30,100
close to a Poisson
random variable.

1189
01:14:30,100 --> 01:14:32,550
But I'd like to see that.

1190
01:14:32,550 --> 01:14:36,490
OK, so the probability that
the minimum of these U sub

1191
01:14:36,490 --> 01:14:41,190
i's, these uniform random
variables, is bigger than S1

1192
01:14:41,190 --> 01:14:44,730
is the product of the
probabilities that each one is

1193
01:14:44,730 --> 01:14:45,800
bigger than S1.

1194
01:14:45,800 --> 01:14:49,440
The only way that the minimum
can be bigger than S1 is that

1195
01:14:49,440 --> 01:14:53,140
each of them are
bigger than S1.

1196
01:14:53,140 --> 01:15:00,380
So this is the product from
I equals 1 to N of 1

1197
01:15:00,380 --> 01:15:03,256
minus S1 over t.

1198
01:15:03,256 --> 01:15:06,240
Then we take the product of
things that are all the same.

1199
01:15:06,240 --> 01:15:11,940
It's 1 minus S1 over t to the
N. And then I go into the

1200
01:15:11,940 --> 01:15:16,100
domain of these arrival
epochs.

1201
01:15:16,100 --> 01:15:22,460
And the probability that the
first arrival epoch, the

1202
01:15:22,460 --> 01:15:26,490
complimentary distribution
function of that, is then 1

1203
01:15:26,490 --> 01:15:30,840
minus S1 over t to the N.

1204
01:15:30,840 --> 01:15:35,630
You can also find the expected
value of S1 just by

1205
01:15:35,630 --> 01:15:37,220
integrating this.

1206
01:15:37,220 --> 01:15:39,600
Namely, integrate the
complimentary distribution

1207
01:15:39,600 --> 01:15:42,820
function to get the
expected value.

1208
01:15:42,820 --> 01:15:43,570
And what do you get?

1209
01:15:43,570 --> 01:15:46,780
You get t over N plus 1.

1210
01:15:46,780 --> 01:15:52,080
A little surprising, but
not as surprising

1211
01:15:52,080 --> 01:15:53,730
as you would think.

1212
01:15:53,730 --> 01:16:00,110
We're looking at an interval
from zero to t.

1213
01:16:00,110 --> 01:16:01,975
We have three arrivals there.

1214
01:16:04,840 --> 01:16:09,390
And then we're asking where
does the first one occur?

1215
01:16:09,390 --> 01:16:14,870
And the argument is this
interval, this interval, this

1216
01:16:14,870 --> 01:16:18,270
interval, and this interval
ought to have the same

1217
01:16:18,270 --> 01:16:20,010
distribution.

1218
01:16:20,010 --> 01:16:23,030
I mean, we haven't talked about
this interval yet, but

1219
01:16:23,030 --> 01:16:25,600
it's there.

1220
01:16:25,600 --> 01:16:28,420
I mean, the last of these
arrivals is not a t, it's

1221
01:16:28,420 --> 01:16:30,250
somewhere before t.

1222
01:16:30,250 --> 01:16:33,420
So I have n plus 1
intervals on the

1223
01:16:33,420 --> 01:16:36,170
side of these n arrivals.

1224
01:16:36,170 --> 01:16:40,020
And what this is saying is the
nice symmetric result at the

1225
01:16:40,020 --> 01:16:46,940
expected place where S1 winds
up is at t over n plus one.

1226
01:16:46,940 --> 01:16:50,440
These intervals are in some
sense, at least as far as

1227
01:16:50,440 --> 01:16:53,700
expectation is concerned,
are uniform.

1228
01:16:53,700 --> 01:16:59,000
And we wind up with t over N
plus 1 as the expected value

1229
01:16:59,000 --> 01:17:01,510
of where the first one is.

1230
01:17:01,510 --> 01:17:02,760
That's nice and clean.

1231
01:17:06,030 --> 01:17:08,540
If you look at this problem
and you think in terms of

1232
01:17:08,540 --> 01:17:13,610
little tiny arrivals coming in
any place, and, in fact, you

1233
01:17:13,610 --> 01:17:21,310
look at this U1 up to U
sub N, one arrival.

1234
01:17:21,310 --> 01:17:23,280
I can think of it
as one arrival

1235
01:17:23,280 --> 01:17:26,400
from each of N processes.

1236
01:17:26,400 --> 01:17:30,950
And these are all Poisson
processes.

1237
01:17:30,950 --> 01:17:35,050
They all add up to give a
joint Poisson process, a

1238
01:17:35,050 --> 01:17:38,670
combined Poisson process,
with N arrivals.

1239
01:17:38,670 --> 01:17:44,470
And then, I order the arrivals
to correspond to the overall

1240
01:17:44,470 --> 01:17:45,970
sum process.

