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PROFESSOR: So last time
we started talking
00:00:24.630 --> 00:00:26.420
about random processes.
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A random process is a random
experiment that
00:00:30.710 --> 00:00:32.570
evolves over time.
00:00:32.570 --> 00:00:35.110
And conceptually, it's important
to realize that it's
00:00:35.110 --> 00:00:37.760
a single probabilistic
experiment
00:00:37.760 --> 00:00:39.250
that has many stages.
00:00:39.250 --> 00:00:42.440
Actually, it has an infinite
number of stages.
00:00:42.440 --> 00:00:47.060
And we discussed the simplest
random process there is, the
00:00:47.060 --> 00:00:50.270
Bernoulli process, which is
nothing but the sequence of
00:00:50.270 --> 00:00:51.570
Bernoulli trials--
00:00:51.570 --> 00:00:54.580
an infinite sequence of
Bernoulli trials.
00:00:54.580 --> 00:00:58.250
For example, flipping a
coin over and over.
00:00:58.250 --> 00:01:01.640
Once we understand what's going
on with that process,
00:01:01.640 --> 00:01:05.519
then what we want is to move
into a continuous time version
00:01:05.519 --> 00:01:06.750
of the Bernoulli process.
00:01:06.750 --> 00:01:09.420
And this is what we will call
the Poisson process.
00:01:09.420 --> 00:01:11.970
And for the Poisson process,
we're going to do exactly the
00:01:11.970 --> 00:01:14.300
same things that we did for
the Bernoulli process.
00:01:14.300 --> 00:01:18.210
That is, talk about the number
of arrivals during a given
00:01:18.210 --> 00:01:21.400
time period, and talk also
about the time between
00:01:21.400 --> 00:01:24.160
consecutive arrivals, and
for the distribution of
00:01:24.160 --> 00:01:27.660
inter-arrival times.
00:01:27.660 --> 00:01:30.660
So let's start with a quick
review of what we
00:01:30.660 --> 00:01:32.680
discussed last time.
00:01:32.680 --> 00:01:35.500
First, a note about language.
00:01:35.500 --> 00:01:38.100
If you think of coin tosses,
we then talk
00:01:38.100 --> 00:01:40.660
about heads and tails.
00:01:40.660 --> 00:01:43.430
If you think of these as a
sequence of trials, you can
00:01:43.430 --> 00:01:47.145
talk about successes
and failures.
00:01:47.145 --> 00:01:50.312
The language that we will be
using will be more the
00:01:50.312 --> 00:01:51.800
language of arrivals.
00:01:51.800 --> 00:01:56.020
That is, if in a given slot you
have a success, you say
00:01:56.020 --> 00:01:57.930
that something arrived.
00:01:57.930 --> 00:02:00.470
If you have a failure,
nothing arrived.
00:02:00.470 --> 00:02:03.090
And that language is a little
more convenient and more
00:02:03.090 --> 00:02:06.820
natural, especially when we talk
about continuous time--
00:02:06.820 --> 00:02:10.250
to talk about arrivals
instead of successes.
00:02:10.250 --> 00:02:12.640
But in any case, for the
Bernoulli process let's keep,
00:02:12.640 --> 00:02:14.990
for a little bit, the language
of successes.
00:02:14.990 --> 00:02:18.530
Whereas working in discrete
time, we have time slots.
00:02:18.530 --> 00:02:20.870
During each time slot,
we have an
00:02:20.870 --> 00:02:22.810
independent Bernoulli trial.
00:02:22.810 --> 00:02:25.750
There is probability p
of having a success.
00:02:25.750 --> 00:02:29.400
Different slots are independent
of each other.
00:02:29.400 --> 00:02:33.190
And this probability p is the
same for any given time slot.
00:02:33.190 --> 00:02:36.540
So for this process we will
discuss the one random
00:02:36.540 --> 00:02:39.240
variable of interest, which
is the following.
00:02:39.240 --> 00:02:43.060
If we have n time slots,
or n trials, how many
00:02:43.060 --> 00:02:44.600
arrivals will there be?
00:02:44.600 --> 00:02:46.990
Or how many successes
will there be?
00:02:46.990 --> 00:02:50.910
Well, this is just given
by the binomial PMF.
00:02:50.910 --> 00:02:54.780
Number of successes in n trials
is a random variable
00:02:54.780 --> 00:02:58.420
that has a binomial PMF, and
we know what this is.
00:02:58.420 --> 00:03:01.150
Then we talked about
inter-arrival times.
00:03:01.150 --> 00:03:04.490
The time until the first
arrival happens has a
00:03:04.490 --> 00:03:07.890
geometric distribution.
00:03:07.890 --> 00:03:11.770
And we have seen that
from some time ago.
00:03:11.770 --> 00:03:14.890
Now if you start thinking
about the time until k
00:03:14.890 --> 00:03:20.040
arrivals happen, and we denote
that by Yk, this is the time
00:03:20.040 --> 00:03:22.190
until the first arrival
happens.
00:03:22.190 --> 00:03:24.860
And then after the first arrival
happens, you have to
00:03:24.860 --> 00:03:27.000
wait some time until
the second arrival
00:03:27.000 --> 00:03:28.730
happens, and so on.
00:03:28.730 --> 00:03:32.650
And then the time from the
(k -1)th arrival, until
00:03:32.650 --> 00:03:34.820
arrival number k.
00:03:34.820 --> 00:03:37.680
The important thing to realize
here is that because the
00:03:37.680 --> 00:03:41.830
process has a memorylessness
property, once the first
00:03:41.830 --> 00:03:45.330
arrival comes, it's as if we're
starting from scratch
00:03:45.330 --> 00:03:48.030
and we will be flipping
our coins until the
00:03:48.030 --> 00:03:49.560
next arrival comes.
00:03:49.560 --> 00:03:52.350
So the time it will take until
the next arrival comes will
00:03:52.350 --> 00:03:54.810
also be a geometric
random variable.
00:03:54.810 --> 00:03:57.790
And because different slots
are independent, whatever
00:03:57.790 --> 00:04:00.580
happens after the first arrival
is independent from
00:04:00.580 --> 00:04:02.640
whatever happened before.
00:04:02.640 --> 00:04:06.250
So T1 and T2 will be independent
random variables.
00:04:06.250 --> 00:04:08.940
And similarly, all
the way up to Tk.
00:04:08.940 --> 00:04:13.460
So the time until the k-th
arrival is a sum of
00:04:13.460 --> 00:04:17.640
independent geometric random
variables, with the same
00:04:17.640 --> 00:04:19.459
parameter p.
00:04:19.459 --> 00:04:23.010
And we saw last time that we
can find the probability
00:04:23.010 --> 00:04:25.850
distribution of Yk.
00:04:25.850 --> 00:04:30.880
The probability that Yk takes
a value of t is equal to--
00:04:30.880 --> 00:04:36.110
there's this combinatorial
factor here, and then you get
00:04:36.110 --> 00:04:41.590
p to the k, (1-p) to the (t-k),
and this formula is
00:04:41.590 --> 00:04:48.020
true for t equal to
k, k+1, and so on.
00:04:48.020 --> 00:04:49.950
And this distribution
has a name.
00:04:49.950 --> 00:04:51.660
It's called the Pascal PMF.
00:04:54.500 --> 00:04:57.080
So this is all there
is to know about
00:04:57.080 --> 00:04:59.130
the Bernoulli process.
00:04:59.130 --> 00:05:02.880
One important comment is to
realize what exactly this
00:05:02.880 --> 00:05:05.540
memorylessness property
is saying.
00:05:05.540 --> 00:05:07.450
So I discussed it a little
bit last time.
00:05:07.450 --> 00:05:10.490
Let me reiterate it.
00:05:10.490 --> 00:05:13.460
So we have a Bernoulli process,
which is a sequence
00:05:13.460 --> 00:05:15.230
of Bernoulli trials.
00:05:15.230 --> 00:05:17.990
And these are (0,1) random
variables that
00:05:17.990 --> 00:05:20.320
keep going on forever.
00:05:20.320 --> 00:05:27.400
So someone is watching this
movie of Bernoulli trials B_t.
00:05:27.400 --> 00:05:31.540
And at some point, they say
they think, or something
00:05:31.540 --> 00:05:33.990
interesting has happened,
why don't you come
00:05:33.990 --> 00:05:36.300
in and start watching?
00:05:36.300 --> 00:05:39.760
So at some time t, they
tell you to come
00:05:39.760 --> 00:05:41.300
in and start watching.
00:05:41.300 --> 00:05:44.780
So what you will see once
you come in will
00:05:44.780 --> 00:05:48.320
be this future trials.
00:05:48.320 --> 00:05:52.610
So actually what you will see
is a random process, whose
00:05:52.610 --> 00:05:57.170
first random variable is going
to be the first one that you
00:05:57.170 --> 00:05:59.340
see, B_(t +1).
00:05:59.340 --> 00:06:03.040
The second one is going
to be this, and so on.
00:06:03.040 --> 00:06:06.850
So this is the process that's
seen by the person who's asked
00:06:06.850 --> 00:06:10.360
to come in and start watching
at that time.
00:06:10.360 --> 00:06:15.220
And the claim is that this
process is itself a Bernoulli
00:06:15.220 --> 00:06:20.380
process, provided that the
person who calls you into the
00:06:20.380 --> 00:06:23.740
room does not look
into the future.
00:06:23.740 --> 00:06:27.100
The person who calls you into
the room decides to call you
00:06:27.100 --> 00:06:31.180
in only on the basis of what
they have seen so far.
