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PROFESSOR: OK, so let's
get started.
00:00:26.680 --> 00:00:30.530
We're going to essentially
finish up on
00:00:30.530 --> 00:00:33.600
Poisson processes today.
00:00:33.600 --> 00:00:38.960
And today, we have the part of
it that's really the fun part.
00:00:38.960 --> 00:00:44.870
What we had until now was the
dry stuff, defining everything
00:00:44.870 --> 00:00:46.150
and so forth.
00:00:46.150 --> 00:00:50.180
As I said before, Poisson
processes are these perfect
00:00:50.180 --> 00:00:54.210
processes where everything that
could be true is true.
00:00:54.210 --> 00:00:58.760
And you have so many different
ways of looking at problems
00:00:58.760 --> 00:01:02.950
that you can solve problems
in a wide variety of ways.
00:01:02.950 --> 00:01:08.900
What you want try to do in the
problem set this week is to
00:01:08.900 --> 00:01:13.650
get out of the mode of starting
to write equations
00:01:13.650 --> 00:01:15.770
before you think about it.
00:01:15.770 --> 00:01:20.060
I mean, write equations, yes,
but while you're doing it,
00:01:20.060 --> 00:01:24.910
think about what the right way
of approaching the problem is.
00:01:24.910 --> 00:01:29.730
As far as I know, every part of
every problem there can be
00:01:29.730 --> 00:01:31.860
solved in one or two lines.
00:01:31.860 --> 00:01:32.800
There's nothing long.
00:01:32.800 --> 00:01:34.360
There's nothing complicated.
00:01:34.360 --> 00:01:38.740
But you have to find exactly
the right way of doing it.
00:01:38.740 --> 00:01:43.010
And that's what you're supposed
to learn because you
00:01:43.010 --> 00:01:48.880
find all sorts of processes in
the world, which are poorly
00:01:48.880 --> 00:01:52.400
modelled as Poisson processes.
00:01:52.400 --> 00:02:00.370
You get a lot of insight about
the real process by looking at
00:02:00.370 --> 00:02:04.370
Poisson processes, but you don't
get the whole story.
00:02:04.370 --> 00:02:08.530
And if you don't understand how
these relationships are
00:02:08.530 --> 00:02:13.000
connected to each other then
you have no hope of getting
00:02:13.000 --> 00:02:18.000
some sense of when Poisson
process theory is telling you
00:02:18.000 --> 00:02:21.580
something and when it isn't.
00:02:21.580 --> 00:02:24.910
That's why engineers are
different than mathematicians.
00:02:24.910 --> 00:02:28.570
Mathematicians live in
this beautiful world.
00:02:28.570 --> 00:02:29.380
And I love it.
00:02:29.380 --> 00:02:30.880
I love to live there.
00:02:30.880 --> 00:02:34.570
Love to go there for vacations
and so on because everything
00:02:34.570 --> 00:02:36.250
is perfectly clean.
00:02:36.250 --> 00:02:39.560
Everything has a
right solution.
00:02:39.560 --> 00:02:41.100
And if it's not a
right solution,
00:02:41.100 --> 00:02:42.545
it's a wrong solution.
00:02:42.545 --> 00:02:47.850
In engineering, everything
is kind of hazy.
00:02:47.850 --> 00:02:50.830
And you get insights about
things when you put a lot of
00:02:50.830 --> 00:02:52.300
insights together.
00:02:52.300 --> 00:02:54.850
You finally make judgments
about things.
00:02:54.850 --> 00:02:57.780
You use all sorts of
models to do this.
00:02:57.780 --> 00:03:03.050
And what you use a course like
this for is to understand what
00:03:03.050 --> 00:03:04.990
all these models are saying.
00:03:04.990 --> 00:03:07.090
And then be able to use them.
00:03:07.090 --> 00:03:11.380
So, the stuff that we'll be
talking about today is stuff
00:03:11.380 --> 00:03:15.090
you will use all the
rest of the term.
00:03:15.090 --> 00:03:18.770
Because everything we do,
surprisingly enough, from now
00:03:18.770 --> 00:03:24.800
on, has some relationships
with Poisson processes.
00:03:24.800 --> 00:03:29.710
It doesn't sound like they do,
but, in fact, they do.
00:03:29.710 --> 00:03:33.110
And this has a lot of the things
that look like tricks,
00:03:33.110 --> 00:03:34.690
but which are more
than tricks.
00:03:34.690 --> 00:03:39.540
They're really what comes from
a basic understanding of
00:03:39.540 --> 00:03:40.850
Poisson processes.
00:03:40.850 --> 00:03:42.660
So first, I'm going to
review a little bit
00:03:42.660 --> 00:03:43.925
what we did last time.
00:03:46.940 --> 00:03:49.710
Poisson process is an
arrival process.
00:03:49.710 --> 00:03:51.800
Remember what an arrival
process is.
00:03:51.800 --> 00:03:57.030
It's just a bunch of arrival
epochs, which have some
00:03:57.030 --> 00:03:59.660
statistics associated
with them.
00:03:59.660 --> 00:04:02.980
And it has IID
exponentially-distributed
00:04:02.980 --> 00:04:04.240
interarrival times.
00:04:04.240 --> 00:04:09.190
So the time between successive
arrivals is independent from
00:04:09.190 --> 00:04:11.700
one arrival to the next.
00:04:11.700 --> 00:04:14.670
And it has this exponential
distribution, which is what
00:04:14.670 --> 00:04:18.512
gives the Poisson process it's
very special characteristic.
00:04:21.620 --> 00:04:24.930
It can be represented by
its arrival epochs.
00:04:24.930 --> 00:04:31.360
These are the things you see
here, S1, S2, S3, and so
00:04:31.360 --> 00:04:32.960
forth, are the arrival epochs.
00:04:32.960 --> 00:04:38.450
If you can specify what the
probability relationship is
00:04:38.450 --> 00:04:42.100
for this set of joint random
variables then you know
00:04:42.100 --> 00:04:45.230
everything there is to know
about a Poisson process.
00:04:45.230 --> 00:04:48.810
If you specify the joint
distribution of these
00:04:48.810 --> 00:04:52.700
interarrival times, and that's
trivial because they're IID
00:04:52.700 --> 00:04:56.045
exponentially-distributed random
variables, then you
00:04:56.045 --> 00:04:58.790
know everything there is to
know about the process.
00:04:58.790 --> 00:05:04.280
And if you can specify N of
t, which is the number of
00:05:04.280 --> 00:05:10.380
arrivals up until time t for
every t then that specifies
00:05:10.380 --> 00:05:12.730
the process completely also.
00:05:12.730 --> 00:05:17.360
So we take the viewpoint here,
I mean, usually we view a
00:05:17.360 --> 00:05:22.130
stochastic process as either a
sequence of random variables
00:05:22.130 --> 00:05:24.940
or a continuum of random
variables.
00:05:24.940 --> 00:05:28.490
Here, we're viewing as this
three ways of looking at the
00:05:28.490 --> 00:05:30.200
same thing.
00:05:30.200 --> 00:05:36.620
So a Poisson process is then
either the sequence of
00:05:36.620 --> 00:05:40.590
interarrival times, the sequence
of arrival epochs,
00:05:40.590 --> 00:05:44.920
or, what we call the counting
process, N of t at each t
00:05:44.920 --> 00:05:46.360
greater than zero.
00:05:46.360 --> 00:05:50.030
The arrival epochs in N of
t are related either
00:05:50.030 --> 00:05:53.180
this way or this way.
00:05:53.180 --> 00:05:55.850
And we talked about that.
00:05:58.800 --> 00:06:01.980
The interarrival times
are memoryless.
00:06:01.980 --> 00:06:05.650
In other words, they satisfy
this relationship here.
00:06:05.650 --> 00:06:08.680
The probability that an
interarrival time Xi is
00:06:08.680 --> 00:06:12.970
greater than t plus x, for any
t and any x, which are
00:06:12.970 --> 00:06:15.850
positive, given that it's
greater than t.
00:06:15.850 --> 00:06:18.730
In other words, given that
you've already wasted time t
00:06:18.730 --> 00:06:23.400
waiting, the probability that
the time from here until the
00:06:23.400 --> 00:06:27.876
actual occurrence occurs
is again exponential.
00:06:32.870 --> 00:06:35.340
This is stated in conditional
form.
00:06:35.340 --> 00:06:41.330
We stated it before in just
joint probability form.
00:06:41.330 --> 00:06:45.380
OK, which says, that if you wait
for a while and nothing
00:06:45.380 --> 00:06:49.180
has happened, you just
keep on waiting.
00:06:49.180 --> 00:06:51.080
You're right where
you started.
00:06:51.080 --> 00:06:52.240
You haven't lost anything.
00:06:52.240 --> 00:06:54.990
You haven't gained anything.
00:06:54.990 --> 00:06:59.100
And we said that other renewal
processes, which are IID
00:06:59.100 --> 00:07:05.790
interarrival random variables,
you can have these heavy
00:07:05.790 --> 00:07:11.090
tailed distributions where if
nothing happens after while
00:07:11.090 --> 00:07:14.700
then you start to really feel
badly because you know
00:07:14.700 --> 00:07:18.290
nothing's going to happen
for an awful lot longer.
00:07:18.290 --> 00:07:21.420
Heavy tailed distribution's best
example is when you're
00:07:21.420 --> 00:07:26.130
trying to catch an airplane and
they say, it's going to be
00:07:26.130 --> 00:07:28.180
10 minutes late.
00:07:28.180 --> 00:07:31.780
That's the worst heavy tailed
distribution there is.
00:07:31.780 --> 00:07:33.300
And it drives you crazy.
00:07:33.300 --> 00:07:36.680
Because I've never caught a
plane that was supposed to be
00:07:36.680 --> 00:07:41.380
10 minutes late that wasn't
at least an hour late.
00:07:41.380 --> 00:07:46.280
And often, it got canceled,
which makes it not a random
00:07:46.280 --> 00:07:47.530
variable at all.
00:07:54.770 --> 00:07:57.330
One of the things we're
interested in now, and we
00:07:57.330 --> 00:08:00.800
talked about it a lot last
time, is you pick some
00:08:00.800 --> 00:08:05.170
arbitrary time t, that can
be any time at all.
00:08:05.170 --> 00:08:10.550
And you ask, how long is it from
time t-- t might be when
00:08:10.550 --> 00:08:13.400
you arrive to wait for
a bus or something--
00:08:13.400 --> 00:08:16.780
how long is it until
the next bus comes?
00:08:16.780 --> 00:08:21.410
So Z is the random variable
that goes.
00:08:21.410 --> 00:08:25.370
And you really should put
some indices on this.
00:08:25.370 --> 00:08:32.360
But what it is is the random
variable from t until this
00:08:32.360 --> 00:08:38.070
slow arrival here that's poking
along finally comes in.
00:08:38.070 --> 00:08:44.200
Now, what we found is that the
interval Z, which is the time
00:08:44.200 --> 00:08:51.760
from this arrival back
to t, is exponential.
00:08:51.760 --> 00:08:58.860
And the way we showed that is
to say, let's condition Z on
00:08:58.860 --> 00:09:00.940
anything we want to
condition it on.
00:09:00.940 --> 00:09:03.780
And the things that's important
too condition it on
00:09:03.780 --> 00:09:07.870
is the value of N of t here,
which is N. And once we
00:09:07.870 --> 00:09:12.520
condition it on the fact that N
of t is n, we then condition
00:09:12.520 --> 00:09:18.100
on S sub n, which is the time
that this last arrival came.
00:09:18.100 --> 00:09:29.750
So if we condition Z on N of t
at time t and the time that
00:09:29.750 --> 00:09:33.570
this last arrival came in, which
is S sub two, in this
00:09:33.570 --> 00:09:39.080
case, then Z turns out to be,
again, just the time left to
00:09:39.080 --> 00:09:40.990
wait after we've already
waited for this
00:09:40.990 --> 00:09:41.990
given amount of time.
00:09:41.990 --> 00:09:45.250
We then find out that Z,
conditional on these two
00:09:45.250 --> 00:09:48.370
things that we don't understand
at all, it's just
00:09:48.370 --> 00:09:50.960
exponential, no matter
what they are.
00:09:50.960 --> 00:09:55.210
And since it's exponential no
matter what they are, we don't
00:09:55.210 --> 00:09:57.720
go to the trouble of trying to
figure out what they are.
00:09:57.720 --> 00:10:02.920
We just say, whatever
distribution they have, Z
00:10:02.920 --> 00:10:07.440
itself, unconditionally, is
just the same exponential
00:10:07.440 --> 00:10:08.900
random variable.
00:10:08.900 --> 00:10:12.930
So that was one of the main
things we did last time.
00:10:15.710 --> 00:10:20.860
Next thing we did was we started
to look at any set of
00:10:20.860 --> 00:10:26.310
time, say t1, t2, up to t sub
k, and then we looked at the
00:10:26.310 --> 00:10:27.720
increments.
00:10:27.720 --> 00:10:31.690
How many arrivals occurred
between zero and t1?
00:10:31.690 --> 00:10:35.060
How many arrivals occurred
between t1 and t2?
00:10:35.060 --> 00:10:37.850
How many between t2 and t3?
00:10:37.850 --> 00:10:44.030
So we looked at these Poisson
process increments, a whole
00:10:44.030 --> 00:10:49.780
bunch of random variables, and
we said, these are stationary
00:10:49.780 --> 00:10:54.080
and independent Poisson counting
processes, over their
00:10:54.080 --> 00:10:54.950
given intervals.
00:10:54.950 --> 00:11:07.410
In other words, if you stop the
first process, you stop
00:11:07.410 --> 00:11:10.370
looking at this at time t1.