1241
01:17:45,970 --> 01:17:48,550
And this is the answer I get.

1242
01:17:48,550 --> 01:17:56,250
So the uniform idea is related
to adding up lots of little

1243
01:17:56,250 --> 01:17:57,500
tiny processes.

1244
01:18:00,540 --> 01:18:04,680
OK, now the next thing I want
to do, and I can see I'm not

1245
01:18:04,680 --> 01:18:08,570
really going to get time to do
it, but I'd like to give you

1246
01:18:08,570 --> 01:18:11,790
the results of it.

1247
01:18:11,790 --> 01:18:23,740
If I look at a box, a little
cube, of area delta squared in

1248
01:18:23,740 --> 01:18:26,870
the S1, S2 space.

1249
01:18:26,870 --> 01:18:32,580
And I look at what that maps
in to, in terms of the

1250
01:18:32,580 --> 01:18:38,370
interarrivals x1 and x2, if you
just map these points into

1251
01:18:38,370 --> 01:18:43,830
this, you see that this square
here is going into a

1252
01:18:43,830 --> 01:18:46,640
parallelepiped.

1253
01:18:46,640 --> 01:18:50,850
And if know a little bit
of linear algebra--

1254
01:18:50,850 --> 01:18:53,190
well, you don't even need to
know any linear algebra to see

1255
01:18:53,190 --> 01:18:57,600
that no matter where you put
this little square anywhere

1256
01:18:57,600 --> 01:19:02,210
around here it's always going to
map into the same kind of a

1257
01:19:02,210 --> 01:19:03,260
parallelepiped.

1258
01:19:03,260 --> 01:19:08,000
So if I take this whole area and
I break it up into a grid,

1259
01:19:08,000 --> 01:19:13,130
this area is going to be broken
up into a grid where

1260
01:19:13,130 --> 01:19:16,080
it's things are going
down this way.

1261
01:19:16,080 --> 01:19:18,300
And this way, we're going to
have a lot of these little

1262
01:19:18,300 --> 01:19:20,320
tiny parallelepipeds.

1263
01:19:20,320 --> 01:19:23,650
If I have uniform probability
density there, I'll have

1264
01:19:23,650 --> 01:19:26,890
uniform probability
density there.

1265
01:19:26,890 --> 01:19:29,890
The place where you need some
linear algebra, if you're

1266
01:19:29,890 --> 01:19:34,520
dealing with n dimensions
instead of 2, is to see that

1267
01:19:34,520 --> 01:19:39,330
the volume of this delta cube
here, for this problem here,

1268
01:19:39,330 --> 01:19:42,110
is going to be identical
to the volume of

1269
01:19:42,110 --> 01:19:45,440
the delta cube here.

1270
01:19:45,440 --> 01:19:48,960
OK, so when you get all done
that, the area in the x1, x2

1271
01:19:48,960 --> 01:19:55,070
space where zero is less than x1
plus x2 is less than t is t

1272
01:19:55,070 --> 01:19:57,020
squared over 2 again.

1273
01:19:57,020 --> 01:20:02,520
And the probability density
of these two interarrival

1274
01:20:02,520 --> 01:20:07,530
intervals is again
2 over t squared.

1275
01:20:07,530 --> 01:20:10,870
It's the same as the arrivals.

1276
01:20:10,870 --> 01:20:12,120
I'm going to skip this slide.

1277
01:20:12,120 --> 01:20:15,200
I want to get the one other
thing I wanted to talk about a

1278
01:20:15,200 --> 01:20:16,980
little bit.

1279
01:20:16,980 --> 01:20:18,230
There's a paradox.

1280
01:20:20,110 --> 01:20:26,000
And the main interarrival time
for Poisson process is one

1281
01:20:26,000 --> 01:20:27,250
over lambda.

1282
01:20:30,370 --> 01:20:34,450
If I come at time t and I start
waiting for the next

1283
01:20:34,450 --> 01:20:40,330
arrival, the mean time I have
to wait is 1 over lambda.

1284
01:20:40,330 --> 01:20:44,450
If I come in and I start looking
back, at the last

1285
01:20:44,450 --> 01:20:50,490
arrival, well, this says it's 1
over lambda times 1 minus e

1286
01:20:50,490 --> 01:20:53,060
to the minus lambda t,
which is what it is.

1287
01:20:53,060 --> 01:20:54,310
We haven't derived that.

1288
01:20:54,310 --> 01:20:56,840
But we know it's something.

1289
01:20:56,840 --> 01:20:59,660
So, what's going on here?