00:06:31.180 --> 00:06:33.860
So for example, who calls you
into the room might have a
00:06:33.860 --> 00:06:39.600
rule that says, as soon as I see
a sequence of 3 heads, I
00:06:39.600 --> 00:06:43.730
ask the other person
to come in.
00:06:43.730 --> 00:06:46.390
So if they use that particular
rule, it means that when
00:06:46.390 --> 00:06:49.900
you're called in, the previous
3 were heads.
00:06:49.900 --> 00:06:53.190
But this doesn't give you any
information about the future.
00:06:53.190 --> 00:06:55.260
And so the future ones
will be just
00:06:55.260 --> 00:06:57.160
independent Bernoulli trials.
00:06:57.160 --> 00:07:00.370
If on the other hand, the person
who calls you in has
00:07:00.370 --> 00:07:04.180
seen the movie before and they
use a rule, such as, for
00:07:04.180 --> 00:07:09.710
example, I call you in just
before 3 heads show up for the
00:07:09.710 --> 00:07:10.820
first time.
00:07:10.820 --> 00:07:13.850
So the person calls you in based
on knowledge that these
00:07:13.850 --> 00:07:15.430
two would be three heads.
00:07:15.430 --> 00:07:17.490
If they have such foresight--
00:07:17.490 --> 00:07:19.820
if they can look into
the future--
00:07:19.820 --> 00:07:25.050
then X1, X2, X3, they're certain
to be three heads, so
00:07:25.050 --> 00:07:27.190
they do not correspond
to random
00:07:27.190 --> 00:07:29.680
independent Bernoulli trials.
00:07:29.680 --> 00:07:33.660
So to rephrase this, the
process is memoryless.
00:07:33.660 --> 00:07:38.460
It does not matter what has
happened in the past.
00:07:38.460 --> 00:07:42.090
And that's true even if you are
called into the room and
00:07:42.090 --> 00:07:45.700
start watching at a random time,
as long as that random
00:07:45.700 --> 00:07:50.950
time is determined in a causal
way on the basis of what has
00:07:50.950 --> 00:07:52.340
happened so far.
00:07:52.340 --> 00:07:55.660
So you are called into the room
in a causal manner, just
00:07:55.660 --> 00:07:57.630
based on what's happened
so far.
00:07:57.630 --> 00:08:00.030
What you're going to see
starting from that time will
00:08:00.030 --> 00:08:03.240
still be a sequence of
independent Bernoulli trials.
00:08:03.240 --> 00:08:06.560
And this is the argument that we
used here, essentially, to
00:08:06.560 --> 00:08:09.190
argue that this T2 is an
independent random
00:08:09.190 --> 00:08:11.030
variable from T1.
00:08:11.030 --> 00:08:14.530
So a person is watching the
movie, sees the first success.
00:08:17.260 --> 00:08:19.810
And on the basis of what
they have seen--
00:08:19.810 --> 00:08:21.620
they have just seen the
first success--
00:08:21.620 --> 00:08:23.590
they ask you to come in.
00:08:23.590 --> 00:08:24.350
You come in.
00:08:24.350 --> 00:08:27.260
What you're going to see is a
sequence of Bernoulli trials.
00:08:27.260 --> 00:08:32.260
And you wait this long until
the next success comes in.
00:08:32.260 --> 00:08:35.390
What you see is a Bernoulli
process, as if the process was
00:08:35.390 --> 00:08:37.299
just starting right now.
00:08:37.299 --> 00:08:40.830
And that convinces us that this
should be a geometric
00:08:40.830 --> 00:08:43.200
random variable of the same
kind as this one, as
00:08:43.200 --> 00:08:47.080
independent from what
happened before.
00:08:47.080 --> 00:08:47.370
All right.
00:08:47.370 --> 00:08:49.610
So this is pretty much all there
is to know about the
00:08:49.610 --> 00:08:50.650
Bernoulli process.
00:08:50.650 --> 00:08:52.860
Plus the two things that we
did at the end of the last
00:08:52.860 --> 00:08:55.640
lecture where we merge two
independent Bernoulli
00:08:55.640 --> 00:08:58.070
processes, we get a
Bernoulli process.
00:08:58.070 --> 00:09:01.240
If we have a Bernoulli process
and we split it by flipping a
00:09:01.240 --> 00:09:05.450
coin and sending things one way
or the other, then we get
00:09:05.450 --> 00:09:07.790
two separate Bernoulli
processes.
00:09:07.790 --> 00:09:10.590
And we see that all of
these carry over to
00:09:10.590 --> 00:09:11.690
the continuous time.
00:09:11.690 --> 00:09:14.700
And our task for today is
basically to work these
00:09:14.700 --> 00:09:18.000
continuous time variations.
00:09:18.000 --> 00:09:21.440
So the Poisson process is a
continuous time version of the
00:09:21.440 --> 00:09:23.480
Bernoulli process.
00:09:23.480 --> 00:09:25.250
Here's the motivation
for considering
00:09:25.250 --> 00:09:26.930
it a Bernoulli process.
00:09:26.930 --> 00:09:29.850
So you have that person whose
job is to sit outside
00:09:29.850 --> 00:09:32.120
the door of a bank.
00:09:32.120 --> 00:09:38.280
And they have this long sheet,
and for every one second slot,
00:09:38.280 --> 00:09:42.560
they mark an X if a person
came in, or they mark
00:09:42.560 --> 00:09:45.760
something else if no one came
in during that slot.
00:09:45.760 --> 00:09:48.500
Now the bank manager is a really
scientifically trained
00:09:48.500 --> 00:09:50.530
person and wants very
accurate results.
00:09:50.530 --> 00:09:53.380
So they tell you, don't use
one second slots, use
00:09:53.380 --> 00:09:54.400
milliseconds slots.
00:09:54.400 --> 00:09:57.160
So you have all those slots
and you keep filling if
00:09:57.160 --> 00:09:59.950
someone arrived or not
during that slot.
00:09:59.950 --> 00:10:01.870
Well then you come
up with an idea.
00:10:01.870 --> 00:10:06.760
Why use millisecond slots and
keep putting crosses or zero's
00:10:06.760 --> 00:10:08.150
into each slot?
00:10:08.150 --> 00:10:12.380
It's much simpler if I just
record the exact times when
00:10:12.380 --> 00:10:14.010
people came in.
00:10:14.010 --> 00:10:16.380
So time is continuous.
00:10:16.380 --> 00:10:20.340
I don't keep doing something
at every time slot.
00:10:20.340 --> 00:10:24.440
But instead of the time axis,
I mark the times at which
00:10:24.440 --> 00:10:26.370
customers arrive.
00:10:26.370 --> 00:10:28.620
So there's no real
need for slots.
00:10:28.620 --> 00:10:32.370
The only information that you
want is when did we have
00:10:32.370 --> 00:10:34.130
arrivals of people.
00:10:34.130 --> 00:10:37.830
And we want to now model a
process of this kind happening
00:10:37.830 --> 00:10:41.880
in continuous time, that has the
same flavor, however, as
00:10:41.880 --> 00:10:44.210
the Bernoulli process.
00:10:44.210 --> 00:10:48.170
So that's the model we
want to develop.
00:10:48.170 --> 00:10:48.550
OK.
00:10:48.550 --> 00:10:52.880
So what are the properties
that we're going to have?
00:10:52.880 --> 00:10:57.190
First, we're going to assume
that intervals over the same
00:10:57.190 --> 00:11:01.340
length behave probabilistically
in an
00:11:01.340 --> 00:11:04.500
identical fashion.
00:11:04.500 --> 00:11:06.740
So what does that mean?
00:11:06.740 --> 00:11:09.640
Think of an interval of
some given length.
00:11:09.640 --> 00:11:12.450
During the interval of that
length, there's going to be a
00:11:12.450 --> 00:11:14.750
random number of arrivals.
00:11:14.750 --> 00:11:17.140
And that random number of
arrivals is going to have a
00:11:17.140 --> 00:11:19.050
probability distribution.
00:11:19.050 --> 00:11:20.630
So that probability
distribution--
00:11:20.630 --> 00:11:24.820
let's denote it by
this notation.
00:11:24.820 --> 00:11:29.410
We fix t, we fix the duration.
00:11:29.410 --> 00:11:31.650
So this is fixed.
00:11:31.650 --> 00:11:34.180
And we look at the
different k's.
00:11:34.180 --> 00:11:37.190
The probability of having 0
arrivals, the probability of 1
00:11:37.190 --> 00:11:40.100
arrival, the probability of
2 arrivals, and so on.
00:11:40.100 --> 00:11:42.630
So this thing is essentially
a PMF.
00:11:42.630 --> 00:11:46.920
So it should have the property
that the sum over all k's of
00:11:46.920 --> 00:11:49.890
this P_(k, tau) should
be equal to 1.
00:11:52.620 --> 00:11:57.490
Now, hidden inside this notation
is an assumption of
00:11:57.490 --> 00:11:59.550
time homogeneity.
00:11:59.550 --> 00:12:03.520
That is, this probability
distribution for the number of
00:12:03.520 --> 00:12:09.020
arrivals only depends on the
length of the interval, but
00:12:09.020 --> 00:12:13.230
not the exact location of the
interval on the time axis.
00:12:13.230 --> 00:12:18.810
That is, if I take an interval
of length tau, and I ask about
00:12:18.810 --> 00:12:21.200
the number of arrivals
in this interval.
00:12:21.200 --> 00:12:24.990
And I take another interval of
length tau, and I ask about
00:12:24.990 --> 00:12:27.790
the number of arrivals
during that interval.