00:11:10.370 --> 00:11:19.620
Then you look at this
from time t1 to t2.
00:11:19.620 --> 00:11:23.480
You look at this one from
tk minus 1 to tk.
00:11:23.480 --> 00:11:26.500
And you look at the last one,
the next last one, all
00:11:26.500 --> 00:11:27.940
the way up to tk.
00:11:27.940 --> 00:11:30.150
And there should be one
where you look at it
00:11:30.150 --> 00:11:34.530
from tk on to t.
00:11:34.530 --> 00:11:37.810
No, you're only looking
at k of them.
00:11:37.810 --> 00:11:40.180
So these are the things that
you're looking at.
00:11:40.180 --> 00:11:42.260
The statement is that these are
00:11:42.260 --> 00:11:44.430
independent random variables.
00:11:44.430 --> 00:11:48.150
The other statement is they're
stationary, which means, if
00:11:48.150 --> 00:11:51.550
you look at the number of
arrivals in this interval,
00:11:51.550 --> 00:11:56.500
here, it's a function
of t2 minus t1.
00:11:56.500 --> 00:11:59.700
But it's not a function
of t1 alone.
00:11:59.700 --> 00:12:03.260
It's only a function of the
length of the interval.
00:12:03.260 --> 00:12:09.540
The number of arrivals in any
interval of length tk minus tk
00:12:09.540 --> 00:12:16.780
minus 1 is a function only of
the length of that interval
00:12:16.780 --> 00:12:19.020
and not of where the
interval is.
00:12:19.020 --> 00:12:23.540
That's a reasonable way to look
at stationarity, I think.
00:12:23.540 --> 00:12:28.240
How many arrivals come in in a
given area is independent of
00:12:28.240 --> 00:12:29.460
where the area is.
00:12:29.460 --> 00:12:34.020
It depends only on how
long the interval is.
00:12:34.020 --> 00:12:38.970
OK, then we found that the
probability mass function for
00:12:38.970 --> 00:12:43.430
N of t, and now we're just
looking from zero to t because
00:12:43.430 --> 00:12:47.440
we know we get the same thing
over any interval, this
00:12:47.440 --> 00:12:52.670
probability mass function is
this nice function, which
00:12:52.670 --> 00:12:54.950
depends only and the
product lambda t.
00:12:54.950 --> 00:12:59.610
It never depends on lambda
alone or t alone.
00:12:59.610 --> 00:13:04.820
Lambda t to n, e to the minus
lambda t over n factorial.
00:13:04.820 --> 00:13:07.210
What the n factorial
is doing there.
00:13:07.210 --> 00:13:10.760
Well, it came out in
the derivation.
00:13:10.760 --> 00:13:15.740
By the time we finish today you
should have more ideas of
00:13:15.740 --> 00:13:18.120
where that factorial
came from.
00:13:18.120 --> 00:13:21.580
And we'll try to understand
that.
00:13:21.580 --> 00:13:23.800
By the stationary and
independent increment
00:13:23.800 --> 00:13:27.390
property, we know that these two
things are independent, N
00:13:27.390 --> 00:13:32.450
of t1 and the number of
arrivals in t1 to t.
00:13:32.450 --> 00:13:36.160
This is a Poisson
random variable.
00:13:36.160 --> 00:13:38.580
This is a Poisson
random variable.
00:13:38.580 --> 00:13:44.190
We know that the number of
arrivals between zero and t is
00:13:44.190 --> 00:13:46.140
also a Poisson random
variable.
00:13:46.140 --> 00:13:48.300
And what does that tell you?
00:13:48.300 --> 00:13:50.900
It tells you, you don't have
to go through all of this
00:13:50.900 --> 00:13:54.190
discrete convolution stuff.
00:13:54.190 --> 00:13:56.690
You probably should go through
it once just for your own
00:13:56.690 --> 00:13:59.410
edification to see that
this all works.
00:13:59.410 --> 00:14:05.180
But for a very lazy person
like me, who likes using
00:14:05.180 --> 00:14:10.000
arguments like this, I say,
well, these two things are
00:14:10.000 --> 00:14:11.620
independent.
00:14:11.620 --> 00:14:14.100
They are Poisson random
variables.
00:14:14.100 --> 00:14:16.040
Their sum is Poisson.
00:14:16.040 --> 00:14:20.400
And therefore, whenever you have
two independent Poisson
00:14:20.400 --> 00:14:23.390
random variables and you add
them together, what you get is
00:14:23.390 --> 00:14:29.930
a Poisson random variable whose
mean is the sum of the
00:14:29.930 --> 00:14:36.070
means of the two individual
random variables.
00:14:36.070 --> 00:14:38.390
In general, sums of independent
Poisson random
00:14:38.390 --> 00:14:40.880
variables are Poisson with
the means adding.
00:14:44.440 --> 00:14:47.500
Them we went through a couple of
alternate definitions of a
00:14:47.500 --> 00:14:48.750
Poisson process.
00:14:51.130 --> 00:14:55.980
And at this point, just from
what I've said so far, and
00:14:55.980 --> 00:14:59.520
from reading the notes and
understanding what I've said
00:14:59.520 --> 00:15:04.060
so far, it ought to be almost
clear that these alternate
00:15:04.060 --> 00:15:06.190
definitions, which we
talked about last
00:15:06.190 --> 00:15:09.030
time have to be valid.
00:15:09.030 --> 00:15:13.020
If an arrival process has the
stationary and independent
00:15:13.020 --> 00:15:17.590
increment properties and if N
of t has the Poisson PMF for
00:15:17.590 --> 00:15:23.520
given lambda, and all t, then
the process is Poisson.
00:15:23.520 --> 00:15:26.640
Now what is that saying?
00:15:26.640 --> 00:15:30.970
I mean, we've said that if we
can specify all of the random
00:15:30.970 --> 00:15:34.030
variables N of t for
all t, then we've
00:15:34.030 --> 00:15:36.170
specified the process.
00:15:36.170 --> 00:15:38.970
What does it mean to
specify a whole
00:15:38.970 --> 00:15:41.650
bunch of random variables?
00:15:41.650 --> 00:15:44.770
It is not sufficient to find the
distribution function of
00:15:44.770 --> 00:15:48.270
all those random variables.
00:15:48.270 --> 00:15:53.190
And one of the problems in the
homeworks at this time is to
00:15:53.190 --> 00:15:57.000
explicitly show that for
a simpler process,
00:15:57.000 --> 00:15:59.240
the Bernoulli process.
00:15:59.240 --> 00:16:07.080
And to actually construct an
example of where N of t is
00:16:07.080 --> 00:16:10.380
specified everywhere.
00:16:10.380 --> 00:16:13.110
But you don't have the
independence between different
00:16:13.110 --> 00:16:16.500
intervals, and therefore,
you don't
00:16:16.500 --> 00:16:18.310
have a Bernoulli process.
00:16:18.310 --> 00:16:22.980
You just have this nice binomial
formula everywhere.
00:16:22.980 --> 00:16:26.400
But it doesn't really
tell you much.
00:16:26.400 --> 00:16:31.080
OK, so, but here we're adding
on the independent increment
00:16:31.080 --> 00:16:36.650
properties, which says over any
set of intervals the joint
00:16:36.650 --> 00:16:39.840
distribution of how many
arrivals there are here, how
00:16:39.840 --> 00:16:54.080
many here, how many here, how
many here, those joint random
00:16:54.080 --> 00:16:59.600
variables are independent of
each other, which is what the
00:16:59.600 --> 00:17:02.210
independent increment
property says.
00:17:02.210 --> 00:17:07.140
So in fact, this tells you
everything you want to know
00:17:07.140 --> 00:17:11.380
because you now know the
relationship between each one
00:17:11.380 --> 00:17:12.630
of these intervals.
00:17:18.970 --> 00:17:21.420
So we see why the process
is Poisson.
00:17:21.420 --> 00:17:23.230
This one's a little trickier.
00:17:23.230 --> 00:17:26.220
If an arrival process has the
stationary and independent
00:17:26.220 --> 00:17:30.190
increment properties and it
satisfies this incremental
00:17:30.190 --> 00:17:33.670
condition, then the
processes Poisson.
00:17:33.670 --> 00:17:36.870
And the incremental condition
says that if you're looking at
00:17:36.870 --> 00:17:46.040
the number of arrivals in some
interval of size delta, the
00:17:46.040 --> 00:17:51.830
probability that this is equal
to N has the form 1 minus
00:17:51.830 --> 00:17:57.030
lambda delta plus o of delta,
where n equals zero.
00:17:57.030 --> 00:18:00.480
Lambda delta plus o of delta,
for n equals 1.
00:18:00.480 --> 00:18:04.480
o of delta for n greater
than or equal to 2.
00:18:04.480 --> 00:18:08.620
This intuitively is only
supposed to talk about very,
00:18:08.620 --> 00:18:10.170
very small delta.
00:18:10.170 --> 00:18:14.442
So if you take the Poisson
distribution lambda t to the
00:18:14.442 --> 00:18:18.660
N, e to the minus lambda t over
n factorial, and you look
00:18:18.660 --> 00:18:22.380
at what happens when t is
very small, this is
00:18:22.380 --> 00:18:24.330
what it turns into.
00:18:24.330 --> 00:18:28.420
When t is very small the
probability that there are no
00:18:28.420 --> 00:18:32.700
arrivals in this interval of
size delta is very large.
00:18:32.700 --> 00:18:42.490
It's 1 minus lambda delta plus
this extra term that says--
00:18:42.490 --> 00:18:45.130
first point of view whenever
you see an o of delta is to
00:18:45.130 --> 00:18:48.320
say, oh, that's not important.
00:18:48.320 --> 00:18:53.890
And for N equals 1, there's
going to be one arrival with
00:18:53.890 --> 00:18:56.680
some fudge factor, which
is not important.
00:18:56.680 --> 00:18:59.770
And there's going to be two or
more arrivals with some fudge
00:18:59.770 --> 00:19:02.400
factor, which is
not important.
00:19:02.400 --> 00:19:07.830
The next thing we talked about
is that o of delta really is
00:19:07.830 --> 00:19:15.650
defined as any function which
is a function of delta where
00:19:15.650 --> 00:19:18.800
the limit of o of
delta divided by
00:19:18.800 --> 00:19:20.960
delta is equal to 0.
00:19:20.960 --> 00:19:23.660
In other words, it's some
function that goes to zero
00:19:23.660 --> 00:19:25.840
faster than delta does.
00:19:25.840 --> 00:19:29.090
So it's something which
is insignificant with
00:19:29.090 --> 00:19:34.950
relationship to this as
delta gets very small.
00:19:34.950 --> 00:19:38.440
Now, how do you use this kind
of statement to make some
00:19:38.440 --> 00:19:41.430
statement about larger
intervals?
00:19:41.430 --> 00:19:44.650
Well, you're clearly
stuck looking at
00:19:44.650 --> 00:19:46.660
differential equations.
00:19:46.660 --> 00:19:48.720
And the text does that.
00:19:48.720 --> 00:19:52.130
I refuse to talk about
differential equations in
00:19:52.130 --> 00:19:53.780
lecture or anyplace else.
00:19:53.780 --> 00:19:57.000
When I retired I said, I will
no longer talk about
00:19:57.000 --> 00:19:58.250
differential equations
anymore.
00:20:01.310 --> 00:20:04.510
And you know, you don't need to
because you can see what's
00:20:04.510 --> 00:20:07.870
happening here.
00:20:07.870 --> 00:20:11.870
And what you see is happening
is, in fact, what's happening.
00:20:11.870 --> 00:20:15.790
OK, so, that's where we
finished up last time.
00:20:15.790 --> 00:20:20.210
And now, we come to the really
fun stuff where we want to
00:20:20.210 --> 00:20:24.000
combine independent
Poisson processes.
00:20:24.000 --> 00:20:26.970
And then, we want to split
Poisson processes.
00:20:26.970 --> 00:20:29.900
And we want to play all sorts of
games with multiple Poisson
00:20:29.900 --> 00:20:33.690
processes, which looks very
hard, and because of this,
00:20:33.690 --> 00:20:34.780
it's very easy.
00:20:34.780 --> 00:20:35.996
Yes?
00:20:35.996 --> 00:20:38.550
AUDIENCE: The previous
definitions, they are if and
00:20:38.550 --> 00:20:39.960
only statements, right?
00:20:39.960 --> 00:20:40.910
PROFESSOR: They are what?
00:20:40.910 --> 00:20:42.480
AUDIENCE: If and only,
all Poisson
00:20:42.480 --> 00:20:45.070
processes satisfy tho--
00:20:45.070 --> 00:20:46.450
PROFESSOR: Yes.
00:20:46.450 --> 00:20:49.590
OK, in other words, what you're
saying is, if you can
00:20:49.590 --> 00:20:53.510
satisfy those properties, this
is a Poisson process.
00:20:53.510 --> 00:20:56.190
I mean, we've already shown
that a Poisson process
00:20:56.190 --> 00:20:57.440
satisfies those properties.
00:21:03.170 --> 00:21:06.510
As a matter of fact, the way I
put it in the notes is these
00:21:06.510 --> 00:21:09.140
are three alternate definitions
where you could
00:21:09.140 --> 00:21:13.940
start out with any one of them
and derive the whole thing.
00:21:13.940 --> 00:21:16.660
Many people like to start out
with this incremental
00:21:16.660 --> 00:21:21.030
definition because it's
very physical.
00:21:21.030 --> 00:21:24.770
But it makes all the mathematics
much, much harder.
00:21:24.770 --> 00:21:28.930
And so, it's just a question
of what you prefer.
00:21:28.930 --> 00:21:32.280
I like to start out with
something clean, then derive
00:21:32.280 --> 00:21:34.530
things, and then say, does
it make any sense for the
00:21:34.530 --> 00:21:36.920
physical situation?