1290
01:20:59,660 --> 01:21:03,780
I come in at time t, the
interval between the last

1291
01:21:03,780 --> 01:21:08,510
arrival and the next arrival,
the mean interval there is

1292
01:21:08,510 --> 01:21:11,650
bigger than the mean interval
between arrivals

1293
01:21:11,650 --> 01:21:14,500
in a Poisson process.

1294
01:21:14,500 --> 01:21:16,715
That is a real paradox.

1295
01:21:16,715 --> 01:21:19,770
And how do I explain that?

1296
01:21:19,770 --> 01:21:23,360
It means that if I'm waiting for
buses, I'm always unlucky.

1297
01:21:23,360 --> 01:21:25,390
And you're unlucky too and
we're all unlucky.

1298
01:21:28,040 --> 01:21:34,200
Because whenever you arrive, the
amount of time between the

1299
01:21:34,200 --> 01:21:39,240
last bus and the next bus is
bigger than it ought to be.

1300
01:21:39,240 --> 01:21:41,960
And how do we explain that?

1301
01:21:41,960 --> 01:21:45,320
Well, this sort of gives
you a really half-assed

1302
01:21:45,320 --> 01:21:47,080
explanation of it.

1303
01:21:47,080 --> 01:21:48,800
It's not very good.

1304
01:21:48,800 --> 01:21:52,960
When we study renewals, we'll
find a good explanation of it.

1305
01:21:57,010 --> 01:22:00,490
First choose a sample path
of a Poisson process.

1306
01:22:00,490 --> 01:22:06,140
I mean, start out with
a sample path.

1307
01:22:06,140 --> 01:22:09,420
And that has arrivals
going along.

1308
01:22:09,420 --> 01:22:12,800
And we want to get rid of zero
because funny things happen

1309
01:22:12,800 --> 01:22:14,340
around zero.

1310
01:22:14,340 --> 01:22:19,910
So we'll start at one eon
and stop at n eons.

1311
01:22:19,910 --> 01:22:22,670
And we'll look over
that interval.

1312
01:22:22,670 --> 01:22:26,470
And then we'll choose t to be
a random variable in this

1313
01:22:26,470 --> 01:22:27,970
interval here.

1314
01:22:27,970 --> 01:22:30,860
OK, so, we choose some
t after we've looked

1315
01:22:30,860 --> 01:22:32,840
at the sample path.

1316
01:22:32,840 --> 01:22:37,900
And then we say for this random
value of t, how big is

1317
01:22:37,900 --> 01:22:42,370
the interval between the
most recent arrival

1318
01:22:42,370 --> 01:22:44,710
and the next arrival.

1319
01:22:44,710 --> 01:22:50,200
And what you see is that since
I have these arrivals of a

1320
01:22:50,200 --> 01:22:57,890
Poisson process laid out here,
the large intervals take up

1321
01:22:57,890 --> 01:23:00,820
proportionally more area than
the little intervals.

1322
01:23:00,820 --> 01:23:14,450
If I have a bunch of little tiny
intervals and some big

1323
01:23:14,450 --> 01:23:18,810
intervals and then I choose a
random t along here, I'm much

1324
01:23:18,810 --> 01:23:22,870
more likely to choose a t in
here, proportionally, than I

1325
01:23:22,870 --> 01:23:26,820
am to choose a t in here.

1326
01:23:26,820 --> 01:23:30,970
This mean time between the
arrivals is the average of

1327
01:23:30,970 --> 01:23:36,450
this, this, this,
this, and this.

1328
01:23:36,450 --> 01:23:40,660
And what I'm looking at here
is I picked some arbitrary

1329
01:23:40,660 --> 01:23:42,320
point in time.

1330
01:23:42,320 --> 01:23:45,910
These arbitrary points in time
are likely to lie in very

1331
01:23:45,910 --> 01:23:47,882
large intervals.

1332
01:23:47,882 --> 01:23:48,794
Yes?

1333
01:23:48,794 --> 01:23:51,165
AUDIENCE: Can you please explain
again what is exactly

1334
01:23:51,165 --> 01:23:52,240
is the paradox?

1335
01:23:52,240 --> 01:23:56,080
PROFESSOR: The paradox is that
the mean time between arrival

1336
01:23:56,080 --> 01:23:58,290
is one over lambda.

1337
01:23:58,290 --> 01:24:03,170
But if I arrive to look for a
bus and the buses are Poisson,

1338
01:24:03,170 --> 01:24:09,300
I arrive in an interval whose
mean length is larger than 1

1339
01:24:09,300 --> 01:24:10,550
over lambda.

1340
01:24:14,870 --> 01:24:16,390
And that seems strange to me.

1341
01:24:19,650 --> 01:24:22,350
Well, I think I'll stop here.

1342
01:24:22,350 --> 01:24:26,650
And maybe we will spend just
a little time sorting

1343
01:24:26,650 --> 01:24:27,900
this out next time.