00:12:27.790 --> 00:12:31.040
Number of arrivals here, and
number of arrivals there have
00:12:31.040 --> 00:12:34.400
the same probability
distribution, which is
00:12:34.400 --> 00:12:36.990
denoted this way.
00:12:36.990 --> 00:12:41.650
So the statistical behavior of
arrivals here is the same as
00:12:41.650 --> 00:12:44.310
the statistical behavioral
of arrivals there.
00:12:44.310 --> 00:12:46.820
What's the relation with
the Bernoulli process?
00:12:46.820 --> 00:12:48.510
It's very much like
the assumption--
00:12:48.510 --> 00:12:49.710
the Bernoulli process--
00:12:49.710 --> 00:12:52.530
that in different slots,
we have the same
00:12:52.530 --> 00:12:54.450
probability of success.
00:12:54.450 --> 00:12:56.770
Every slot looks
probabilistically
00:12:56.770 --> 00:12:58.400
as any other slot.
00:12:58.400 --> 00:13:02.760
So similarly here, any interval
of length tau looks
00:13:02.760 --> 00:13:06.790
probabilistically as any other
interval of length tau.
00:13:06.790 --> 00:13:09.710
And the number of arrivals
during that interval is a
00:13:09.710 --> 00:13:12.350
random variable described
by these probabilities.
00:13:12.350 --> 00:13:15.570
Number of arrivals here is a
random variable described by
00:13:15.570 --> 00:13:18.270
these same probabilities.
00:13:18.270 --> 00:13:19.900
So that's our first
assumption.
00:13:19.900 --> 00:13:21.030
Then what else?
00:13:21.030 --> 00:13:23.340
In the Bernoulli process we
had the assumption that
00:13:23.340 --> 00:13:27.710
different time slots were
independent of each other.
00:13:27.710 --> 00:13:32.970
Here we do not have time slots,
but we can still think
00:13:32.970 --> 00:13:37.490
in a similar way and impose the
following assumption, that
00:13:37.490 --> 00:13:40.970
these joint time intervals are
statistically independent.
00:13:40.970 --> 00:13:42.640
What does that mean?
00:13:42.640 --> 00:13:45.530
Does a random number of arrivals
during this interval,
00:13:45.530 --> 00:13:48.270
and the random number of
arrivals during this interval,
00:13:48.270 --> 00:13:49.830
and the random number of
00:13:49.830 --> 00:13:51.730
arrivals during this interval--
00:13:51.730 --> 00:13:55.110
so these are three different
random variables--
00:13:55.110 --> 00:13:58.900
these three random variables are
independent of each other.
00:13:58.900 --> 00:14:02.170
How many arrivals we got here
is independent from how many
00:14:02.170 --> 00:14:04.040
arrivals we got there.
00:14:04.040 --> 00:14:07.410
So this is similar to saying
that different time slots were
00:14:07.410 --> 00:14:08.110
independent.
00:14:08.110 --> 00:14:10.020
That's what we did
in discrete time.
00:14:10.020 --> 00:14:13.190
The continuous time analog is
this independence assumption.
00:14:13.190 --> 00:14:16.400
So for example, in particular,
number of arrivals here is
00:14:16.400 --> 00:14:19.440
independent from the number
of arrivals there.
00:14:19.440 --> 00:14:22.980
So these are two basic
assumptions about the process.
00:14:25.620 --> 00:14:30.270
Now in order to write down a
formula, eventually, about
00:14:30.270 --> 00:14:33.050
this probability
distribution--
00:14:33.050 --> 00:14:36.860
which is our next objective, we
would like to say something
00:14:36.860 --> 00:14:38.790
specific about this
distribution
00:14:38.790 --> 00:14:40.310
of number of arrivals--
00:14:40.310 --> 00:14:43.980
we need to add a little more
structure into the problem.
00:14:43.980 --> 00:14:47.380
And we're going to make the
following assumption.
00:14:47.380 --> 00:14:51.140
If we look at the time interval
of length delta--
00:14:51.140 --> 00:14:54.090
and delta now is supposed to
be a small number, so a
00:14:54.090 --> 00:14:55.900
picture like this--
00:14:55.900 --> 00:15:00.790
during a very small time
interval, there is a
00:15:00.790 --> 00:15:06.140
probability that we get exactly
one arrival, which is
00:15:06.140 --> 00:15:07.750
lambda times delta.
00:15:07.750 --> 00:15:10.630
Delta is the length of the
interval and lambda is a
00:15:10.630 --> 00:15:14.510
proportionality factor, which
is sort of the intensity of
00:15:14.510 --> 00:15:16.190
the arrival process.
00:15:16.190 --> 00:15:21.000
Bigger lambda means that a
little interval is more likely
00:15:21.000 --> 00:15:24.140
to get an arrival.
00:15:24.140 --> 00:15:25.595
So there's a probability lambda
00:15:25.595 --> 00:15:27.570
times delta of 1 arrival.
00:15:27.570 --> 00:15:31.740
The remaining probability
goes to 0 arrivals.
00:15:31.740 --> 00:15:35.560
And when delta is small, the
probability of 2 arrivals can
00:15:35.560 --> 00:15:39.660
be approximated by 0.
00:15:39.660 --> 00:15:42.460
So this is a description
of what happens during
00:15:42.460 --> 00:15:45.240
a small, tiny slot.
00:15:45.240 --> 00:15:48.200
Now this is something that's
supposed to be true in some
00:15:48.200 --> 00:15:51.530
limiting sense, when delta
is very small.
00:15:51.530 --> 00:15:56.070
So the exact version of this
statement would be that this
00:15:56.070 --> 00:16:02.050
is an equality, plus order
of delta squared terms.
00:16:02.050 --> 00:16:04.060
So this is an approximate
equality.
00:16:04.060 --> 00:16:07.850
And what approximation means is
that in the limit of small
00:16:07.850 --> 00:16:11.970
deltas, the dominant terms--
00:16:11.970 --> 00:16:15.900
the constant and the first order
term are given by this.
00:16:15.900 --> 00:16:19.760
Now when delta is very small,
second order terms in delta do
00:16:19.760 --> 00:16:21.130
not matter.
00:16:21.130 --> 00:16:24.380
They are small compared
to first order terms.
00:16:24.380 --> 00:16:26.190
So we ignore this.
00:16:26.190 --> 00:16:30.280
So you can either think in terms
of an exact relation,
00:16:30.280 --> 00:16:34.640
which is the probabilities are
given by this, plus delta
00:16:34.640 --> 00:16:35.990
squared terms.
00:16:35.990 --> 00:16:38.730
Or if you want to be a little
more loose, you just write
00:16:38.730 --> 00:16:41.000
here, as an approximate
equality.
00:16:41.000 --> 00:16:44.080
And the understanding is that
this equality holds--
00:16:44.080 --> 00:16:50.850
approximately becomes more
and more correct as
00:16:50.850 --> 00:16:53.410
delta goes to 0.
00:16:53.410 --> 00:16:57.250
So another version of that
statement would be that if you
00:16:57.250 --> 00:17:03.280
take the limit as delta goes to
0, of p, the probability of
00:17:03.280 --> 00:17:06.829
having 1 arrival in an interval
of length delta,
00:17:06.829 --> 00:17:10.300
divided by delta, this
is equal to lambda.
00:17:10.300 --> 00:17:16.010
So that would be one version of
an exact statement of what
00:17:16.010 --> 00:17:19.250
we are assuming here.
00:17:19.250 --> 00:17:22.750
So this lambda, we call it the
arrival rate, or the intensity
00:17:22.750 --> 00:17:23.930
of the process.
00:17:23.930 --> 00:17:27.349
And clearly, if you double
lambda, then a little interval
00:17:27.349 --> 00:17:29.340
is likely --
00:17:29.340 --> 00:17:31.630
you expect to get --
00:17:31.630 --> 00:17:34.200
the probability of obtaining
an arrival during that
00:17:34.200 --> 00:17:35.740
interval has doubled.
00:17:35.740 --> 00:17:40.400
So in some sense we have twice
as intense arrival process.
00:17:40.400 --> 00:17:46.470
If you look at the number
of arrivals during delta
00:17:46.470 --> 00:17:54.490
interval, what is the expected
value of that random variable?
00:17:54.490 --> 00:18:00.100
Well with probability lambda
delta we get 1 arrival.
00:18:00.100 --> 00:18:01.240
And with the remaining
00:18:01.240 --> 00:18:03.610
probability, we get 0 arrivals.
00:18:03.610 --> 00:18:06.680
So it's just lambda
times delta.
00:18:06.680 --> 00:18:10.640
So expected number of arrivals
during a little interval is
00:18:10.640 --> 00:18:12.170
lambda times delta.
00:18:12.170 --> 00:18:15.460
So expected number of arrivals
is proportional to lambda, and
00:18:15.460 --> 00:18:19.050
that's again why we call lambda
the arrival rate.
00:18:19.050 --> 00:18:22.820
If you send delta to the
denominator in this equality,
00:18:22.820 --> 00:18:26.380
it tells you that lambda is
the expected number of
00:18:26.380 --> 00:18:30.000
arrivals per unit time.
00:18:30.000 --> 00:18:37.010
So the arrival rate is expected
number of arrivals
00:18:37.010 --> 00:18:38.750
per unit time.
00:18:38.750 --> 00:18:42.580
And again, that justifies why
we call lambda the intensity
00:18:42.580 --> 00:18:43.830
of this process.
00:18:46.316 --> 00:18:46.760
All right.
00:18:46.760 --> 00:18:49.680
So where are we now?