00:21:36.920 --> 00:21:38.970
And that's what we usually do.
00:21:38.970 --> 00:21:42.340
We don't usually start out with
a physical situation and
00:21:42.340 --> 00:21:48.120
analyze the hell out of it and
say, aha, this is a Poisson
00:21:48.120 --> 00:21:51.100
process because it satisfies
all these properties.
00:21:51.100 --> 00:21:54.470
It never satisfies all
those properties.
00:21:54.470 --> 00:21:57.810
I mean, you say it's a Poisson
process because a Poisson
00:21:57.810 --> 00:22:02.340
process is simple and you can
get some insight from it, not
00:22:02.340 --> 00:22:06.760
because it really is
a Poisson process.
00:22:06.760 --> 00:22:12.890
Let's talk about taking two
independent Poisson processes.
00:22:12.890 --> 00:22:17.340
Just to be a little more
precise, two counting
00:22:17.340 --> 00:22:24.030
processes, N1 of t and N2 of t
are independent if for all t1
00:22:24.030 --> 00:22:29.840
up to t sub n the random
variables N1 of t1 to N1 of tn
00:22:29.840 --> 00:22:35.280
are independent of N1 of
t1 up to N2 of tn.
00:22:35.280 --> 00:22:38.590
Why don't I just say that for
all t they're independent?
00:22:38.590 --> 00:22:40.930
Because I don't even know
what that means.
00:22:40.930 --> 00:22:45.010
I mean, we've never defined
independence for an infinite
00:22:45.010 --> 00:22:47.100
number of things.
00:22:47.100 --> 00:22:51.760
So all we can do is say, for all
finite sets, we have this
00:22:51.760 --> 00:22:53.460
independence.
00:22:53.460 --> 00:22:58.210
Now, give you a short
pop quiz.
00:22:58.210 --> 00:23:03.375
Suppose that instead of doing it
this way, I say, for all t1
00:23:03.375 --> 00:23:09.390
to tn the random variables
N1 of t1 to N1 of tn are
00:23:09.390 --> 00:23:17.325
independent of for all tao1,
tao2, tao3, tao4, N2 of tao1
00:23:17.325 --> 00:23:20.300
up to N2 of tao sub n?
00:23:20.300 --> 00:23:22.250
That sounds much more
general, doesn't it?
00:23:22.250 --> 00:23:26.600
Because it's saying that I can
count one process at one set
00:23:26.600 --> 00:23:31.610
of times and the other process
at another set of times.
00:23:31.610 --> 00:23:34.490
Now, why isn't it any more
general to do it that way?
00:23:41.700 --> 00:23:45.470
Well, it's an unfair pop quiz
because if you can answer that
00:23:45.470 --> 00:23:49.520
question in the short one
sentence answer that you'd be
00:23:49.520 --> 00:23:54.570
willing to give in a class like
this, I would just give
00:23:54.570 --> 00:23:59.250
you A plus and tell
you go away.
00:23:59.250 --> 00:24:02.800
And you already know all the
things you should know.
00:24:02.800 --> 00:24:06.870
The argument is the following,
if you order these different
00:24:06.870 --> 00:24:14.760
times, first tao1 of tao1 is
less than t1 then t1, if
00:24:14.760 --> 00:24:18.560
that's the next one, and you
order them all along.
00:24:18.560 --> 00:24:22.440
And then you apply this
definition to
00:24:22.440 --> 00:24:25.060
that ordered set.
00:24:25.060 --> 00:24:33.830
t1, tao1, t2, t3, t4, tao2, t5,
tao3, and so forth, and
00:24:33.830 --> 00:24:37.200
you apply this definition, and
then you get the other
00:24:37.200 --> 00:24:38.320
definition.
00:24:38.320 --> 00:24:40.690
So one is not more general
than the other.
00:24:44.940 --> 00:24:49.980
The theorem then, is that if
N1 of t and N2 of t are
00:24:49.980 --> 00:24:53.140
independent Poisson processes,
one of them has
00:24:53.140 --> 00:24:54.740
a rate lambda 1.
00:24:54.740 --> 00:24:57.120
One of them has a
rate lambda 2.
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
00:25:02.530 --> 00:25:08.230
means for every t this random
variable N of t is the sum of
00:25:08.230 --> 00:25:11.190
the random variable N1
of t plus the random
00:25:11.190 --> 00:25:12.960
variable N2 of 2.
00:25:12.960 --> 00:25:14.530
This is true for all.
00:25:14.530 --> 00:25:18.775
This is, definition, for all t
greater than 0, then the sum N
00:25:18.775 --> 00:25:24.920
of t is a Poisson process of
rate lambda equals lambda 1
00:25:24.920 --> 00:25:26.600
plus lambda 2.
00:25:26.600 --> 00:25:30.680
Looks almost obvious,
doesn't it?
00:25:30.680 --> 00:25:33.380
I said that today there
was lots of fun stuff.
00:25:33.380 --> 00:25:37.360
There's also a little bit of
ugly stuff and this is one of
00:25:37.360 --> 00:25:42.160
those half obvious things
that's ugly.
00:25:42.160 --> 00:25:44.370
And I'm not going to waste
a lot of time on it.
00:25:44.370 --> 00:25:47.290
You can read all about the
details in the notes.
00:25:47.290 --> 00:25:49.610
But I will spend a little
bit of time on it
00:25:49.610 --> 00:25:52.850
because it's important.
00:25:52.850 --> 00:25:57.620
The idea is that if you look at
any small increment, t to t
00:25:57.620 --> 00:26:02.757
plus delta, the number of
arrivals in the interval t to
00:26:02.757 --> 00:26:07.280
t plus delta is equal to the
number of arrivals in the
00:26:07.280 --> 00:26:10.970
interval t to t plus delta from
the first process plus
00:26:10.970 --> 00:26:12.870
that in the second process.
00:26:12.870 --> 00:26:17.230
So the probability that there's
one arrival in this
00:26:17.230 --> 00:26:21.600
combined process is the
probability that there's one
00:26:21.600 --> 00:26:27.940
arrival in the first process and
no arrivals in the second
00:26:27.940 --> 00:26:31.890
process, or that there's no
arrivals in the first process
00:26:31.890 --> 00:26:34.270
and one arrival in the
second process.
00:26:34.270 --> 00:26:37.315
That's just a very simple
case of convolution.
00:26:39.870 --> 00:26:42.020
Those are the only ways
you can get one in
00:26:42.020 --> 00:26:44.270
the combined process.
00:26:44.270 --> 00:26:51.310
This term here is delta
times lambda 1--
00:26:51.310 --> 00:26:52.420
too early in the morning.
00:26:52.420 --> 00:26:55.586
I'm confusing my deltas
and lambdas.
00:26:55.586 --> 00:26:57.820
There are too many
of each of them.
00:26:57.820 --> 00:26:59.310
--plus o of delta.
00:26:59.310 --> 00:27:02.680
And this term here, probability
that there's zero
00:27:02.680 --> 00:27:05.620
for the second process
is 1 minus delta
00:27:05.620 --> 00:27:08.300
lambda 2 plus o of delta.
00:27:08.300 --> 00:27:11.370
And then this term is just
the opposite term
00:27:11.370 --> 00:27:13.010
corresponding to this.
00:27:13.010 --> 00:27:15.670
Now I multiply these
terms out.
00:27:15.670 --> 00:27:17.560
And what do I get?
00:27:17.560 --> 00:27:22.750
Well, this 1 here combines
with a delta lambda 1.
00:27:22.750 --> 00:27:28.660
Then there's a delta lambda 2
times delta lambda 1, which is
00:27:28.660 --> 00:27:29.860
delta squared.
00:27:29.860 --> 00:27:32.800
It's a delta squared term,
so that's really
00:27:32.800 --> 00:27:34.980
an o of delta term.
00:27:34.980 --> 00:27:39.540
It's negligible as delta
goes to zero.
00:27:39.540 --> 00:27:41.490
So we forget about that.
00:27:41.490 --> 00:27:45.710
There's an o of delta times 1,
that's still an o of delta.
00:27:45.710 --> 00:27:48.380
There's an o of delta times
a delta lambda.
00:27:48.380 --> 00:27:53.610
And that's sort of an o of delta
squared if you wish.
00:27:53.610 --> 00:27:54.950
But it's still an o of delta.
00:27:54.950 --> 00:27:58.140
It goes to zero as
delta gets large.
00:27:58.140 --> 00:28:00.160
And goes to zero faster
than delta.
00:28:00.160 --> 00:28:02.770
What we're trying to do is to
find the terms that are
00:28:02.770 --> 00:28:06.480
significant in terms of delta.
00:28:06.480 --> 00:28:10.130
Namely, when delta gets very
small, I want to find things
00:28:10.130 --> 00:28:13.990
that are at least proportional
to delta and not of lower
00:28:13.990 --> 00:28:15.450
order than delta.
00:28:15.450 --> 00:28:18.790
So when I get done with all of
that, this is delta times
00:28:18.790 --> 00:28:22.500
lambda 1 plus lambda
2 plus o of delta.
00:28:22.500 --> 00:28:27.040
That's the incremental property
that we want a
00:28:27.040 --> 00:28:28.370
Poisson process to have.
00:28:28.370 --> 00:28:30.895
So it has that incremental
property.
00:28:35.190 --> 00:28:42.150
And those are the same sort of
argument if you want to for
00:28:42.150 --> 00:28:47.160
the number of arrivals in
t, that t plus delta.
00:28:47.160 --> 00:28:48.820
Maybe a picture of
this would help.
00:28:52.320 --> 00:28:56.030
Pictures always help
in the morning.
00:28:56.030 --> 00:28:57.280
Here we have two processes.
00:29:00.220 --> 00:29:06.815
We're looking at some interval
of time, t to t plus delta.
00:29:09.530 --> 00:29:14.160
t to plus delta.
00:29:14.160 --> 00:29:15.790
And we might have
an arrival here.
00:29:18.580 --> 00:29:22.120
We might have an arrival
here, an arrival
00:29:22.120 --> 00:29:25.460
here, an arrival here.
00:29:25.460 --> 00:29:29.040
Well, the probability of an
arrival here and an arrival
00:29:29.040 --> 00:29:32.990
here is something of order
delta squared.
00:29:32.990 --> 00:29:36.050
So that's something we ignore.
00:29:36.050 --> 00:29:38.760
So it says, we might have
an arrival here
00:29:38.760 --> 00:29:41.810
and no arrival here.
00:29:41.810 --> 00:29:43.910
This is a lambda 1.
00:29:43.910 --> 00:29:45.960
This is lambda 2 here.
00:29:48.840 --> 00:29:51.700
We might have an arrival here
and none here, or we might
00:29:51.700 --> 00:29:54.470
have an arrival here
and none there.
00:29:54.470 --> 00:29:58.080
Two arrivals is just too
unlikely to worry about, so we
00:29:58.080 --> 00:30:00.784
forget about it at least
for the time being.
00:30:05.160 --> 00:30:09.190
Now, after going through this
incremental argument, if you
00:30:09.190 --> 00:30:13.050
go back and say, let's forget
about all these o of deltas
00:30:13.050 --> 00:30:15.670
because they're very confusing,
let's just do the
00:30:15.670 --> 00:30:18.370
convolution knowing
what N of t is in
00:30:18.370 --> 00:30:20.110
both of these intervals.
00:30:20.110 --> 00:30:24.150
If you do that, it's much easier
to find out that the
00:30:24.150 --> 00:30:29.940
number of arrivals in this
sum of intervals.
00:30:29.940 --> 00:30:33.760
Number arrivals here, plus the
number of arrivals here, this
00:30:33.760 --> 00:30:34.810
is Poisson.
00:30:34.810 --> 00:30:35.870
This is Poisson.
00:30:35.870 --> 00:30:39.040
You add two Poisson,
you get Poisson.
00:30:39.040 --> 00:30:44.294
The rate of the sum of the
two Poisson is Poisson.
00:30:44.294 --> 00:30:49.290
So it's Poisson with lambda
1 plus lambda 2.
00:30:49.290 --> 00:30:52.870
Now how many of you saw that
and says, why is this idiot
00:30:52.870 --> 00:30:55.650
going through this incremental
argument?
00:30:55.650 --> 00:30:56.930
Anyone?
00:30:56.930 --> 00:30:58.880
I won't be embarrassed.
00:30:58.880 --> 00:30:59.740
I knew it anyway.
00:30:59.740 --> 00:31:00.990
And I did it for a reason.
00:31:04.120 --> 00:31:07.060
But I must confess, when I wrote
the first edition of
00:31:07.060 --> 00:31:09.980
this book I didn't
recognize that.
00:31:09.980 --> 00:31:13.710
So I went through this terribly
tedious argument.
00:31:13.710 --> 00:31:22.630
Anyway, the more important issue
is if you have the sum
00:31:22.630 --> 00:31:25.870
of many, many small independent
arrival
00:31:25.870 --> 00:31:27.810
processes--
00:31:27.810 --> 00:31:32.110
OK, in other words, you
have the internet.
00:31:32.110 --> 00:31:37.680
And a node in the internet is
getting jobs, or messages,
00:31:37.680 --> 00:31:41.920
from all sorts of people, and
all sorts of processes, and
00:31:41.920 --> 00:31:45.400
all sorts of nonsense going to
all sorts of people, all sorts
00:31:45.400 --> 00:31:49.540
of processes, and all sorts
of nonsense and those are
00:31:49.540 --> 00:31:51.360
independent of each
other, not really
00:31:51.360 --> 00:31:53.240
independent of each other.
00:31:53.240 --> 00:32:00.750
But relative to the data rate
that's travelling over this
00:32:00.750 --> 00:32:03.760
internet, each of
those processes
00:32:03.760 --> 00:32:06.190
are very, very small.