00:18:49.680 --> 00:18:53.740
For the Bernoulli process, the
number of arrivals during a
00:18:53.740 --> 00:19:00.210
given interval of length n had
the PMF that we knew it was
00:19:00.210 --> 00:19:01.545
the binomial PMF.
00:19:04.190 --> 00:19:07.530
What is the formula for the
corresponding PMF for the
00:19:07.530 --> 00:19:09.190
continuous time process?
00:19:09.190 --> 00:19:12.570
Somehow we would like to use
our assumptions and come up
00:19:12.570 --> 00:19:16.100
with the formula for
this quantity.
00:19:16.100 --> 00:19:19.110
So this tells us about the
distribution of number of
00:19:19.110 --> 00:19:23.750
arrivals during an interval
of some general length.
00:19:23.750 --> 00:19:27.690
We have made assumptions about
the number of arrivals during
00:19:27.690 --> 00:19:30.410
an interval of small length.
00:19:30.410 --> 00:19:34.350
An interval of big length is
composed of many intervals of
00:19:34.350 --> 00:19:37.830
small length, so maybe this
is the way to go.
00:19:37.830 --> 00:19:43.120
Take a big interval, and split
it into many intervals of
00:19:43.120 --> 00:19:44.970
small length.
00:19:44.970 --> 00:19:48.410
So we have here our time axis.
00:19:48.410 --> 00:19:51.480
And we have an interval
of length tau.
00:19:51.480 --> 00:19:55.240
And I'm going to split it into
lots of little intervals of
00:19:55.240 --> 00:19:56.580
length delta.
00:19:56.580 --> 00:19:59.000
So how many intervals are
we going to have?
00:19:59.000 --> 00:20:03.060
The number of intervals is going
to be the total time,
00:20:03.060 --> 00:20:04.620
divided by delta.
00:20:07.520 --> 00:20:12.960
Now what happens during each one
of these little intervals?
00:20:12.960 --> 00:20:22.380
As long as the intervals are
small, what you have is that
00:20:22.380 --> 00:20:24.240
during an interval, you're
going to have
00:20:24.240 --> 00:20:27.220
either 0 or 1 arrival.
00:20:27.220 --> 00:20:29.940
The probability of more than
1 arrival during a little
00:20:29.940 --> 00:20:31.950
interval is negligible.
00:20:31.950 --> 00:20:35.380
So with this picture, you have
essentially a Bernoulli
00:20:35.380 --> 00:20:39.970
process that consists
of so many trials.
00:20:39.970 --> 00:20:43.160
And during each one of those
trials, we have a probability
00:20:43.160 --> 00:20:46.730
of success, which is
lambda times delta.
00:20:51.845 --> 00:20:54.330
Different little intervals
here are
00:20:54.330 --> 00:20:56.140
independent of each other.
00:20:56.140 --> 00:20:58.670
That's one of our assumptions,
that these joint time
00:20:58.670 --> 00:21:00.380
intervals are independent.
00:21:00.380 --> 00:21:05.590
So approximately, what we have
is a Bernoulli process.
00:21:05.590 --> 00:21:06.980
We have independence.
00:21:06.980 --> 00:21:09.250
We have the number of
slots of interest.
00:21:09.250 --> 00:21:11.450
And during each one of the
slots we have a certain
00:21:11.450 --> 00:21:13.530
probability of success.
00:21:13.530 --> 00:21:17.300
So if we think of this as
another good approximation of
00:21:17.300 --> 00:21:18.870
the Poisson process--
00:21:18.870 --> 00:21:21.090
with the approximation becoming
more and more
00:21:21.090 --> 00:21:23.595
accurate as delta goes to 0 --
00:21:23.595 --> 00:21:28.150
what we should do would be to
take the formula for the PMF
00:21:28.150 --> 00:21:32.320
of number of arrivals in a
Bernoulli process, and then
00:21:32.320 --> 00:21:37.230
take the limit as
delta goes to 0.
00:21:37.230 --> 00:21:45.260
So in the Bernoulli process, the
probability of k arrivals
00:21:45.260 --> 00:21:52.730
is n choose k, and then
you have p to the k.
00:21:52.730 --> 00:21:57.610
Now in our case, we have here
lambda times delta, delta is
00:21:57.610 --> 00:21:59.340
tau over n.
00:22:02.190 --> 00:22:08.410
Delta is tau over n, so p is
lambda times tau divided by n.
00:22:08.410 --> 00:22:11.010
So here's our p --
00:22:11.010 --> 00:22:13.690
Lambda tau over n --
00:22:13.690 --> 00:22:22.760
to the power k, and then times
one minus this-- this is our
00:22:22.760 --> 00:22:24.540
one minus p--
00:22:24.540 --> 00:22:25.790
to the power n-k.
00:22:30.010 --> 00:22:35.730
So this is the exact formula
for the Bernoulli process.
00:22:35.730 --> 00:22:39.830
For the Poisson process, what we
do is we take that formula
00:22:39.830 --> 00:22:43.360
and we let delta go to 0.
00:22:43.360 --> 00:22:48.150
As delta goes to 0, n
goes to infinity.
00:22:48.150 --> 00:22:51.280
So that's the limit
that we're taking.
00:22:51.280 --> 00:22:55.580
On the other hand, this
expression lambda times tau--
00:22:59.730 --> 00:23:03.740
lambda times tau, what
is it going to be?
00:23:03.740 --> 00:23:06.860
Lambda times tau is equal
to n times p.
00:23:09.900 --> 00:23:11.990
n times p, is that
what I want?
00:23:21.300 --> 00:23:22.550
No, let's see.
00:23:26.600 --> 00:23:28.110
Lambda tau is np.
00:23:28.110 --> 00:23:29.990
Yeah.
00:23:29.990 --> 00:23:32.060
So lambda tau is np.
00:23:53.370 --> 00:23:54.030
All right.
00:23:54.030 --> 00:23:59.320
So we have this relation,
lambda tau equals np.
00:23:59.320 --> 00:24:03.070
These two numbers being equal
kind of makes sense. np is the
00:24:03.070 --> 00:24:05.890
expected number of successes
you're going to get in the
00:24:05.890 --> 00:24:07.750
Bernoulli process.
00:24:07.750 --> 00:24:08.780
Lambda tau--
00:24:08.780 --> 00:24:11.800
since lambda is the arrival rate
and you have a total time
00:24:11.800 --> 00:24:15.710
of tau, lambda tau you can think
of it as the number of
00:24:15.710 --> 00:24:19.750
expected arrivals in the
Bernoulli process.
00:24:19.750 --> 00:24:22.000
We're doing a Bernoulli
approximation
00:24:22.000 --> 00:24:23.250
to the Poisson process.
00:24:23.250 --> 00:24:26.150
We take the formula for the
Bernoulli, and now take the
00:24:26.150 --> 00:24:30.060
limit as n goes to infinity.
00:24:30.060 --> 00:24:35.330
Now lambda tau over n is equal
to p, so it's clear what this
00:24:35.330 --> 00:24:37.040
term is going to give us.
00:24:37.040 --> 00:24:39.695
This is just p to the power k.
00:24:48.820 --> 00:24:53.230
It will actually take a little
more work than that.
00:24:53.230 --> 00:24:58.210
Now I'm not going to do the
algebra, but I'm just telling
00:24:58.210 --> 00:25:03.390
you that one can take the limit
in this formula here, as
00:25:03.390 --> 00:25:05.000
n goes to infinity.
00:25:05.000 --> 00:25:08.860
And that will give you another
formula, the final formula for
00:25:08.860 --> 00:25:10.320
the Poisson PMF.
00:25:10.320 --> 00:25:13.400
One thing to notice is that here
you have something like 1
00:25:13.400 --> 00:25:17.460
minus a constant over
n, to the power n.
00:25:17.460 --> 00:25:21.630
And you may recall from calculus
a formula of this
00:25:21.630 --> 00:25:26.540
kind, that this converges
to e to the minus c.
00:25:26.540 --> 00:25:29.560
If you remember that formula
from calculus, then you will
00:25:29.560 --> 00:25:32.750
expect that here, in the limit,
you are going to get
00:25:32.750 --> 00:25:36.520
something like an e to
the minus lambda tau.
00:25:36.520 --> 00:25:39.180
So indeed, we will
get such a term.
00:25:39.180 --> 00:25:42.230
There is some work that needs
to be done to find the limit
00:25:42.230 --> 00:25:45.880
of this expression, times
that expression.
00:25:45.880 --> 00:25:48.690
The algebra is not hard,
it's in the text.
00:25:48.690 --> 00:25:51.340
Let's not spend more
time doing this.
00:25:51.340 --> 00:25:53.820
But let me just give you
the formula of what
00:25:53.820 --> 00:25:55.620
comes at the end.
00:25:55.620 --> 00:25:59.720
And the formula that comes at
the end is of this form.
00:25:59.720 --> 00:26:03.710
So what matters here is not so
much the specific algebra that
00:26:03.710 --> 00:26:07.690
you will do to go from this
formula to that one.
00:26:07.690 --> 00:26:09.370
It's kind of straightforward.
00:26:09.370 --> 00:26:14.535
What's important is the idea
that the Poisson process, by
00:26:14.535 --> 00:26:19.710
definition, can be approximated
by a Bernoulli
00:26:19.710 --> 00:26:25.040
process in which we have a very
large number of slots--
00:26:25.040 --> 00:26:27.970
n goes to infinity.
00:26:27.970 --> 00:26:32.600
Whereas we have a very small
probability of success during
00:26:32.600 --> 00:26:34.420
each time slot.