00:32:06.190 --> 00:32:11.330
And what happens is the sum of
many, many small independent
00:32:11.330 --> 00:32:15.820
arrival processes tend to be
Poisson even if the small
00:32:15.820 --> 00:32:18.230
processes are not.
00:32:18.230 --> 00:32:21.690
In a sense, the independence
between the processes
00:32:21.690 --> 00:32:25.860
overcomes the dependence between
successive arrivals in
00:32:25.860 --> 00:32:27.660
each process.
00:32:27.660 --> 00:32:29.930
Now, I look at that and I say,
well, it's sort of a
00:32:29.930 --> 00:32:31.900
plausibility argument.
00:32:31.900 --> 00:32:35.230
You look at the argument in
the text, and you say, ah,
00:32:35.230 --> 00:32:37.185
it's sort of a plausibility
argument.
00:32:39.990 --> 00:32:42.290
I mean, proving this statement,
you need to put a
00:32:42.290 --> 00:32:44.080
lot of conditions on it.
00:32:44.080 --> 00:32:46.450
And you need to really go
through an awful lot of work.
00:32:49.060 --> 00:32:53.020
It's like proving the central
limit theorem, but it's
00:32:53.020 --> 00:32:55.510
probably harder than that.
00:32:55.510 --> 00:33:00.150
So, if you read the text,
and say you don't really
00:33:00.150 --> 00:33:04.120
understand the argument there,
I don't understand it either
00:33:04.120 --> 00:33:06.980
because I don't think
it's exactly right.
00:33:06.980 --> 00:33:10.520
And I was just trying to say
something to give some idea of
00:33:10.520 --> 00:33:11.770
why this is plausible.
00:33:14.980 --> 00:33:16.230
It should probably be changed.
00:33:18.830 --> 00:33:23.760
Next we want to talk about
splitting a Poisson process.
00:33:23.760 --> 00:33:26.440
So we start out with a
Poisson process here
00:33:26.440 --> 00:33:29.350
and a t of rate lambda.
00:33:29.350 --> 00:33:30.390
And what happens?
00:33:30.390 --> 00:33:33.690
These arrivals come
to some point.
00:33:33.690 --> 00:33:36.110
And some character is
standing there.
00:33:38.670 --> 00:33:41.720
It's like when you're having
your passport checked when you
00:33:41.720 --> 00:33:44.100
come back to Boston
after being away.
00:33:47.390 --> 00:33:52.630
In some places you press a
button, and if the button come
00:33:52.630 --> 00:33:56.790
up one way, you're sent off to
one line to be interrogated
00:33:56.790 --> 00:33:57.840
and all sorts of junk.
00:33:57.840 --> 00:33:59.490
And it's supposed
to be random.
00:33:59.490 --> 00:34:02.440
I have no idea whether it's
random or not, but it's
00:34:02.440 --> 00:34:04.820
supposed to be random.
00:34:04.820 --> 00:34:06.890
And otherwise, you go
through and you get
00:34:06.890 --> 00:34:08.790
through very quickly.
00:34:08.790 --> 00:34:11.110
So it's the same sort
of thing here.
00:34:11.110 --> 00:34:13.530
You have a bunch of arrivals.
00:34:13.530 --> 00:34:20.310
And each arrival is effectively
randomly shoveled
00:34:20.310 --> 00:34:23.050
this way or shoveled this way.
00:34:23.050 --> 00:34:26.330
With probability p, it's
shoveled this way.
00:34:26.330 --> 00:34:30.360
With probability 1 minus p,
it's shoveled this way.
00:34:30.360 --> 00:34:38.889
So you can characterize this
switch as a Bernoulli process.
00:34:38.889 --> 00:34:42.330
It's a Bernoulli process
because it's random and
00:34:42.330 --> 00:34:48.030
independent from, not
time to time now,
00:34:48.030 --> 00:34:50.150
but arrival to arrival.
00:34:50.150 --> 00:34:53.540
When we first looked at a
Poisson process, we said it's
00:34:53.540 --> 00:34:56.110
a sequence of random
variables.
00:34:56.110 --> 00:34:58.820
Sometimes we look at
in terms of time.
00:34:58.820 --> 00:35:01.680
Time doesn't really make
any difference there.
00:35:01.680 --> 00:35:05.940
It's just a sequence of
IID random variables.
00:35:05.940 --> 00:35:09.290
So you have a sequence of IID
random variables doing this
00:35:09.290 --> 00:35:10.450
switching here.
00:35:10.450 --> 00:35:13.520
You have a Poisson process
coming in.
00:35:13.520 --> 00:35:17.130
And when you look at the
combination of the Poisson
00:35:17.130 --> 00:35:21.990
process and the Bernoulli
process, you get some kind of
00:35:21.990 --> 00:35:24.870
process of things
coming out here.
00:35:24.870 --> 00:35:28.840
And another kind of process
of things coming out here.
00:35:28.840 --> 00:35:35.120
And the theorem says, that when
you combine this Poisson
00:35:35.120 --> 00:35:39.500
process with this independent
Bernoulli process, what you
00:35:39.500 --> 00:35:45.240
get is a Poisson process here
and an independent Poisson
00:35:45.240 --> 00:35:48.040
process here.
00:35:48.040 --> 00:35:51.600
And, of course, you need to know
what the probability is
00:35:51.600 --> 00:35:55.970
of being switched this way and
being switched that way.
00:35:55.970 --> 00:35:59.790
Each new process clearly has a
stationary and independent
00:35:59.790 --> 00:36:01.360
increment property.
00:36:01.360 --> 00:36:04.450
Why is that?
00:36:04.450 --> 00:36:10.010
Well, you look at some increment
of time and this
00:36:10.010 --> 00:36:15.680
process here is independent
from one period of time to
00:36:15.680 --> 00:36:18.650
another period of time.
00:36:18.650 --> 00:36:23.930
The Bernoulli process is just
switching the terms within
00:36:23.930 --> 00:36:27.220
that interval of time, which
is independent of all other
00:36:27.220 --> 00:36:29.060
intervals of time.
00:36:29.060 --> 00:36:31.680
So that when you look at the
combination of the Bernoulli
00:36:31.680 --> 00:36:35.580
process and the Poisson
process, you have the
00:36:35.580 --> 00:36:40.680
stationary and independent
increment property.
00:36:40.680 --> 00:36:44.890
And each satisfies the small
increment property.
00:36:44.890 --> 00:36:48.410
If you look at very
small delta here,
00:36:48.410 --> 00:36:49.960
unless each is Poisson.
00:36:49.960 --> 00:36:52.930
There's a more careful argument
in the notes.
00:36:52.930 --> 00:36:56.150
What I'm trying to do in the
lecture is not to give you
00:36:56.150 --> 00:36:59.580
careful proofs of things, but to
give you some insight into
00:36:59.580 --> 00:37:01.520
why they're true.
00:37:01.520 --> 00:37:04.570
So that you can read
the proof.
00:37:04.570 --> 00:37:07.580
And instead of going through
each line and saying, yes, I
00:37:07.580 --> 00:37:10.410
agree with that, yes, I agree
with that and you finally come
00:37:10.410 --> 00:37:13.780
to the end of the proof
and you think what?
00:37:13.780 --> 00:37:17.020
Which I've done all the time
because I don't really know
00:37:17.020 --> 00:37:19.960
what's going on.
00:37:19.960 --> 00:37:23.680
And you don't learn anything
from that kind of proof.
00:37:23.680 --> 00:37:28.200
If you're surprised when it's
done it means you ought to go
00:37:28.200 --> 00:37:30.270
back and look at it more
carefully because there's
00:37:30.270 --> 00:37:36.540
something there that you didn't
understand while it was
00:37:36.540 --> 00:37:37.790
wandering past you.
00:37:41.230 --> 00:37:44.250
The small increment property
really doesn't make it clear
00:37:44.250 --> 00:37:48.090
that the split processes
are independent.
00:37:48.090 --> 00:37:54.000
And for independence both
processes must sometimes have
00:37:54.000 --> 00:37:56.560
arrivals in the same
small increment.
00:37:56.560 --> 00:37:58.470
And this independence
is hidden in
00:37:58.470 --> 00:38:00.350
those o of delta terms.
00:38:00.350 --> 00:38:03.220
And if you want to resolve that
for yourself, you really
00:38:03.220 --> 00:38:06.440
have to look at the text.
00:38:06.440 --> 00:38:09.380
And you have to do
some work too.
00:38:09.380 --> 00:38:13.030
The nice thing about combining
and splitting is that you
00:38:13.030 --> 00:38:15.350
typically do them together.
00:38:15.350 --> 00:38:19.600
Most of the places where you
use combining and splitting
00:38:19.600 --> 00:38:22.690
you use both of them
repeatedly.
00:38:22.690 --> 00:38:26.400
The typical thing you do is
first, you look at separate
00:38:26.400 --> 00:38:29.000
independent Poisson processes.
00:38:29.000 --> 00:38:32.610
And you take those separate
independent Poisson processes,
00:38:32.610 --> 00:38:37.440
and you say, I want to look at
those as a combined process.
00:38:37.440 --> 00:38:40.760
And after you look at them as
a combined process, you then
00:38:40.760 --> 00:38:42.880
split them again.
00:38:42.880 --> 00:38:46.540
And what you're doing, when you
then split them again, is
00:38:46.540 --> 00:38:50.740
you're saying these two
independent Poisson processes
00:38:50.740 --> 00:38:54.720
that I started with, I can
view them as one Poisson
00:38:54.720 --> 00:38:58.310
process plus a Bernoulli
process.
00:38:58.310 --> 00:39:01.630
And you'd be amazed at how many
problems you can solve by
00:39:01.630 --> 00:39:03.130
doing that.
00:39:03.130 --> 00:39:06.560
You will be amazed when you
do the homework this time.
00:39:06.560 --> 00:39:12.220
If you don't use that property
at least 10 times, you haven't
00:39:12.220 --> 00:39:18.005
really understood what's being
asked for in those problems.
00:39:21.370 --> 00:39:25.060
Let's look at a simple
example.
00:39:25.060 --> 00:39:28.030
Let's look at a last come,
first served queue.
00:39:28.030 --> 00:39:31.980
Last come, first served queues
are rather peculiar.
00:39:31.980 --> 00:39:38.310
They are queues where arrivals
come in, and for some reason
00:39:38.310 --> 00:39:42.580
or other, when a new arrival
comes in that arrival goes
00:39:42.580 --> 00:39:44.930
into the server immediately.
00:39:44.930 --> 00:39:47.010
And the queue gets backed up.
00:39:47.010 --> 00:39:49.795
Whatever was being served
gets backed up.
00:39:49.795 --> 00:39:53.680
And the new arrival gets served,
which is fine for the
00:39:53.680 --> 00:39:57.260
new arrival, unless another
arrival comes
00:39:57.260 --> 00:40:00.410
in before it finishes.
00:40:00.410 --> 00:40:02.500
And then it gets
backed up too.
00:40:02.500 --> 00:40:05.518
So things get backed
up in the queue.
00:40:05.518 --> 00:40:07.380
And those that are lucky
enough to get
00:40:07.380 --> 00:40:08.880
through, get through.
00:40:08.880 --> 00:40:11.400
And those that aren't, don't.
00:40:11.400 --> 00:40:13.950
Anybody have any
idea why that--
00:40:13.950 --> 00:40:17.470
I mean, it sounds like a very
unfair thing to do.
00:40:17.470 --> 00:40:22.570
But, in fact, it's not unfair
because you're not singling
00:40:22.570 --> 00:40:25.270
out any particular group
or anything.
00:40:25.270 --> 00:40:29.120
You're just that's
the rule you use.
00:40:29.120 --> 00:40:32.540
And it applies to everyone
equally, so it's not unfair.
00:40:32.540 --> 00:40:35.880
But why might it make
sense sometimes?
00:40:35.880 --> 00:40:37.122
Yeah?
00:40:37.122 --> 00:40:40.098
AUDIENCE: If you're doing a
first come, first served
00:40:40.098 --> 00:40:43.580
you're going to have a
certain distribution
00:40:43.580 --> 00:40:47.250
for your serve time.
00:40:47.250 --> 00:40:50.948
Whereas, if you do last
come, first served--
00:40:50.948 --> 00:40:51.894
I don't know.
00:40:51.894 --> 00:40:54.505
I'm just trying to think that
there might some situations
00:40:54.505 --> 00:40:57.110
where the distribution of
service times in that is
00:40:57.110 --> 00:41:00.427
favorable overall, even though
some people, every once in a
00:41:00.427 --> 00:41:02.850
while, are feeling screwed.
00:41:02.850 --> 00:41:06.080
PROFESSOR: It's a good try.
00:41:06.080 --> 00:41:07.790
But it's not quite right.
00:41:07.790 --> 00:41:11.250
And, in fact, it's not right for
a Poisson process because
00:41:11.250 --> 00:41:16.700
for a Poisson process the server
is just chunking things
00:41:16.700 --> 00:41:21.490
out at rate mu, no
matter what.
00:41:21.490 --> 00:41:23.980
So it doesn't really
help anyone.
00:41:23.980 --> 00:41:27.450
If you have a heavy tail
distribution, if somebody who
00:41:27.450 --> 00:41:31.180
comes in and requires an
enormous amount of service,
00:41:31.180 --> 00:41:35.340
then you get everybody else
done first because that
00:41:35.340 --> 00:41:40.430
customer with a huge service
requirement keeps getting
00:41:40.430 --> 00:41:44.210
pushed back every time the queue
is empty he gets some
00:41:44.210 --> 00:41:45.220
service again.
00:41:45.220 --> 00:41:47.100
And then other people come in.