00:26:34.420 --> 00:26:38.640
So a large number of slots,
but tiny probability of
00:26:38.640 --> 00:26:40.480
success during each slot.
00:26:40.480 --> 00:26:42.370
And we take the limit
as the slots
00:26:42.370 --> 00:26:44.680
become smaller and smaller.
00:26:44.680 --> 00:26:47.170
So with this approximation
we end up with
00:26:47.170 --> 00:26:49.030
this particular formula.
00:26:49.030 --> 00:26:51.890
And this is the so-called
Poisson PMF.
00:26:51.890 --> 00:26:53.930
Now this function P here --
00:26:53.930 --> 00:26:55.190
has two arguments.
00:26:55.190 --> 00:26:58.320
The important thing to realize
is that when you think of this
00:26:58.320 --> 00:27:02.900
as a PMF, you fix t to tau.
00:27:02.900 --> 00:27:06.010
And for a fixed tau,
now this is a PMF.
00:27:06.010 --> 00:27:11.260
As I said before, the sum over
k has to be equal to 1.
00:27:11.260 --> 00:27:15.510
So for a given tau, these
probabilities add up to 1.
00:27:15.510 --> 00:27:20.590
The formula is moderately messy,
but not too messy.
00:27:20.590 --> 00:27:24.570
One can work with it without
too much pain.
00:27:24.570 --> 00:27:28.460
And what's the mean and
variance of this PMF?
00:27:28.460 --> 00:27:31.560
Well what's the expected
number of arrivals?
00:27:31.560 --> 00:27:35.680
If you think of this Bernoulli
analogy, we know that the
00:27:35.680 --> 00:27:37.940
expected number of arrivals
in the Bernoulli
00:27:37.940 --> 00:27:41.250
process is n times p.
00:27:41.250 --> 00:27:44.500
In the approximation that
we're using in these
00:27:44.500 --> 00:27:48.170
procedure, n times p is the
same as lambda tau.
00:27:48.170 --> 00:27:52.490
And that's why we get lambda tau
to be the expected number
00:27:52.490 --> 00:27:53.480
of arrivals.
00:27:53.480 --> 00:27:56.140
Here I'm using t
instead of tau.
00:27:56.140 --> 00:28:01.670
The expected number of
arrivals is lambda t.
00:28:01.670 --> 00:28:05.450
So if you double the time,
you expect to get
00:28:05.450 --> 00:28:07.290
twice as many arrivals.
00:28:07.290 --> 00:28:10.960
If you double the arrival rate,
you expect to get twice
00:28:10.960 --> 00:28:12.760
as many arrivals.
00:28:12.760 --> 00:28:15.290
How about the formula
for the variance?
00:28:15.290 --> 00:28:18.930
The variance of the Bernoulli
process is np,
00:28:18.930 --> 00:28:22.720
times one minus p.
00:28:22.720 --> 00:28:25.800
What does this go
to in the limit?
00:28:25.800 --> 00:28:31.170
In the limit that we're taking,
as delta goes to zero,
00:28:31.170 --> 00:28:33.550
then p also goes to zero.
00:28:33.550 --> 00:28:37.270
The probability of success in
any given slot goes to zero.
00:28:37.270 --> 00:28:39.700
So this term becomes
insignificant.
00:28:39.700 --> 00:28:45.980
So this becomes n times p, which
is again lambda t, or
00:28:45.980 --> 00:28:47.710
lambda tau.
00:28:47.710 --> 00:28:50.960
So the variance, instead of
having this more complicated
00:28:50.960 --> 00:28:54.290
formula of the variance is the
Bernoulli process, here it
00:28:54.290 --> 00:28:56.840
gets simplified and
it's lambda t.
00:28:56.840 --> 00:29:00.580
So interestingly, the variance
in the Poisson process is
00:29:00.580 --> 00:29:03.200
exactly the same as the
expected value.
00:29:03.200 --> 00:29:06.360
So you can look at this as
just some interesting
00:29:06.360 --> 00:29:07.780
coincidence.
00:29:07.780 --> 00:29:10.260
So now we're going to take
this formula and
00:29:10.260 --> 00:29:11.370
see how to use it.
00:29:11.370 --> 00:29:14.060
First we're going to do
a completely trivial,
00:29:14.060 --> 00:29:16.560
straightforward example.
00:29:16.560 --> 00:29:24.630
So 15 years ago when that
example was made, email was
00:29:24.630 --> 00:29:27.230
coming at a rate of five
messages per hour.
00:29:27.230 --> 00:29:30.660
I wish that was the
case today.
00:29:30.660 --> 00:29:38.450
And now emails that are coming
in, let's say during the day--
00:29:38.450 --> 00:29:41.750
the arrival rates of emails
are probably different in
00:29:41.750 --> 00:29:42.960
different times of the day.
00:29:42.960 --> 00:29:46.840
But if you fix a time slot,
let's say 1:00 to 2:00 in the
00:29:46.840 --> 00:29:49.370
afternoon, there's probably
a constant rate.
00:29:49.370 --> 00:29:53.050
And email arrivals are
reasonably well modeled by a
00:29:53.050 --> 00:29:54.790
Poisson process.
00:29:54.790 --> 00:29:58.220
Speaking of modeling, it's
not just email arrivals.
00:29:58.220 --> 00:30:02.290
Whenever arrivals happen in a
completely random way, without
00:30:02.290 --> 00:30:05.370
any additional structure, the
Poisson process is a good
00:30:05.370 --> 00:30:07.010
model of these arrivals.
00:30:07.010 --> 00:30:10.200
So the times at which car
accidents will happen, that's
00:30:10.200 --> 00:30:11.450
a Poisson processes.
00:30:15.530 --> 00:30:19.550
If you have a very, very weak
light source that's shooting
00:30:19.550 --> 00:30:24.290
out photons, just one at a time,
the times at which these
00:30:24.290 --> 00:30:27.240
photons will go out is
well modeled again
00:30:27.240 --> 00:30:28.670
by a Poisson process.
00:30:28.670 --> 00:30:30.540
So it's completely random.
00:30:30.540 --> 00:30:35.230
Or if you have a radioactive
material where one atom at a
00:30:35.230 --> 00:30:43.720
time changes at random times.
00:30:43.720 --> 00:30:45.920
So it's a very slow
radioactive decay.
00:30:45.920 --> 00:30:48.900
The time at which these alpha
particles, or whatever we get
00:30:48.900 --> 00:30:51.580
emitted, again is going
to be described
00:30:51.580 --> 00:30:53.200
by a Poisson process.
00:30:53.200 --> 00:30:58.220
So if you have arrivals, or
emissions, that happen at
00:30:58.220 --> 00:31:02.660
completely random times, and
once in a while you get an
00:31:02.660 --> 00:31:07.500
arrival or an event, then the
Poisson process is a very good
00:31:07.500 --> 00:31:10.070
model for these events.
00:31:10.070 --> 00:31:12.200
So back to emails.
00:31:12.200 --> 00:31:16.350
Get them at a rate of five
messages per day, per hour.
00:31:16.350 --> 00:31:19.420
In 30 minutes this
is half an hour.
00:31:19.420 --> 00:31:23.770
So what we have is that
lambda t, total
00:31:23.770 --> 00:31:26.520
number of arrivals is--
00:31:26.520 --> 00:31:29.020
the expected number
of arrivals is--
00:31:29.020 --> 00:31:33.810
lambda is five, t is one-half,
if we talk about hours.
00:31:33.810 --> 00:31:36.480
So lambda t is two to the 0.5.
00:31:36.480 --> 00:31:41.220
The probability of no new
messages is the probability of
00:31:41.220 --> 00:31:48.560
zero, in time interval of length
t, which, in our case,
00:31:48.560 --> 00:31:51.790
is one-half.
00:31:51.790 --> 00:31:55.510
And then we look back into the
formula from the previous
00:31:55.510 --> 00:31:59.550
slide, and the probability of
zero arrivals is lambda t to
00:31:59.550 --> 00:32:03.770
the power zero, divided by zero
factorial, and then an e
00:32:03.770 --> 00:32:05.450
to the lambda t.
00:32:05.450 --> 00:32:07.840
And you plug in the numbers
that we have.
00:32:07.840 --> 00:32:10.380
Lambda t to the zero
power is one.
00:32:10.380 --> 00:32:12.040
Zero factorial is one.
00:32:12.040 --> 00:32:15.500
So we're left with e
to the minus 2.5.
00:32:15.500 --> 00:32:18.860
And that number is 0.08.
00:32:18.860 --> 00:32:22.090
Similarly, you can ask for the
probability that you get
00:32:22.090 --> 00:32:24.850
exactly one message
in half an hour.
00:32:24.850 --> 00:32:27.420
And that would be-- the
probability of one message in
00:32:27.420 --> 00:32:28.680
one-half an hour--
00:32:28.680 --> 00:32:32.590
is going to be lambda t to the
first power, divided by 1
00:32:32.590 --> 00:32:38.230
factorial, e to the minus
lambda t, which--
00:32:38.230 --> 00:32:41.900
as we now get the extra lambda t
factor-- is going to be 2.5,
00:32:41.900 --> 00:32:43.650
e to the minus 2.5.
00:32:43.650 --> 00:32:46.930
And the numerical
answer is 0.20.
00:32:46.930 --> 00:32:50.450
So this is how you use the PMF
formula for the Poisson
00:32:50.450 --> 00:32:55.540
distribution that we had
in the previous slide.
00:32:55.540 --> 00:32:55.890
All right.
00:32:55.890 --> 00:33:00.010
So this was all about
the distribution of
00:33:00.010 --> 00:33:01.780
the number of arrivals.
00:33:01.780 --> 00:33:03.350
What else did we do last time?
00:33:03.350 --> 00:33:08.250
Last time we also talked about
the time it takes until the
00:33:08.250 --> 00:33:09.500
k-th arrival.
00:33:12.390 --> 00:33:12.790
OK.
00:33:12.790 --> 00:33:16.020
So let's try to figure out
something about this
00:33:16.020 --> 00:33:18.260
particular distribution.
00:33:18.260 --> 00:33:21.180
We can derive the distribution
of the time of the k-th
00:33:21.180 --> 00:33:24.730
arrival by using the
exact same argument
00:33:24.730 --> 00:33:27.360
as we did last time.
00:33:27.360 --> 00:33:31.650
So now the time of the
k-th arrival is a
00:33:31.650 --> 00:33:33.830
continuous random variable.
00:33:33.830 --> 00:33:36.160
So it has a PDF.
00:33:36.160 --> 00:33:38.430
Since we are in continuous
time, arrivals can
00:33:38.430 --> 00:33:39.900
happen at any time.
00:33:39.900 --> 00:33:42.310
So Yk is a continuous
random variable.
00:33:45.200 --> 00:33:48.160
But now let's think of
a time interval of
00:33:48.160 --> 00:33:49.410
length little delta.
00:33:52.370 --> 00:33:58.620
And use our usual interpretation
of PDFs.
00:33:58.620 --> 00:34:03.180
The PDF of a random variable
evaluated at a certain time
00:34:03.180 --> 00:34:08.010
times delta, this is the
probability that the Yk falls
00:34:08.010 --> 00:34:09.514
in this little interval.
00:34:13.460 --> 00:34:16.639
So as I've said before, this
is the best way of thinking
00:34:16.639 --> 00:34:18.420
about PDFs.
00:34:18.420 --> 00:34:22.179
PDFs give you probabilities
of little intervals.
00:34:22.179 --> 00:34:25.540
So now let's try to calculate
this probability.
00:34:25.540 --> 00:34:29.880
For the k-th arrival to happen
inside this little interval,
00:34:29.880 --> 00:34:31.550
we need two things.
00:34:31.550 --> 00:34:35.790
We need an arrival to happen in
this interval, and we need
00:34:35.790 --> 00:34:41.530
k minus one arrivals to happen
during that interval.
00:34:41.530 --> 00:34:41.880
OK.
00:34:41.880 --> 00:34:45.469
You'll tell me, but it's
possible that we might have
00:34:45.469 --> 00:34:50.130
the k minus one arrival happen
here, and the k-th arrival to
00:34:50.130 --> 00:34:51.219
happen here.
00:34:51.219 --> 00:34:53.050
In principle, that's possible.
00:34:53.050 --> 00:34:56.139
But in the limit, when we take
delta very small, the
00:34:56.139 --> 00:34:59.850
probability of having two
arrivals in the same little
00:34:59.850 --> 00:35:01.830
slot is negligible.
00:35:01.830 --> 00:35:06.870
So assuming that no two arrivals
can happen in the
00:35:06.870 --> 00:35:10.940
same mini slot, then for the
k-th one to happen here, we
00:35:10.940 --> 00:35:15.710
must have k minus one during
this interval.
00:35:15.710 --> 00:35:20.210
Now because we have assumed that
these joint intervals are
00:35:20.210 --> 00:35:23.900
independent of each other,
this breaks down into the
00:35:23.900 --> 00:35:33.070
probability that we have exactly
k minus one arrivals,
00:35:33.070 --> 00:35:37.600
during the interval from zero to
t, times the probability of
00:35:37.600 --> 00:35:41.410
exactly one arrival during that
little interval, which is
00:35:41.410 --> 00:35:43.420
lambda delta.
00:35:43.420 --> 00:35:51.010
We do have a formula for this
from the previous slide, which
00:35:51.010 --> 00:35:59.340
is lambda t, to the k minus 1,
over k minus one factorial,
00:35:59.340 --> 00:36:07.190
times e to minus lambda t.
00:36:07.190 --> 00:36:09.070
And then lambda times delta.
00:36:14.910 --> 00:36:16.160
Did I miss something?
00:36:24.680 --> 00:36:26.310
Yeah, OK.
00:36:26.310 --> 00:36:26.970
All right.
00:36:26.970 --> 00:36:30.220
And now you cancel this
delta with that delta.
00:36:30.220 --> 00:36:36.820
And that gives us a formula for
the PDF of the time until
00:36:36.820 --> 00:36:39.170
the k-th arrival.
00:36:39.170 --> 00:36:43.290
This PDF, of course, depends
on the number k.
00:36:43.290 --> 00:36:46.850
The first arrival is going
to happen somewhere in
00:36:46.850 --> 00:36:48.040
this range of time.
00:36:48.040 --> 00:36:50.140
So this is the PDF
that it has.
00:36:50.140 --> 00:36:53.170
The second arrival, of course,
is going to happen later.
00:36:53.170 --> 00:36:54.860
And the PDF is this.
00:36:54.860 --> 00:36:57.880
So it's more likely to happen
around these times.
00:36:57.880 --> 00:37:01.410
The third arrival has this PDF,
so it's more likely to
00:37:01.410 --> 00:37:03.690
happen around those times.
00:37:03.690 --> 00:37:08.020
And if you were to take
k equal to 100,
00:37:08.020 --> 00:37:10.470
you might get a PDF--
00:37:10.470 --> 00:37:13.260
it's extremely unlikely that
the k-th arrival happens in
00:37:13.260 --> 00:37:18.060
the beginning, and it might
happen somewhere down there,
00:37:18.060 --> 00:37:20.010
far into the future.
00:37:20.010 --> 00:37:22.230
So depending on which particular
arrival we're
00:37:22.230 --> 00:37:25.510
talking about, it has a
different probability
00:37:25.510 --> 00:37:26.350
distribution.
00:37:26.350 --> 00:37:30.340
The time of the 100th arrival,
of course, is expected to be a
00:37:30.340 --> 00:37:34.100
lot larger than the time
of the first arrival.
00:37:34.100 --> 00:37:38.550
Incidentally, the time of the
first arrival has a PDF whose
00:37:38.550 --> 00:37:40.160
form is quite simple.
00:37:40.160 --> 00:37:43.850
If you let k equal to one here,
this term disappears.
00:37:43.850 --> 00:37:46.310
That term becomes a one.
00:37:46.310 --> 00:37:49.880
You're left with just lambda,
e to the minus lambda.
00:37:49.880 --> 00:37:53.210
And you recognize it, it's the
exponential distribution.
00:37:53.210 --> 00:37:57.210
So the time until the first
arrival in a Poisson process
00:37:57.210 --> 00:38:00.160
is an exponential
distribution.
00:38:00.160 --> 00:38:02.150
What was the time of the
first arrival in
00:38:02.150 --> 00:38:03.970
the Bernoulli process?
00:38:03.970 --> 00:38:07.060
It was a geometric
distribution.
00:38:07.060 --> 00:38:11.170
Well, not coincidentally, these
two look quite a bit
00:38:11.170 --> 00:38:13.030
like the other.
00:38:13.030 --> 00:38:17.980
A geometric distribution
has this kind of shape.
00:38:17.980 --> 00:38:21.900
The exponential distribution
has that kind of shape.
00:38:21.900 --> 00:38:25.560
The geometric is just a discrete
version of the
00:38:25.560 --> 00:38:27.090
exponential.
00:38:27.090 --> 00:38:29.860
In the Bernoulli case, we
are in discrete time.
00:38:29.860 --> 00:38:32.540
We have a PMF for the
time of the first
00:38:32.540 --> 00:38:35.080
arrival, which is geometric.
00:38:35.080 --> 00:38:38.540
In the Poisson case, what we
get is the limit of the
00:38:38.540 --> 00:38:41.560
geometric as you let those
lines become closer and
00:38:41.560 --> 00:38:46.480
closer, which gives you the
exponential distribution.
00:38:46.480 --> 00:38:50.430
Now the Poisson process shares
all the memorylessness
00:38:50.430 --> 00:38:52.870
properties of the Bernoulli
process.
00:38:52.870 --> 00:38:56.750
And the way one can argue is
just in terms of this picture.
00:38:56.750 --> 00:39:00.250
Since the Poisson process is
the limit of Bernoulli
00:39:00.250 --> 00:39:03.570
processes, whatever qualitative
processes you have
00:39:03.570 --> 00:39:07.340
in the Bernoulli process
remain valid
00:39:07.340 --> 00:39:08.360
for the Poisson process.
00:39:08.360 --> 00:39:11.470
In particular we have this
memorylessness property.
00:39:11.470 --> 00:39:15.120
You let the Poisson process run
for some time, and then
00:39:15.120 --> 00:39:16.600
you start watching it.
00:39:16.600 --> 00:39:18.520
What ever happened in
the past has no
00:39:18.520 --> 00:39:20.220
bearing about the future.
00:39:20.220 --> 00:39:23.150
Starting from right now, what's
going to happen in the
00:39:23.150 --> 00:39:27.330
future is described again by a
Poisson process, in the sense
00:39:27.330 --> 00:39:30.530
that during every little slot of
length delta, there's going
00:39:30.530 --> 00:39:33.790
to be a probability of lambda
delta of having an arrival.
00:39:33.790 --> 00:39:36.590
And that probably lambda
delta is the same-- is
00:39:36.590 --> 00:39:38.070
always lambda delta--
00:39:38.070 --> 00:39:41.270
no matter what happened in
the past of the process.