00:41:47.100 --> 00:41:48.440
And people with small service
00:41:48.440 --> 00:41:50.200
requirements get served quickly.
00:41:53.700 --> 00:41:55.200
And what that means
is it's not quite
00:41:55.200 --> 00:41:57.550
as crazy as it sounds.
00:41:57.550 --> 00:42:02.450
But the reason we want to look
at it here is because it's a
00:42:02.450 --> 00:42:04.470
nice example of this
combining and
00:42:04.470 --> 00:42:07.430
splitting of Poisson processes.
00:42:07.430 --> 00:42:09.970
So how does that happen?
00:42:09.970 --> 00:42:14.530
Well, you view services
as a Poisson process.
00:42:14.530 --> 00:42:18.640
Namely, we have an exponential
server here, where the time
00:42:18.640 --> 00:42:22.040
for each service is
exponentially distributed.
00:42:22.040 --> 00:42:24.660
Now, if you're awake, you point
out, well, what happens
00:42:24.660 --> 00:42:27.850
when the server has
nothing to do?
00:42:27.850 --> 00:42:32.710
Well, just suppose that there's
some very low priority
00:42:32.710 --> 00:42:36.870
set of jobs that the server
is doing, which also are
00:42:36.870 --> 00:42:40.200
exponentially distributed that
the server goes to when it has
00:42:40.200 --> 00:42:41.510
nothing to do.
00:42:41.510 --> 00:42:47.520
So the server is still doing
things exponentially at rate
00:42:47.520 --> 00:42:51.770
lambda, but if there's nothing
real to do then
00:42:51.770 --> 00:42:53.570
the output is wasted.
00:42:53.570 --> 00:42:55.910
So that's a Poisson process.
00:42:55.910 --> 00:42:59.940
The income is Poisson also.
00:42:59.940 --> 00:43:04.880
So the arrival process, plus the
service process, they're
00:43:04.880 --> 00:43:06.030
independent.
00:43:06.030 --> 00:43:06.980
So they're Poisson.
00:43:06.980 --> 00:43:10.900
And the rate is lambda
plus mu.
00:43:10.900 --> 00:43:16.020
Now, interesting question,
what's the probability that an
00:43:16.020 --> 00:43:20.030
arrival completes service before
being interrupted?
00:43:20.030 --> 00:43:22.120
I'm lucky, I get into
the server.
00:43:22.120 --> 00:43:23.830
I start getting served.
00:43:23.830 --> 00:43:26.950
What's the probability that
I finish before I get
00:43:26.950 --> 00:43:28.200
interrupted?
00:43:33.380 --> 00:43:37.200
Well, I get into service
at a particular time t.
00:43:37.200 --> 00:43:41.640
I look at this combined Poisson
process of arrivals
00:43:41.640 --> 00:43:47.630
and services and if the first
arrival in this combined
00:43:47.630 --> 00:43:53.740
process gets switched to
service, I'm done.
00:43:53.740 --> 00:43:58.460
If gets switched to arrival,
I'm interrupted.
00:43:58.460 --> 00:44:02.180
So the question is, what's the
probability that I get
00:44:02.180 --> 00:44:04.380
switched to--
00:44:08.360 --> 00:44:10.940
I want to find the probability
that I'm interrupted, so
00:44:10.940 --> 00:44:14.240
what's the probability that a
new arrival in the combined
00:44:14.240 --> 00:44:20.480
process gets switched to
arrivals because that's the
00:44:20.480 --> 00:44:22.270
case where I get interrupted.
00:44:22.270 --> 00:44:23.520
And it's just lambda.
00:44:29.470 --> 00:44:32.730
The probability that I get
interrupted is lambda divided
00:44:32.730 --> 00:44:34.290
by lambda plus mu.
00:44:34.290 --> 00:44:39.180
And the probability that I
complete my service is mu over
00:44:39.180 --> 00:44:40.440
lambda plus mu.
00:44:40.440 --> 00:44:44.450
When mu is very large, when the
server is going very, very
00:44:44.450 --> 00:44:47.640
fast, I have a good chance
of finishing before being
00:44:47.640 --> 00:44:48.600
interrupted.
00:44:48.600 --> 00:44:51.120
When it's the other
way, I have a much
00:44:51.120 --> 00:44:54.120
smaller chance of finishing.
00:44:54.120 --> 00:44:57.340
More interesting, and more
difficult case, given that
00:44:57.340 --> 00:45:01.410
you're interrupted, what is the
probability that you have
00:45:01.410 --> 00:45:03.730
no further interruptions?
00:45:03.730 --> 00:45:07.360
In other words, I'm being
served, I get interrupted, so
00:45:07.360 --> 00:45:10.760
now I'm sitting at the
front of the queue.
00:45:10.760 --> 00:45:14.260
Everybody else that came in
before me is in back of me.
00:45:14.260 --> 00:45:16.590
I'm sitting there at the
front of the queue.
00:45:16.590 --> 00:45:21.360
This interrupting customer
is being served.
00:45:21.360 --> 00:45:33.700
What's the probability that I'm
going to finish my service
00:45:33.700 --> 00:45:36.110
before any further interruption
occurs?
00:45:36.110 --> 00:45:39.430
Have to be careful about
how to state this.
00:45:39.430 --> 00:45:42.900
The probability that there is
no further interruption is
00:45:42.900 --> 00:45:48.700
that two services occur before
the next arrival.
00:45:52.580 --> 00:45:56.710
And the probability of that is
mu over lambda plus mu the
00:45:56.710 --> 00:45:57.960
quantity squared.
00:46:01.990 --> 00:46:04.380
Now, whether you agree with
that number or not is
00:46:04.380 --> 00:46:05.050
immaterial.
00:46:05.050 --> 00:46:08.670
The thing I want you to
understand is that this is a
00:46:08.670 --> 00:46:14.140
method which you can use
in quite complicated
00:46:14.140 --> 00:46:15.390
circumstances.
00:46:17.100 --> 00:46:23.380
And it's something that applies
in so many places that
00:46:23.380 --> 00:46:25.820
it's amazing.
00:46:25.820 --> 00:46:28.060
OK, let's talk a little
bit about
00:46:28.060 --> 00:46:31.325
non-homogeneous Poisson processes.
00:46:35.990 --> 00:46:38.960
Maybe the most important
application of this is optical
00:46:38.960 --> 00:46:40.260
transmission.
00:46:40.260 --> 00:46:44.410
There's an optical stream of
photons that's modulated by
00:46:44.410 --> 00:46:46.230
variable power.
00:46:46.230 --> 00:46:49.590
Photon stream is reasonably
modeled as a Poisson process,
00:46:49.590 --> 00:46:51.770
not perfectly modeled
that way.
00:46:51.770 --> 00:46:55.380
But again, what we're doing here
is saying, let's look at
00:46:55.380 --> 00:46:56.540
this model.
00:46:56.540 --> 00:46:59.850
And then, see whether the
consequences of the model
00:46:59.850 --> 00:47:02.010
apply to the physical
situation.
00:47:02.010 --> 00:47:05.720
The modulation converts the
steady photon rate into a
00:47:05.720 --> 00:47:07.370
variable rate.
00:47:07.370 --> 00:47:13.210
So the arrivals are being
served, namely, they're being
00:47:13.210 --> 00:47:16.190
transmitted at some
rate, lambda of
00:47:16.190 --> 00:47:19.570
t, where t is varying.
00:47:19.570 --> 00:47:22.800
I mean, the photons have
nothing to do the
00:47:22.800 --> 00:47:25.230
information at all.
00:47:25.230 --> 00:47:27.350
They're just random photons.
00:47:27.350 --> 00:47:30.640
And the information is
represented by lambda t.
00:47:30.640 --> 00:47:32.500
The question is can
you actually send
00:47:32.500 --> 00:47:34.400
anything this way?
00:47:34.400 --> 00:47:38.710
We model the number of photons
in any interval t and t plus
00:47:38.710 --> 00:47:43.660
delta as a Poisson random
variable whose rate parameter
00:47:43.660 --> 00:47:48.030
over t and t plus delta, or
very small delta, is the
00:47:48.030 --> 00:47:52.210
average photon rate over t and
t plus delta times delta.
00:47:52.210 --> 00:47:55.780
So we go through this small
increment model again.
00:47:55.780 --> 00:47:59.140
But in the small increment
model, we have a lambda t
00:47:59.140 --> 00:48:04.100
instead of a lambda and
everything else is the same.
00:48:04.100 --> 00:48:10.510
And when you carry this
out, what you find--
00:48:10.510 --> 00:48:13.740
and I'm not going to show
this or anything.
00:48:13.740 --> 00:48:16.550
I'm just going to tell it to you
because it's in the notes
00:48:16.550 --> 00:48:19.280
if you want to read
more about it.
00:48:19.280 --> 00:48:24.530
The probability that you have a
given number of arrivals in
00:48:24.530 --> 00:48:30.320
some given interval is a Poisson
random variable again.
00:48:30.320 --> 00:48:33.010
Namely, whether the process
is homogeneous or not
00:48:33.010 --> 00:48:42.320
homogeneous, if the original
photons are Poisson then what
00:48:42.320 --> 00:48:47.340
you wind up with is
this parameter
00:48:47.340 --> 00:48:49.680
of the Poisson process.
00:48:49.680 --> 00:48:56.370
m hat to the N e the minus m
hat divided by n factorial.
00:48:56.370 --> 00:48:58.820
And this parameter
here is what?
00:48:58.820 --> 00:49:02.080
It's the overall arrival
rate in that interval.
00:49:02.080 --> 00:49:05.770
So it's exactly what you'd
expect it to be.
00:49:05.770 --> 00:49:08.810
So what this is saying is
combining and splitting
00:49:08.810 --> 00:49:12.980
non-homogeneous processes
still works like in the
00:49:12.980 --> 00:49:14.680
homogeneous case.
00:49:14.680 --> 00:49:17.630
The independent exponential
in arrivals don't work.
00:49:24.280 --> 00:49:27.280
OK, let me say that
another way.
00:49:27.280 --> 00:49:30.490
When you're looking at
non-homogeneous Poisson
00:49:30.490 --> 00:49:35.310
processes, looking at them in
terms of a Poisson counting
00:49:35.310 --> 00:49:38.240
process is really a
neat thing to do.
00:49:38.240 --> 00:49:41.930
Because all of the Poisson
counting process formulas
00:49:41.930 --> 00:49:43.100
still work.
00:49:43.100 --> 00:49:45.040
And all you have to do when
you want to look at the
00:49:45.040 --> 00:49:48.430
distribution of the number
arrivals in an interval is
00:49:48.430 --> 00:49:50.950
look at the average arrival
rate over that interval.
00:49:50.950 --> 00:49:53.390
And that tells you everything.
00:49:53.390 --> 00:49:56.460
When you look at the
interarrival times and things
00:49:56.460 --> 00:50:00.180
like that they aren't
exponential anymore.
00:50:00.180 --> 00:50:01.430
And none of that works.
00:50:04.790 --> 00:50:09.050
So you get half the pie, but
not the whole thing.
00:50:09.050 --> 00:50:15.100
Now, the other thing, which is
really fun, and which will
00:50:15.100 --> 00:50:22.690
take us a bit of time, is to
study, now it's not a small
00:50:22.690 --> 00:50:25.380
increment but a bigger
increment, of a Poisson
00:50:25.380 --> 00:50:29.900
process when you condition
on N of t.
00:50:29.900 --> 00:50:32.970
In other words, if somebody
tells you how many arrivals
00:50:32.970 --> 00:50:37.590
have occurred by time t, how do
you analyze where those are
00:50:37.590 --> 00:50:41.240
arrivals occur, where the
arrival epochs are,
00:50:41.240 --> 00:50:42.820
and things like that?
00:50:42.820 --> 00:50:47.340
Well, since we're conditioning
on N of t, let's start out
00:50:47.340 --> 00:50:51.460
with N of t equals 23 because
that's obviously the simplest
00:50:51.460 --> 00:50:53.830
number to start out with.
00:50:53.830 --> 00:50:54.950
Obviously not.
00:50:54.950 --> 00:50:58.950
We start out with N of t equals
1 because that clearly
00:50:58.950 --> 00:51:01.130
is a simpler thing
to start with.
00:51:01.130 --> 00:51:03.500
So here's the situation
we have.
00:51:03.500 --> 00:51:06.200
We're assuming that there's
a one arrival
00:51:06.200 --> 00:51:07.670
up until this time.
00:51:07.670 --> 00:51:11.380
That one arrival has to
occur somewhere in
00:51:11.380 --> 00:51:12.660
this interval here.
00:51:12.660 --> 00:51:14.300
We don't know where it is.
00:51:14.300 --> 00:51:19.040
So we say, well, let's suppose
that it's in the interval s1
00:51:19.040 --> 00:51:24.460
to s1 plus delta and try to find
the probability that it
00:51:24.460 --> 00:51:26.840
actually is in that interval.
00:51:26.840 --> 00:51:30.070
So we start out with something
that looks ugly.
00:51:30.070 --> 00:51:32.770
But we'll see it isn't as
ugly as it looks like.
00:51:32.770 --> 00:51:38.297
We'll look at the conditional
density of s1 given that N of
00:51:38.297 --> 00:51:40.480
t is equal to 1.
00:51:40.480 --> 00:51:41.710
That's what this is.
00:51:41.710 --> 00:51:44.110
This is a conditional
density here.
00:51:44.110 --> 00:51:47.340
And we say, well, how do we find
the conditional density?
00:51:47.340 --> 00:51:49.710
This a one dimensional
density.
00:51:49.710 --> 00:51:51.680
So what we're going to do is
we're going to look at the
00:51:51.680 --> 00:51:56.540
probability that there's one
arrival in this tiny interval.