00:39:41.270 --> 00:39:47.040
And in particular, we could use
this argument to say that
00:39:47.040 --> 00:39:50.460
the time until the k-th arrival
is the time that it
00:39:50.460 --> 00:39:53.720
takes for the first
arrival to happen.
00:39:53.720 --> 00:39:56.380
OK, let me do it for
k equal to two.
00:39:56.380 --> 00:39:59.630
And then after the first arrival
happens, you wait a
00:39:59.630 --> 00:40:02.600
certain amount of time until
the second arrival happens.
00:40:02.600 --> 00:40:06.410
Now once the first arrival
happened, that's in the past.
00:40:06.410 --> 00:40:07.400
You start watching.
00:40:07.400 --> 00:40:10.690
From now on you have mini slots
of length delta, each
00:40:10.690 --> 00:40:13.230
one having a probability of
success lambda delta.
00:40:13.230 --> 00:40:16.230
It's as if we started the
Poisson process from scratch.
00:40:16.230 --> 00:40:19.310
So starting from that time,
the time until the next
00:40:19.310 --> 00:40:22.840
arrival is going to be again an
exponential distribution,
00:40:22.840 --> 00:40:26.010
which doesn't care about what
happened in the past, how long
00:40:26.010 --> 00:40:28.000
it took you for the
first arrival.
00:40:28.000 --> 00:40:33.410
So these two random variables
are going to be independent
00:40:33.410 --> 00:40:38.140
and exponential, with the
same parameter lambda.
00:40:38.140 --> 00:40:42.570
So among other things, what we
have done here is we have
00:40:42.570 --> 00:40:48.130
essentially derived the PDF of
the sum of k independent
00:40:48.130 --> 00:40:49.320
exponentials.
00:40:49.320 --> 00:40:53.990
The time of the k-th arrival
is the sum of k
00:40:53.990 --> 00:40:56.230
inter-arrival times.
00:40:56.230 --> 00:40:59.380
The inter-arrival times are all
independent of each other
00:40:59.380 --> 00:41:01.450
because of memorylessness.
00:41:01.450 --> 00:41:04.245
And they all have the same
exponential distribution.
00:41:07.130 --> 00:41:08.980
And by the way, this
gives you a way to
00:41:08.980 --> 00:41:11.080
simulate the Poisson process.
00:41:11.080 --> 00:41:14.070
If you wanted to simulate it
on your computer, you would
00:41:14.070 --> 00:41:20.140
have one option to break time
into tiny, tiny slots.
00:41:20.140 --> 00:41:24.030
And for every tiny slot, use
your random number generator
00:41:24.030 --> 00:41:27.520
to decide whether there
was an arrival or not.
00:41:27.520 --> 00:41:29.810
To get it very accurate,
you would have to
00:41:29.810 --> 00:41:32.090
use tiny, tiny slots.
00:41:32.090 --> 00:41:35.280
So that would be a lot
of computation.
00:41:35.280 --> 00:41:38.530
The more clever way of
simulating the Poisson process
00:41:38.530 --> 00:41:42.320
is you use your random number
generator to generate a sample
00:41:42.320 --> 00:41:45.280
from an exponential distribution
and call that
00:41:45.280 --> 00:41:47.240
your first arrival time.
00:41:47.240 --> 00:41:50.050
Then go back to the random
number generator, generate
00:41:50.050 --> 00:41:53.040
another independent sample,
again from the same
00:41:53.040 --> 00:41:54.780
exponential distribution.
00:41:54.780 --> 00:41:58.490
That's the time between the
first and the second arrival,
00:41:58.490 --> 00:42:01.390
and you keep going that way.
00:42:01.390 --> 00:42:03.260
So as a sort of a
quick summary,
00:42:03.260 --> 00:42:04.910
this is the big picture.
00:42:04.910 --> 00:42:08.630
This table doesn't tell
you anything new.
00:42:08.630 --> 00:42:12.230
But it's good to have it as a
reference, and to look at it,
00:42:12.230 --> 00:42:14.740
and to make sure you understand
what all the
00:42:14.740 --> 00:42:16.300
different boxes are.
00:42:16.300 --> 00:42:18.930
Basically the Bernoulli process
runs in discrete time.
00:42:18.930 --> 00:42:20.960
The Poisson process runs
in continuous time.
00:42:20.960 --> 00:42:25.140
There's an analogy of arrival
rates, p per trial, or
00:42:25.140 --> 00:42:27.270
intensity per unit time.
00:42:27.270 --> 00:42:32.190
We did derive, or sketched the
derivation for the PMF of the
00:42:32.190 --> 00:42:33.610
number of arrivals.
00:42:33.610 --> 00:42:37.810
And the Poisson distribution,
which is the distribution that
00:42:37.810 --> 00:42:40.450
we get, this Pk of t.
00:42:40.450 --> 00:42:44.220
Pk and t is the limit of the
binomial when we take the
00:42:44.220 --> 00:42:49.710
limit in this particular way,
as delta goes to zero, and n
00:42:49.710 --> 00:42:51.220
goes to infinity.
00:42:51.220 --> 00:42:54.270
The geometric becomes an
exponential in the limit.
00:42:54.270 --> 00:42:56.960
And the distribution of the
time of the k-th arrival--
00:42:56.960 --> 00:42:59.600
we had a closed form formula
last time for
00:42:59.600 --> 00:43:01.050
the Bernoulli process.
00:43:01.050 --> 00:43:03.930
We got the closed form
formula this time
00:43:03.930 --> 00:43:05.230
for the Poisson process.
00:43:05.230 --> 00:43:08.940
And we actually used exactly the
same argument to get these
00:43:08.940 --> 00:43:12.320
two closed form formulas.
00:43:12.320 --> 00:43:12.650
All right.
00:43:12.650 --> 00:43:18.280
So now let's talk about adding
or merging Poisson processes.
00:43:18.280 --> 00:43:21.060
And there's two statements
that we can make here.
00:43:21.060 --> 00:43:25.970
One has to do with adding
Poisson random variables, just
00:43:25.970 --> 00:43:26.820
random variables.
00:43:26.820 --> 00:43:28.290
There's another statement about
00:43:28.290 --> 00:43:30.770
adding Poisson processes.
00:43:30.770 --> 00:43:34.540
And the second is a bigger
statement than the first.
00:43:34.540 --> 00:43:36.140
But this is a warm up.
00:43:36.140 --> 00:43:39.140
Let's work with the
first statement.
00:43:39.140 --> 00:43:42.460
So the claim is that the sum of
independent Poisson random
00:43:42.460 --> 00:43:45.340
variables is Poisson.
00:43:45.340 --> 00:43:45.990
OK.
00:43:45.990 --> 00:43:50.490
So suppose that we have a
Poisson process with rate--
00:43:50.490 --> 00:43:51.760
just for simplicity--
00:43:51.760 --> 00:43:53.170
lambda one.
00:43:53.170 --> 00:43:56.240
And I take the interval
from zero to two.
00:43:56.240 --> 00:44:00.620
And that take then the interval
from two until five.
00:44:00.620 --> 00:44:03.720
The number of arrivals during
this interval--
00:44:03.720 --> 00:44:06.730
let's call it n from
zero to two--
00:44:06.730 --> 00:44:13.920
is going to be a Poisson
random variable, with
00:44:13.920 --> 00:44:18.240
parameter, or with mean, two.
00:44:18.240 --> 00:44:24.340
The number of arrivals during
this interval is n from time
00:44:24.340 --> 00:44:26.340
two until five.
00:44:26.340 --> 00:44:31.120
This is again a Poisson random
variable with mean equal to
00:44:31.120 --> 00:44:34.690
three, because the arrival rate
is 1 and the duration of
00:44:34.690 --> 00:44:36.990
the interval is three.
00:44:36.990 --> 00:44:41.320
These two random variables
are independent.
00:44:41.320 --> 00:44:43.760
They obey the Poisson
distribution
00:44:43.760 --> 00:44:45.640
that we derived before.
00:44:45.640 --> 00:44:50.930
If you add them, what you get
is the number of arrivals
00:44:50.930 --> 00:44:53.850
during the interval
from zero to five.
00:44:53.850 --> 00:44:56.290
Now what kind of distribution
does this
00:44:56.290 --> 00:44:57.910
random variable have?
00:44:57.910 --> 00:45:00.760
Well this is the number of
arrivals over an interval of a
00:45:00.760 --> 00:45:03.600
certain length in a
Poisson process.
00:45:03.600 --> 00:45:08.580
Therefore, this is also Poisson
with mean five.
00:45:16.520 --> 00:45:19.040
Because for the Poisson process
we know that this
00:45:19.040 --> 00:45:23.300
number of arrivals is Poisson,
this is Poisson, but also the
00:45:23.300 --> 00:45:26.610
number of overall arrivals
is also Poisson.
00:45:26.610 --> 00:45:30.040
This establishes that the sum
of a Poisson plus a Poisson
00:45:30.040 --> 00:45:32.200
random variable gives
us another
00:45:32.200 --> 00:45:33.630
Poisson random variable.
00:45:33.630 --> 00:45:37.110
So adding Poisson random
variables gives us a Poisson
00:45:37.110 --> 00:45:38.720
random variable.
00:45:38.720 --> 00:45:42.660
But now I'm going to make a more
general statement that
00:45:42.660 --> 00:45:44.940
it's not just number
of arrivals during
00:45:44.940 --> 00:45:46.415
a fixed time interval--
00:45:50.420 --> 00:45:53.240
it's not just numbers of
arrivals for given time
00:45:53.240 --> 00:45:54.260
intervals--
00:45:54.260 --> 00:45:57.770
but rather if you take two
different Poisson processes
00:45:57.770 --> 00:46:02.330
and add them up, the process
itself is Poisson in the sense
00:46:02.330 --> 00:46:05.930
that this process is going to
satisfy all the assumptions of
00:46:05.930 --> 00:46:07.510
a Poisson process.
00:46:07.510 --> 00:46:11.060
So the story is that you have
a red bulb that flashes at
00:46:11.060 --> 00:46:13.350
random times at the rate
of lambda one.
00:46:13.350 --> 00:46:14.980
It's a Poisson process.
00:46:14.980 --> 00:46:19.080
You have an independent process
where a green bulb
00:46:19.080 --> 00:46:21.230
flashes at random times.
00:46:21.230 --> 00:46:24.800
And you happen to be color
blind, so you just see when
00:46:24.800 --> 00:46:26.630
something is flashing.
00:46:26.630 --> 00:46:29.920
So these two are assumed to be
independent Poisson processes.
00:46:29.920 --> 00:46:34.968
What can we say about the
process that you observe?
00:46:34.968 --> 00:46:40.250
So in the processes that you
observe, if you take a typical
00:46:40.250 --> 00:46:45.170
time interval of length little
delta, what can happen during
00:46:45.170 --> 00:46:48.380
that little time interval?
00:46:48.380 --> 00:46:55.280
The red process may have
something flashing.
00:46:55.280 --> 00:46:56.815
So red flashes.
00:46:59.850 --> 00:47:01.580
Or the red does not.
00:47:06.610 --> 00:47:12.170
And for the other bulb, the
green bulb, there's two
00:47:12.170 --> 00:47:13.020
possibilities.
00:47:13.020 --> 00:47:17.910
The green one flashes.
00:47:17.910 --> 00:47:20.900
And the other possibility is
that the green does not.
00:47:24.990 --> 00:47:25.330
OK.
00:47:25.330 --> 00:47:29.070
So there's four possibilities
about what can happen during a
00:47:29.070 --> 00:47:31.170
little slot.
00:47:31.170 --> 00:47:36.080
The probability that the red one
flashes and the green one
00:47:36.080 --> 00:47:39.750
flashes, what is this
probability?
00:47:39.750 --> 00:47:43.510
It's lambda one delta that the
first one flashes, and lambda
00:47:43.510 --> 00:47:47.290
two delta that the
second one does.
00:47:47.290 --> 00:47:50.280
I'm multiplying probabilities
here because I'm making the
00:47:50.280 --> 00:47:52.645
assumption that the two
processes are independent.
00:47:55.330 --> 00:47:57.330
OK.
00:47:57.330 --> 00:48:00.130
Now the probability that
the red one flashes
00:48:00.130 --> 00:48:01.440
is lambda one delta.
00:48:01.440 --> 00:48:08.210
But the green one doesn't is
one, minus lambda two delta.
00:48:08.210 --> 00:48:12.840
Here the probability would be
that the red one does not,
00:48:12.840 --> 00:48:16.400
times the probability that
the green one does.
00:48:16.400 --> 00:48:20.750
And then here we have the
probability that none of them
00:48:20.750 --> 00:48:26.790
flash, which is whatever
is left.
00:48:26.790 --> 00:48:29.600
But it's one minus lambda
one delta, times one
00:48:29.600 --> 00:48:33.160
minus lambda two delta.
00:48:33.160 --> 00:48:36.920
Now we're thinking about
delta as small.
00:48:36.920 --> 00:48:43.260
So think of the case where delta
goes to zero, but in a
00:48:43.260 --> 00:48:49.840
way that we keep the
first order terms.
00:48:49.840 --> 00:48:54.070
We keep the delta terms, but
we throw away the delta
00:48:54.070 --> 00:48:55.020
squared terms.
00:48:55.020 --> 00:48:58.170
Delta squared terms are much
smaller than the delta terms
00:48:58.170 --> 00:49:00.260
when delta becomes small.
00:49:00.260 --> 00:49:01.920
If we do that--
00:49:01.920 --> 00:49:05.650
if we only keep the order
of delta terms--
00:49:05.650 --> 00:49:07.940
this term effectively
disappears.
00:49:07.940 --> 00:49:09.110
This is delta squared.
00:49:09.110 --> 00:49:11.550
So we make it zero.
00:49:11.550 --> 00:49:14.550
So the probability of having
simultaneously a red and a
00:49:14.550 --> 00:49:17.940
green flash during a little
interval is negligible.
00:49:17.940 --> 00:49:20.150
What do we get here?
00:49:20.150 --> 00:49:23.200
Lambda delta times
one survives, but
00:49:23.200 --> 00:49:24.910
this times that doesn't.
00:49:24.910 --> 00:49:28.820
So we can throw that away.
00:49:28.820 --> 00:49:32.190
So the approximation that we
get is lambda one delta.
00:49:32.190 --> 00:49:34.010
Similarly here, this
goes away.
00:49:34.010 --> 00:49:36.420
We're left with a lambda
two delta.
00:49:36.420 --> 00:49:42.140
And this is whatever remains,
whatever is left.
00:49:42.140 --> 00:49:45.000
So what do we have?
00:49:45.000 --> 00:49:51.400
That there is a probability of
seeing a flash, either a red
00:49:51.400 --> 00:49:54.360
or a green, which is
lambda one delta,
00:49:54.360 --> 00:49:57.020
plus lambda two delta.
00:49:57.020 --> 00:50:03.590
So if we take a little interval
of length delta here,
00:50:03.590 --> 00:50:11.780
it's going to see an arrival
with probability approximately
00:50:11.780 --> 00:50:15.100
lambda one, plus lambda
two, delta.
00:50:15.100 --> 00:50:20.940
So every slot in this merged
process has an arrival
00:50:20.940 --> 00:50:25.780
probability with a rate which
is the sum of the rates of
00:50:25.780 --> 00:50:27.600
these two processes.
00:50:27.600 --> 00:50:29.640
So this is one part
of the definition
00:50:29.640 --> 00:50:31.680
of the Poisson process.
00:50:31.680 --> 00:50:34.890
There's a few more things that
one would need to verify.
00:50:34.890 --> 00:50:37.980
Namely, that intervals of the
same length have the same
00:50:37.980 --> 00:50:41.000
probability distribution and
that different slots are
00:50:41.000 --> 00:50:42.710
independent of each other.
00:50:42.710 --> 00:50:50.780
This can be argued by starting
from here because different
00:50:50.780 --> 00:50:53.620
intervals in this process are
independent from each other.
00:50:53.620 --> 00:50:56.900
Different intervals here are
independent from each other.
00:50:56.900 --> 00:50:59.900
It's not hard to argue that
different intervals in the
00:50:59.900 --> 00:51:03.580
merged process will also be
independent of each other.
00:51:03.580 --> 00:51:06.480
So the conclusion that comes
at the end is that this
00:51:06.480 --> 00:51:10.130
process is a Poisson process,
with a total rate which is
00:51:10.130 --> 00:51:13.210
equal to the sum of the rate
of the two processes.
00:51:13.210 --> 00:51:17.010
And now if I tell you that an
arrival happened in the merged
00:51:17.010 --> 00:51:20.530
process at a certain time,
how likely is it that
00:51:20.530 --> 00:51:23.470
it came from here?
00:51:23.470 --> 00:51:24.950
How likely is it?
00:51:24.950 --> 00:51:26.980
We go to this picture.
00:51:26.980 --> 00:51:30.140
Given that an arrival
occurred--
00:51:30.140 --> 00:51:36.050
which is the event that this
or that happened--
00:51:36.050 --> 00:51:39.330
what is the probability that
it came from the first
00:51:39.330 --> 00:51:42.060
process, the red one?
00:51:42.060 --> 00:51:45.190
Well it's the probability
of this divided by the
00:51:45.190 --> 00:51:48.030
probability of this,
times that.
00:51:48.030 --> 00:51:52.760
Given that this event occurred,
you want to find the
00:51:52.760 --> 00:51:56.560
conditional probability
of that sub event.
00:51:56.560 --> 00:51:58.960
So we're asking the question,
out of the total probability
00:51:58.960 --> 00:52:00.660
of these two, what
fraction of that
00:52:00.660 --> 00:52:02.790
probability is assigned here?
00:52:02.790 --> 00:52:05.300
And this is lambda one
delta, after we
00:52:05.300 --> 00:52:07.040
ignore the other terms.
00:52:07.040 --> 00:52:09.170
This is lambda two delta.
00:52:09.170 --> 00:52:15.040
So that fraction is going to be
lambda one, over lambda one
00:52:15.040 --> 00:52:16.770
plus lambda two.
00:52:16.770 --> 00:52:17.640
What does this tell you?
00:52:17.640 --> 00:52:21.820
If lambda one and lambda two are
equal, given that I saw an
00:52:21.820 --> 00:52:25.580
arrival here, it's equally
likely to be red or green.
00:52:25.580 --> 00:52:29.716
But if the reds have a much
higher arrival rate, when I
00:52:29.716 --> 00:52:32.700
see an arrival here, it's
more likely this
00:52:32.700 --> 00:52:34.050
number will be large.
00:52:34.050 --> 00:52:38.390
So it's more likely to have
come from the red process.
00:52:38.390 --> 00:52:40.830
OK so we'll continue with
this story and do some
00:52:40.830 --> 00:52:42.080
applications next time.