00:51:56.540 --> 00:51:58.840
And then we're going to divide
by delta and that gives us the
00:51:58.840 --> 00:52:01.700
density, if everything
is well defined.
00:52:01.700 --> 00:52:06.070
So the density is going to be
the limit as delta goes to
00:52:06.070 --> 00:52:11.200
zero, which is the way you
always define densities, of
00:52:11.200 --> 00:52:19.300
the probability that there
is no arrivals in here.
00:52:19.300 --> 00:52:21.560
That there's one arrival here.
00:52:21.560 --> 00:52:23.910
And that there's no
arrivals here.
00:52:23.910 --> 00:52:27.840
So probability zero
arrivals here.
00:52:27.840 --> 00:52:30.950
Probability of one
arrival here.
00:52:30.950 --> 00:52:33.560
And probability of zero
arrivals here.
00:52:33.560 --> 00:52:35.580
We want to find that
probability.
00:52:35.580 --> 00:52:38.740
We want to divide for the
conditioning by the
00:52:38.740 --> 00:52:44.240
probability that there was one
arrival in the whole interval.
00:52:44.240 --> 00:52:47.360
This delta here is because we
want to go to the limit and
00:52:47.360 --> 00:52:49.540
get a density.
00:52:49.540 --> 00:52:52.350
OK, when we look at this and
we calculate this, e to the
00:52:52.350 --> 00:52:56.100
minus lambda s1 is
what this is.
00:52:56.100 --> 00:52:59.780
Lambda delta times e to
the minus lambda delta
00:52:59.780 --> 00:53:02.200
is what this is.
00:53:02.200 --> 00:53:06.340
And e to the minus lambda
t minus s1 minus delta
00:53:06.340 --> 00:53:07.290
is what this is.
00:53:07.290 --> 00:53:14.665
Because this last interval is
t minus s1 minus delta.
00:53:18.070 --> 00:53:22.795
Now, the remarkable thing is
when I add, I multiply e to
00:53:22.795 --> 00:53:28.560
the minus lambda s1 times e to
the minus lambda delta times e
00:53:28.560 --> 00:53:31.940
to the minus lambda t minus
s1 minus delta.
00:53:31.940 --> 00:53:33.480
There's an s1 here.
00:53:33.480 --> 00:53:34.830
An s1 one here.
00:53:34.830 --> 00:53:36.190
There's a delta here.
00:53:36.190 --> 00:53:37.650
And a delta here.
00:53:37.650 --> 00:53:41.050
This whole thing is e to
the minus lambda delta.
00:53:41.050 --> 00:53:44.440
And it cancels out with this e
to the minus lambda delta.
00:53:44.440 --> 00:53:47.150
That's going to happen
no matter how many
00:53:47.150 --> 00:53:48.390
arrivals I put in here.
00:53:48.390 --> 00:53:51.740
So that's an interesting
thing to observe.
00:53:51.740 --> 00:53:57.150
So what I wind up with is lambda
delta up here, lambda
00:53:57.150 --> 00:54:00.330
delta t here, and
my probability
00:54:00.330 --> 00:54:04.050
density is 1 over t.
00:54:04.050 --> 00:54:07.530
And now we look at that and we
say, my God, how strange.
00:54:07.530 --> 00:54:08.930
And then we look at it
and we say, oh, of
00:54:08.930 --> 00:54:09.970
course, it's obvious.
00:54:09.970 --> 00:54:10.360
Yes?
00:54:10.360 --> 00:54:14.792
AUDIENCE: Why do you use N s1
and then the N tilde for the
00:54:14.792 --> 00:54:18.700
other two in the first
part in the limit?
00:54:18.700 --> 00:54:18.980
PROFESSOR: Here?
00:54:18.980 --> 00:54:21.590
AUDIENCE: You have p and s1 and
it's e N tilde on the top.
00:54:21.590 --> 00:54:23.920
Why do you use the N
and the N tilde?
00:54:27.615 --> 00:54:30.090
PROFESSOR: N of t is not
shown in the figure.
00:54:30.090 --> 00:54:35.220
N of t just says there is one
arrival up until time t, which
00:54:35.220 --> 00:54:36.980
is what lets me draw
the figure.
00:54:36.980 --> 00:54:38.852
AUDIENCE: I meant in
the limit as delta
00:54:38.852 --> 00:54:41.200
approaches zero, Pn, Professor.
00:54:41.200 --> 00:54:44.943
You used Pn and then Pn tilde
for the other two.
00:54:44.943 --> 00:54:46.899
Does that signify anything?
00:54:51.300 --> 00:54:56.560
PROFESSOR: OK, this term is the
probability that we have
00:54:56.560 --> 00:54:59.110
no arrivals between
zero and s1.
00:55:03.130 --> 00:55:08.440
This N tilde means number of
arrivals in an interval, which
00:55:08.440 --> 00:55:11.550
starts at s1 and goes
to s1 plus delta.
00:55:11.550 --> 00:55:15.240
So this is the probability we
have one arrival somewhere in
00:55:15.240 --> 00:55:16.590
this interval.
00:55:16.590 --> 00:55:20.420
And this term here is the
probability that we have no
00:55:20.420 --> 00:55:27.360
arrivals in s1 plus
delta up until t.
00:55:27.360 --> 00:55:31.130
And when we write all of those
things out everything cancels
00:55:31.130 --> 00:55:33.370
out except the 1 over t.
00:55:33.370 --> 00:55:35.320
And then we look at that
and we say, of course.
00:55:43.960 --> 00:55:47.940
If you look at the small
increment definition of a
00:55:47.940 --> 00:55:54.950
Poisson process, it says that
arrivals are equally likely at
00:55:54.950 --> 00:55:57.250
any point along here.
00:55:57.250 --> 00:56:00.290
If I condition on the fact that
there's been one arrival
00:56:00.290 --> 00:56:03.500
in this whole interval, and
they're equally likely to
00:56:03.500 --> 00:56:07.750
occur anywhere along here, the
only sensible thing that can
00:56:07.750 --> 00:56:12.320
happen is that the probability
density for where that arrival
00:56:12.320 --> 00:56:16.350
happens is uniform over
this whole interval.
00:56:20.966 --> 00:56:23.780
Now, the important point
is that this
00:56:23.780 --> 00:56:26.000
doesn't depend on s1.
00:56:26.000 --> 00:56:28.260
And it doesn't depend
on lambda.
00:56:28.260 --> 00:56:31.030
That's a little surprising
also.
00:56:31.030 --> 00:56:33.580
I mean, we have this Poisson
process where arrivals are
00:56:33.580 --> 00:56:38.340
dependent on lambda, but you
see, as soon as you say N of t
00:56:38.340 --> 00:56:43.410
equals 1, you sort
of wash that out.
00:56:43.410 --> 00:56:46.300
Because you're already
conditioning on the fact that
00:56:46.300 --> 00:56:49.280
there was only one arrival
by this time.
00:56:49.280 --> 00:56:52.710
So you have this nice
result here.
00:56:52.710 --> 00:56:58.200
But a result which looks
a little suspicious.
00:56:58.200 --> 00:57:00.560
Well, we were successful
with that.
00:57:00.560 --> 00:57:04.160
Let's go on to N
of t equals 2.
00:57:04.160 --> 00:57:07.370
We'll get the general
picture with this.
00:57:07.370 --> 00:57:17.290
We want to look at the
probability density of
00:57:17.290 --> 00:57:19.830
arrivals at s1.
00:57:19.830 --> 00:57:23.360
This s1 and an arrival here,
given that there were only two
00:57:23.360 --> 00:57:26.840
arrivals overall.
00:57:26.840 --> 00:57:32.930
And again, we take the limit
as delta goes to zero.
00:57:32.930 --> 00:57:36.420
And we take a little delta
interval here, a little delta
00:57:36.420 --> 00:57:37.240
interval here.
00:57:37.240 --> 00:57:41.600
So we're finding the probability
that there are two
00:57:41.600 --> 00:57:46.280
arrivals, one between s1 and
s1 plus delta and the other
00:57:46.280 --> 00:57:50.700
one between s2 and
s2 plus delta.
00:57:50.700 --> 00:57:55.025
So that probability should be
proportional to delta squared.
00:57:55.025 --> 00:57:58.080
And we're dividing by delta
squared to find the joint
00:57:58.080 --> 00:58:01.220
probability density.
00:58:01.220 --> 00:58:04.440
You don't believe that, go back
and look at how joint
00:58:04.440 --> 00:58:07.360
probability densities
are the defined.
00:58:07.360 --> 00:58:10.480
I mean, you have to define
them somehow.
00:58:10.480 --> 00:58:15.620
So you have to look at the
probability in a small area.
00:58:15.620 --> 00:58:18.380
And then shrink the
area to zero.
00:58:18.380 --> 00:58:22.920
And the area of the area
is delta squared.
00:58:22.920 --> 00:58:25.040
And you have to divide
by delta squared as
00:58:25.040 --> 00:58:26.490
you go to the limit.
00:58:26.490 --> 00:58:29.800
OK, so, we do the same thing
that we did before.
00:58:29.800 --> 00:58:34.090
The probability of no arrivals
in here is e to the
00:58:34.090 --> 00:58:35.430
minus lambda s1.
00:58:35.430 --> 00:58:40.710
In other words, we are taking
the unconditional probability
00:58:40.710 --> 00:58:44.450
and then we're dividing by the
conditioning probability.
00:58:44.450 --> 00:58:47.170
So the probability that there's
nothing arriving in
00:58:47.170 --> 00:58:50.170
here is that.
00:58:50.170 --> 00:58:53.890
The probability there's one
arrival in here is that.
00:58:53.890 --> 00:58:56.330
Probability that there's
nothing in
00:58:56.330 --> 00:58:58.720
this interval is this.
00:58:58.720 --> 00:59:02.600
Probability there's one in this
interval is lambda delta
00:59:02.600 --> 00:59:04.750
times e to the minus
lambda delta.
00:59:04.750 --> 00:59:09.350
And finally, this last term
here, we have to divide by
00:59:09.350 --> 00:59:11.470
delta squared to go
to a density.
00:59:11.470 --> 00:59:15.200
We have to divide by the
probability that there were
00:59:15.200 --> 00:59:18.620
actually just two
arrivals here.
00:59:18.620 --> 00:59:21.490
Well, again, the same
thing happens.
00:59:21.490 --> 00:59:26.100
If we take all the exponentials,
lambda s1,
00:59:26.100 --> 00:59:34.110
lambda delta, lambda 2
in here, this term
00:59:34.110 --> 00:59:36.350
cancels out this term.
00:59:36.350 --> 00:59:40.420
This term here cancels
out this term.
00:59:40.420 --> 00:59:47.750
And all we're left with is e to
the minus lambda t up here,
00:59:47.750 --> 00:59:50.620
e to the minus lambda
t down here.
00:59:50.620 --> 00:59:58.590
And because the PMF function for
N of t is e to the minus
00:59:58.590 --> 01:00:06.120
lambda t over n factorial, n
factorial for n equals 2 is 2.
01:00:06.120 --> 01:00:10.220
We wind up with 2
over t squared.
01:00:10.220 --> 01:00:12.670
Now that's a little
bit surprising.
01:00:12.670 --> 01:00:14.740
I'm going to show you a picture
in just a little bit,
01:00:14.740 --> 01:00:19.510
which makes it clear what
that 2 is doing there.
01:00:19.510 --> 01:00:22.470
But let's not worry about it too
much for the time being.
01:00:22.470 --> 01:00:25.880
The important thing
here is, again,
01:00:25.880 --> 01:00:28.580
it's independent lambda.
01:00:28.580 --> 01:00:34.020
And it's a independent
of s1 and s2.
01:00:34.020 --> 01:00:37.390
Namely, given that there are two
arrivals in here doesn't
01:00:37.390 --> 01:00:40.150
make any difference
where they are.
01:00:40.150 --> 01:00:45.060
This is, in some sense, uniform
over s1 and s2.
01:00:45.060 --> 01:00:49.250
And I have to be more careful
about what uniform means here.
01:00:49.250 --> 01:00:52.320
But we'll do that in
just a little bit.
01:00:52.320 --> 01:00:55.170
Now we can do the same thing
for N of t equals N, for
01:00:55.170 --> 01:00:59.690
arbitrary N. And when I put
in an arbitrary number of
01:00:59.690 --> 01:01:04.720
arrivals here I still get this
property that when I take e to
01:01:04.720 --> 01:01:08.420
the minus lambda of this region,
e to the minus lambda
01:01:08.420 --> 01:01:15.250
times this region, this, this,
this, this, this, this and so
01:01:15.250 --> 01:01:19.070
forth, when I get all done, it's
e to the minus lambda t.
01:01:19.070 --> 01:01:23.890
And it cancels out with
this term down here.
01:01:23.890 --> 01:01:25.440
We've done that down here.
01:01:29.150 --> 01:01:35.850
And what you come up with is
this joint density is equal to
01:01:35.850 --> 01:01:42.740
N factorial over t to the N. So
again, it's the same story.
01:01:42.740 --> 01:01:44.080
It's uniform.
01:01:44.080 --> 01:01:48.230
It doesn't depend on where the
s's are, except for the fact
01:01:48.230 --> 01:01:52.080
that s1 is less than
s2 is less than s3.
01:01:52.080 --> 01:01:53.950
We have to sort that out.
01:01:53.950 --> 01:01:58.520
It has this peculiar N factorial
here that doesn't
01:01:58.520 --> 01:02:00.080
depend on lambda, as I said.
01:02:00.080 --> 01:02:03.120
It does depend N in this way.
01:02:03.120 --> 01:02:07.410
That's a uniform N dimensional
probability density over the
01:02:07.410 --> 01:02:11.980
volume t to the N over N
factorial corresponding to the
01:02:11.980 --> 01:02:17.060
constraint region zero less than
s1 less than blah, blah,
01:02:17.060 --> 01:02:24.150
blah, less than s
sub N. Now that
01:02:24.150 --> 01:02:27.410
region is a little peculiar.
01:02:27.410 --> 01:02:30.310
This N factorial is
a little peculiar.
01:02:30.310 --> 01:02:33.200
The t to the N is
pretty obvious.
01:02:33.200 --> 01:02:38.220
Because when you have a joint
density over t things, each
01:02:38.220 --> 01:02:41.470
bounded between zero and
t, you expect to see
01:02:41.470 --> 01:02:42.720
a t to the N there.
01:02:45.280 --> 01:02:49.930
So if you ask, what's
going on?
01:02:49.930 --> 01:02:53.780
In a very simple minded way, the
question is how did this
01:02:53.780 --> 01:02:59.050
derivation know that the volume
of s1 to sn over zero
01:02:59.050 --> 01:03:04.360
less than s1, less than sn, less
than s2, is N factorial
01:03:04.360 --> 01:03:05.930
over t to the N?
01:03:05.930 --> 01:03:08.950
I have a uniform probability
density over
01:03:08.950 --> 01:03:11.320
some strange volume.
01:03:11.320 --> 01:03:13.790
And it's a strange volume
which satisfies this
01:03:13.790 --> 01:03:15.360
constraint.
01:03:15.360 --> 01:03:20.030
How do I know that this
is N factorial here?
01:03:20.030 --> 01:03:21.750
Very confusing at first.
01:03:21.750 --> 01:03:27.050
We saw the same question when
we derived the airline
01:03:27.050 --> 01:03:28.810
density, remember?
01:03:28.810 --> 01:03:31.600
There was an N factorial
there.
01:03:31.600 --> 01:03:35.520
And that N factorial when
we derived the Poisson
01:03:35.520 --> 01:03:39.850
distribution led to an
N factorial there.
01:03:39.850 --> 01:03:43.370
And, in fact, the fact that
we were using the Poisson
01:03:43.370 --> 01:03:48.540
distribution there is why that
N factorial appears here.
01:03:48.540 --> 01:03:52.370
It was because we stuck that
in when we were doing the
01:03:52.370 --> 01:03:54.790
derivation of the Poisson PMF.
01:03:54.790 --> 01:03:56.900
But it still seems a
little mystifying.
01:03:56.900 --> 01:03:58.800
So I want to try to resolve
that mystery.
01:04:04.020 --> 01:04:07.090
And to resolve it
let's look at a
01:04:07.090 --> 01:04:10.730
supposedly unrelated problem.
01:04:10.730 --> 01:04:17.810
And the unrelated problem is the
following, suppose U1 up
01:04:17.810 --> 01:04:22.790
to U sub n are n IID random
variables, independent and
01:04:22.790 --> 01:04:26.900
identically distributed random
variables, each uniform
01:04:26.900 --> 01:04:29.320
over zero to t.
01:04:29.320 --> 01:04:32.430
Well, now, the probability
density for those n random
01:04:32.430 --> 01:04:34.600
variables is very,
very simple.
01:04:34.600 --> 01:04:38.380
Because each one of them is
uniform over one to t.
01:04:38.380 --> 01:04:43.850
The joint density is
1 over t to the n.
01:04:43.850 --> 01:04:45.440
OK, no problem there.
01:04:45.440 --> 01:04:46.760
They're each independent.
01:04:46.760 --> 01:04:49.440
So the probability densities
multiply.
01:04:49.440 --> 01:04:53.400
Each one has a probability
density 1 over t.
01:04:53.400 --> 01:04:56.230
No matter where it is because
it's uniform.
01:04:56.230 --> 01:04:57.930
So you multiply them together.
01:04:57.930 --> 01:05:02.050
And no matter where you are in
this n dimensional cube, the
01:05:02.050 --> 01:05:07.090
probability density is
1 over t to the n.
01:05:07.090 --> 01:05:11.380
The next thing I want
to do is define--
01:05:11.380 --> 01:05:15.380
now these are supposedly not the
same random variables as
01:05:15.380 --> 01:05:17.470
these arrival epochs
we had before.
01:05:17.470 --> 01:05:21.340
These are just defined
in terms of these
01:05:21.340 --> 01:05:22.805
uniform random variables.
01:05:25.570 --> 01:05:30.610
You're seeing a good example
here of why in probability
01:05:30.610 --> 01:05:35.450
theory we start out with axioms,
we derive properties.
01:05:35.450 --> 01:05:39.620
and we don't lock ourselves in
completely to some physical
01:05:39.620 --> 01:05:42.030
problem that we're
trying to solve.
01:05:42.030 --> 01:05:44.280
Because if we had locked
ourselves completely into a
01:05:44.280 --> 01:05:47.320
Poisson process, we wouldn't
even be able to look at this.
01:05:47.320 --> 01:05:50.150
We'd say, ah, that's nothing
to do with our problem.
01:05:50.150 --> 01:05:51.020
But it does.
01:05:51.020 --> 01:05:52.950
So wait.
01:05:52.950 --> 01:05:56.090
We're going to define
s1 as the minimum of
01:05:56.090 --> 01:05:58.470
U1 up to U sub n.
01:05:58.470 --> 01:06:01.770
In other words, these are
the order statistics of
01:06:01.770 --> 01:06:04.300
U1 up to U sub n.
01:06:04.300 --> 01:06:08.760
I choose U1 to be anything
between zero and t, U2 to be
01:06:08.760 --> 01:06:11.170
anything between zero and t.
01:06:11.170 --> 01:06:15.350
And then I choose s1 to be the
smaller of those two and s2 to
01:06:15.350 --> 01:06:16.730
be the larger of those two.
01:06:24.698 --> 01:06:28.070
This is zero to t.
01:06:28.070 --> 01:06:30.770
And here is say, U1.
01:06:35.295 --> 01:06:38.340
And here is U2.
01:06:38.340 --> 01:06:45.650
And then I transfer and call
this s1 and this s2.
01:06:45.650 --> 01:06:54.570
And if I have a U3 out here then
this becomes s2 and this
01:06:54.570 --> 01:06:56.690
becomes s3.
01:06:56.690 --> 01:07:00.220
So the order statistics are
just, you take these
01:07:00.220 --> 01:07:02.145
arrivals and you--
01:07:02.145 --> 01:07:03.670
well, they don't have
to be arrivals.
01:07:03.670 --> 01:07:04.680
They're whatever they are.
01:07:04.680 --> 01:07:06.780
They're just uniform
random variables.
01:07:06.780 --> 01:07:09.650
And I just order them.
01:07:09.650 --> 01:07:17.790
So now my question is, the
region of volume t to the n,
01:07:17.790 --> 01:07:24.470
name I have a volume t to the n,
for the possible values of
01:07:24.470 --> 01:07:26.195
U1 up to U sub n.
01:07:29.940 --> 01:07:35.620
That volume where the density of
Un is non-zero, I can think
01:07:35.620 --> 01:07:39.685
of it as being partitioned
into n factorial regions.
01:07:42.370 --> 01:07:46.470
First region is U1 less
than U2, and so
01:07:46.470 --> 01:07:48.930
forth up to U sub n.
01:07:48.930 --> 01:07:53.410
The second one is U2 is less
than U1, less than U3
01:07:53.410 --> 01:07:54.990
and so forth up.
01:07:54.990 --> 01:08:03.380
And for every permutation of
U1 to U sub n, I get a
01:08:03.380 --> 01:08:05.610
different ordering here.
01:08:05.610 --> 01:08:07.880
I don't even care about
the ordering anymore.
01:08:07.880 --> 01:08:15.110
All I care about is that the
number of permutations of U1
01:08:15.110 --> 01:08:16.520
to U sub n--
01:08:16.520 --> 01:08:18.890
or the integers one
to n, really is
01:08:18.890 --> 01:08:19.840
what I'm talking about.
01:08:19.840 --> 01:08:25.189
Number of permutations of
1 to n is n factorial.
01:08:25.189 --> 01:08:31.720
And for each one of those
permutations, I can talk about
01:08:31.720 --> 01:08:36.689
the region in which the first
permutation is less than the
01:08:36.689 --> 01:08:39.920
second permutation less than
the third and so forth.
01:08:39.920 --> 01:08:41.170
Those are disjoint--
01:08:43.560 --> 01:08:47.075
they have to be the same
volume by symmetry.
01:08:53.310 --> 01:08:56.729
Now how does that symmetry
argument work there?
01:08:56.729 --> 01:09:00.439
I have n integers.
01:09:00.439 --> 01:09:03.380
There's no way to tell the
difference between them,
01:09:03.380 --> 01:09:07.380
except by giving them names.
01:09:07.380 --> 01:09:14.390
And each of them corresponds
to a IID random variable.
01:09:17.279 --> 01:09:25.140
And now, what I'm saying is that
the region which U1 is
01:09:25.140 --> 01:09:31.490
less than U2 has the same area
as the region where U2
01:09:31.490 --> 01:09:32.930
is less than U1.
01:09:32.930 --> 01:09:38.029
And the argument is whatever you
want to do with U1 and U2
01:09:38.029 --> 01:09:42.979
to find the area of that, I
can take your argument and
01:09:42.979 --> 01:09:46.840
every time you've written a U1,
I'll substitute U2 for it.
01:09:46.840 --> 01:09:48.740
And every time you've
written a U2 I'll
01:09:48.740 --> 01:09:51.240
substitute U1 for it.
01:09:51.240 --> 01:09:53.740
And it's the same argument.
01:09:53.740 --> 01:09:58.900
So that if you find the area
using your permutation, I get
01:09:58.900 --> 01:10:02.580
the same area just using
your argument again.
01:10:02.580 --> 01:10:07.060
And I can do this for as many
terms as I want to.
01:10:07.060 --> 01:10:12.790
OK, so this says that from
symmetry each volume is the
01:10:12.790 --> 01:10:16.930
same and thus, each volume
is t to the N
01:10:16.930 --> 01:10:19.860
divided by N factorial.
01:10:19.860 --> 01:10:24.410
Now the region where s sub N is
non-zero, s sub N are these
01:10:24.410 --> 01:10:25.950
ordering statistics.
01:10:25.950 --> 01:10:29.140
These are the terms in which you
wanted less than U2 less
01:10:29.140 --> 01:10:33.250
than U sub n after
I reordered them.
01:10:35.810 --> 01:10:43.040
The volume of the region where
S1 is less than S2 less than
01:10:43.040 --> 01:10:48.000
Sn less than or equal to t, the
region of that volume is
01:10:48.000 --> 01:10:52.110
exactly t to the N divided
by N factorial.
01:10:52.110 --> 01:10:55.630
Because it's one over
N factorial
01:10:55.630 --> 01:10:58.750
of the entire region.
01:10:58.750 --> 01:11:01.310
Now, you just have to think
about that symmetry argument
01:11:01.310 --> 01:11:02.560
for a while.
01:11:04.810 --> 01:11:07.940
The other thing you can do is
just integrate, which is what
01:11:07.940 --> 01:11:13.370
we did when we were deriving
the airline distribution.
01:11:13.370 --> 01:11:14.960
I mean, you just integrate
the thing out.
01:11:14.960 --> 01:11:18.250
And you find out that
that N factorial
01:11:18.250 --> 01:11:21.160
magically appears there.
01:11:21.160 --> 01:11:24.000
I think you get a better idea
of what's going on here.
01:11:31.540 --> 01:11:36.700
Let me go on to this slide,
which will explain a little
01:11:36.700 --> 01:11:40.820
bit what's going on here.
01:11:40.820 --> 01:11:43.740
Suppose I'm looking
at S1 and S2.
01:11:43.740 --> 01:11:52.170
The area where S1 is less
than S2 is this cross
01:11:52.170 --> 01:11:53.420
hatched area here.
01:11:57.350 --> 01:12:00.110
All the points in here
are points where S2
01:12:00.110 --> 01:12:01.830
is bigger than S1.
01:12:01.830 --> 01:12:07.250
As advertised, the region where
S2 is bigger than S1 has
01:12:07.250 --> 01:12:14.470
the same area as the region
where S2 is less than S1.
01:12:14.470 --> 01:12:18.550
And all I'm saying is this same
property happens for N
01:12:18.550 --> 01:12:22.850
equals 3, N equals
4, and so forth.
01:12:22.850 --> 01:12:29.180
And the argument there really
is a symmetry argument.
01:12:29.180 --> 01:12:30.720
OK, I don't want to worry
about what the
01:12:30.720 --> 01:12:32.670
x's are at this point.
01:12:32.670 --> 01:12:40.940
I'm just showing you that a
little tiny area volume there
01:12:40.940 --> 01:12:43.100
has to be part of
this triangle.
01:12:43.100 --> 01:12:46.740
And I get that factor
of 2 because as S2
01:12:46.740 --> 01:12:47.990
is bigger than S1.
01:13:06.540 --> 01:13:11.460
Let me give you an example of
using the ordering statistics,
01:13:11.460 --> 01:13:12.880
which we just talked about.
01:13:15.740 --> 01:13:19.800
Namely, looking at N IID random
variables, and then
01:13:19.800 --> 01:13:22.560
choosing S1 to be the smallest,
S2 to be the next
01:13:22.560 --> 01:13:24.380
smallest and so forth.
01:13:24.380 --> 01:13:29.560
What we've seen is that this S1
to S sub N is identically
01:13:29.560 --> 01:13:34.480
distributed to the problem of
the epochs of arrivals in a
01:13:34.480 --> 01:13:39.420
Poisson process conditional on
N arrivals up until time t.
01:13:39.420 --> 01:13:47.110
So we can use either the uniform
distribution and
01:13:47.110 --> 01:13:50.740
ordering or we can use
Poisson results.
01:13:50.740 --> 01:13:54.840
Either one can give us results
about either process.
01:13:54.840 --> 01:13:58.430
Here what I'm going to do is use
order statistics to find
01:13:58.430 --> 01:14:04.350
the distribution function of S1
conditional on N arrivals
01:14:04.350 --> 01:14:08.410
until time N. Conditional in
the fact that I found 1,000
01:14:08.410 --> 01:14:11.150
arrivals up until time 1,000.
01:14:11.150 --> 01:14:14.380
I want to find the distribution
function of when
01:14:14.380 --> 01:14:16.360
the first arrival occurs.
01:14:16.360 --> 01:14:21.110
Now, if I know that 1,000
arrivals have occurred by time
01:14:21.110 --> 01:14:25.490
1,000, the first arrival is
probably going to be pretty
01:14:25.490 --> 01:14:30.100
close to a Poisson
random variable.
01:14:30.100 --> 01:14:32.550
But I'd like to see that.
01:14:32.550 --> 01:14:36.490
OK, so the probability that
the minimum of these U sub
01:14:36.490 --> 01:14:41.190
i's, these uniform random
variables, is bigger than S1
01:14:41.190 --> 01:14:44.730
is the product of the
probabilities that each one is
01:14:44.730 --> 01:14:45.800
bigger than S1.
01:14:45.800 --> 01:14:49.440
The only way that the minimum
can be bigger than S1 is that
01:14:49.440 --> 01:14:53.140
each of them are
bigger than S1.
01:14:53.140 --> 01:15:00.380
So this is the product from
I equals 1 to N of 1
01:15:00.380 --> 01:15:03.256
minus S1 over t.
01:15:03.256 --> 01:15:06.240
Then we take the product of
things that are all the same.
01:15:06.240 --> 01:15:11.940
It's 1 minus S1 over t to the
N. And then I go into the
01:15:11.940 --> 01:15:16.100
domain of these arrival
epochs.
01:15:16.100 --> 01:15:22.460
And the probability that the
first arrival epoch, the
01:15:22.460 --> 01:15:26.490
complimentary distribution
function of that, is then 1
01:15:26.490 --> 01:15:30.840
minus S1 over t to the N.
01:15:30.840 --> 01:15:35.630
You can also find the expected
value of S1 just by
01:15:35.630 --> 01:15:37.220
integrating this.
01:15:37.220 --> 01:15:39.600
Namely, integrate the
complimentary distribution
01:15:39.600 --> 01:15:42.820
function to get the
expected value.
01:15:42.820 --> 01:15:43.570
And what do you get?
01:15:43.570 --> 01:15:46.780
You get t over N plus 1.
01:15:46.780 --> 01:15:52.080
A little surprising, but
not as surprising
01:15:52.080 --> 01:15:53.730
as you would think.
01:15:53.730 --> 01:16:00.110
We're looking at an interval
from zero to t.
01:16:00.110 --> 01:16:01.975
We have three arrivals there.
01:16:04.840 --> 01:16:09.390
And then we're asking where
does the first one occur?
01:16:09.390 --> 01:16:14.870
And the argument is this
interval, this interval, this
01:16:14.870 --> 01:16:18.270
interval, and this interval
ought to have the same
01:16:18.270 --> 01:16:20.010
distribution.
01:16:20.010 --> 01:16:23.030
I mean, we haven't talked about
this interval yet, but
01:16:23.030 --> 01:16:25.600
it's there.
01:16:25.600 --> 01:16:28.420
I mean, the last of these
arrivals is not a t, it's
01:16:28.420 --> 01:16:30.250
somewhere before t.
01:16:30.250 --> 01:16:33.420
So I have n plus 1
intervals on the
01:16:33.420 --> 01:16:36.170
side of these n arrivals.
01:16:36.170 --> 01:16:40.020
And what this is saying is the
nice symmetric result at the
01:16:40.020 --> 01:16:46.940
expected place where S1 winds
up is at t over n plus one.
01:16:46.940 --> 01:16:50.440
These intervals are in some
sense, at least as far as
01:16:50.440 --> 01:16:53.700
expectation is concerned,
are uniform.
01:16:53.700 --> 01:16:59.000
And we wind up with t over N
plus 1 as the expected value
01:16:59.000 --> 01:17:01.510
of where the first one is.
01:17:01.510 --> 01:17:02.760
That's nice and clean.
01:17:06.030 --> 01:17:08.540
If you look at this problem
and you think in terms of
01:17:08.540 --> 01:17:13.610
little tiny arrivals coming in
any place, and, in fact, you
01:17:13.610 --> 01:17:21.310
look at this U1 up to U
sub N, one arrival.
01:17:21.310 --> 01:17:23.280
I can think of it
as one arrival
01:17:23.280 --> 01:17:26.400
from each of N processes.
01:17:26.400 --> 01:17:30.950
And these are all Poisson
processes.
01:17:30.950 --> 01:17:35.050
They all add up to give a
joint Poisson process, a
01:17:35.050 --> 01:17:38.670
combined Poisson process,
with N arrivals.
01:17:38.670 --> 01:17:44.470
And then, I order the arrivals
to correspond to the overall
01:17:44.470 --> 01:17:45.970
sum process.
01:17:45.970 --> 01:17:48.550
And this is the answer I get.
01:17:48.550 --> 01:17:56.250
So the uniform idea is related
to adding up lots of little
01:17:56.250 --> 01:17:57.500
tiny processes.
01:18:00.540 --> 01:18:04.680
OK, now the next thing I want
to do, and I can see I'm not
01:18:04.680 --> 01:18:08.570
really going to get time to do
it, but I'd like to give you
01:18:08.570 --> 01:18:11.790
the results of it.
01:18:11.790 --> 01:18:23.740
If I look at a box, a little
cube, of area delta squared in
01:18:23.740 --> 01:18:26.870
the S1, S2 space.
01:18:26.870 --> 01:18:32.580
And I look at what that maps
in to, in terms of the
01:18:32.580 --> 01:18:38.370
interarrivals x1 and x2, if you
just map these points into
01:18:38.370 --> 01:18:43.830
this, you see that this square
here is going into a
01:18:43.830 --> 01:18:46.640
parallelepiped.
01:18:46.640 --> 01:18:50.850
And if know a little bit
of linear algebra--
01:18:50.850 --> 01:18:53.190
well, you don't even need to
know any linear algebra to see
01:18:53.190 --> 01:18:57.600
that no matter where you put
this little square anywhere
01:18:57.600 --> 01:19:02.210
around here it's always going to
map into the same kind of a
01:19:02.210 --> 01:19:03.260
parallelepiped.
01:19:03.260 --> 01:19:08.000
So if I take this whole area and
I break it up into a grid,
01:19:08.000 --> 01:19:13.130
this area is going to be broken
up into a grid where
01:19:13.130 --> 01:19:16.080
it's things are going
down this way.
01:19:16.080 --> 01:19:18.300
And this way, we're going to
have a lot of these little
01:19:18.300 --> 01:19:20.320
tiny parallelepipeds.
01:19:20.320 --> 01:19:23.650
If I have uniform probability
density there, I'll have
01:19:23.650 --> 01:19:26.890
uniform probability
density there.
01:19:26.890 --> 01:19:29.890
The place where you need some
linear algebra, if you're
01:19:29.890 --> 01:19:34.520
dealing with n dimensions
instead of 2, is to see that
01:19:34.520 --> 01:19:39.330
the volume of this delta cube
here, for this problem here,
01:19:39.330 --> 01:19:42.110
is going to be identical
to the volume of
01:19:42.110 --> 01:19:45.440
the delta cube here.
01:19:45.440 --> 01:19:48.960
OK, so when you get all done
that, the area in the x1, x2
01:19:48.960 --> 01:19:55.070
space where zero is less than x1
plus x2 is less than t is t
01:19:55.070 --> 01:19:57.020
squared over 2 again.
01:19:57.020 --> 01:20:02.520
And the probability density
of these two interarrival
01:20:02.520 --> 01:20:07.530
intervals is again
2 over t squared.
01:20:07.530 --> 01:20:10.870
It's the same as the arrivals.
01:20:10.870 --> 01:20:12.120
I'm going to skip this slide.
01:20:12.120 --> 01:20:15.200
I want to get the one other
thing I wanted to talk about a
01:20:15.200 --> 01:20:16.980
little bit.
01:20:16.980 --> 01:20:18.230
There's a paradox.
01:20:20.110 --> 01:20:26.000
And the main interarrival time
for Poisson process is one
01:20:26.000 --> 01:20:27.250
over lambda.
01:20:30.370 --> 01:20:34.450
If I come at time t and I start
waiting for the next
01:20:34.450 --> 01:20:40.330
arrival, the mean time I have
to wait is 1 over lambda.
01:20:40.330 --> 01:20:44.450
If I come in and I start looking
back, at the last
01:20:44.450 --> 01:20:50.490
arrival, well, this says it's 1
over lambda times 1 minus e
01:20:50.490 --> 01:20:53.060
to the minus lambda t,
which is what it is.
01:20:53.060 --> 01:20:54.310
We haven't derived that.
01:20:54.310 --> 01:20:56.840
But we know it's something.
01:20:56.840 --> 01:20:59.660
So, what's going on here?
01:20:59.660 --> 01:21:03.780
I come in at time t, the
interval between the last
01:21:03.780 --> 01:21:08.510
arrival and the next arrival,
the mean interval there is
01:21:08.510 --> 01:21:11.650
bigger than the mean interval
between arrivals
01:21:11.650 --> 01:21:14.500
in a Poisson process.
01:21:14.500 --> 01:21:16.715
That is a real paradox.
01:21:16.715 --> 01:21:19.770
And how do I explain that?
01:21:19.770 --> 01:21:23.360
It means that if I'm waiting for
buses, I'm always unlucky.
01:21:23.360 --> 01:21:25.390
And you're unlucky too and
we're all unlucky.
01:21:28.040 --> 01:21:34.200
Because whenever you arrive, the
amount of time between the
01:21:34.200 --> 01:21:39.240
last bus and the next bus is
bigger than it ought to be.
01:21:39.240 --> 01:21:41.960
And how do we explain that?
01:21:41.960 --> 01:21:45.320
Well, this sort of gives
you a really half-assed
01:21:45.320 --> 01:21:47.080
explanation of it.
01:21:47.080 --> 01:21:48.800
It's not very good.
01:21:48.800 --> 01:21:52.960
When we study renewals, we'll
find a good explanation of it.
01:21:57.010 --> 01:22:00.490
First choose a sample path
of a Poisson process.
01:22:00.490 --> 01:22:06.140
I mean, start out with
a sample path.
01:22:06.140 --> 01:22:09.420
And that has arrivals
going along.
01:22:09.420 --> 01:22:12.800
And we want to get rid of zero
because funny things happen
01:22:12.800 --> 01:22:14.340
around zero.
01:22:14.340 --> 01:22:19.910
So we'll start at one eon
and stop at n eons.
01:22:19.910 --> 01:22:22.670
And we'll look over
that interval.
01:22:22.670 --> 01:22:26.470
And then we'll choose t to be
a random variable in this
01:22:26.470 --> 01:22:27.970
interval here.
01:22:27.970 --> 01:22:30.860
OK, so, we choose some
t after we've looked
01:22:30.860 --> 01:22:32.840
at the sample path.
01:22:32.840 --> 01:22:37.900
And then we say for this random
value of t, how big is
01:22:37.900 --> 01:22:42.370
the interval between the
most recent arrival
01:22:42.370 --> 01:22:44.710
and the next arrival.
01:22:44.710 --> 01:22:50.200
And what you see is that since
I have these arrivals of a
01:22:50.200 --> 01:22:57.890
Poisson process laid out here,
the large intervals take up
01:22:57.890 --> 01:23:00.820
proportionally more area than
the little intervals.
01:23:00.820 --> 01:23:14.450
If I have a bunch of little tiny
intervals and some big
01:23:14.450 --> 01:23:18.810
intervals and then I choose a
random t along here, I'm much
01:23:18.810 --> 01:23:22.870
more likely to choose a t in
here, proportionally, than I
01:23:22.870 --> 01:23:26.820
am to choose a t in here.
01:23:26.820 --> 01:23:30.970
This mean time between the
arrivals is the average of
01:23:30.970 --> 01:23:36.450
this, this, this,
this, and this.
01:23:36.450 --> 01:23:40.660
And what I'm looking at here
is I picked some arbitrary
01:23:40.660 --> 01:23:42.320
point in time.
01:23:42.320 --> 01:23:45.910
These arbitrary points in time
are likely to lie in very
01:23:45.910 --> 01:23:47.882
large intervals.
01:23:47.882 --> 01:23:48.794
Yes?
01:23:48.794 --> 01:23:51.165
AUDIENCE: Can you please explain
again what is exactly
01:23:51.165 --> 01:23:52.240
is the paradox?
01:23:52.240 --> 01:23:56.080
PROFESSOR: The paradox is that
the mean time between arrival
01:23:56.080 --> 01:23:58.290
is one over lambda.
01:23:58.290 --> 01:24:03.170
But if I arrive to look for a
bus and the buses are Poisson,
01:24:03.170 --> 01:24:09.300
I arrive in an interval whose
mean length is larger than 1
01:24:09.300 --> 01:24:10.550
over lambda.
01:24:14.870 --> 01:24:16.390
And that seems strange to me.
01:24:19.650 --> 01:24:22.350
Well, I think I'll stop here.
01:24:22.350 --> 01:24:26.650
And maybe we will spend just
a little time sorting
01:24:26.650 --> 01:24:27.900
this out next time.