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PROFESSOR: So last time
we started talking
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about random processes.
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00:00:26,420 --> 00:00:30,710
A random process is a random
experiment that
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00:00:30,710 --> 00:00:32,570
evolves over time.
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00:00:32,570 --> 00:00:35,110
And conceptually, it's important
to realize that it's
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00:00:35,110 --> 00:00:37,760
a single probabilistic
experiment
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00:00:37,760 --> 00:00:39,250
that has many stages.
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00:00:39,250 --> 00:00:42,440
Actually, it has an infinite
number of stages.
18
00:00:42,440 --> 00:00:47,060
And we discussed the simplest
random process there is, the
19
00:00:47,060 --> 00:00:50,270
Bernoulli process, which is
nothing but the sequence of
20
00:00:50,270 --> 00:00:51,570
Bernoulli trials--
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00:00:51,570 --> 00:00:54,580
an infinite sequence of
Bernoulli trials.
22
00:00:54,580 --> 00:00:58,250
For example, flipping a
coin over and over.
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00:00:58,250 --> 00:01:01,640
Once we understand what's going
on with that process,
24
00:01:01,640 --> 00:01:05,519
then what we want is to move
into a continuous time version
25
00:01:05,519 --> 00:01:06,750
of the Bernoulli process.
26
00:01:06,750 --> 00:01:09,420
And this is what we will call
the Poisson process.
27
00:01:09,420 --> 00:01:11,970
And for the Poisson process,
we're going to do exactly the
28
00:01:11,970 --> 00:01:14,300
same things that we did for
the Bernoulli process.
29
00:01:14,300 --> 00:01:18,210
That is, talk about the number
of arrivals during a given
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00:01:18,210 --> 00:01:21,400
time period, and talk also
about the time between
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00:01:21,400 --> 00:01:24,160
consecutive arrivals, and
for the distribution of
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00:01:24,160 --> 00:01:27,660
inter-arrival times.
33
00:01:27,660 --> 00:01:30,660
So let's start with a quick
review of what we
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00:01:30,660 --> 00:01:32,680
discussed last time.
35
00:01:32,680 --> 00:01:35,500
First, a note about language.
36
00:01:35,500 --> 00:01:38,100
If you think of coin tosses,
we then talk
37
00:01:38,100 --> 00:01:40,660
about heads and tails.
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00:01:40,660 --> 00:01:43,430
If you think of these as a
sequence of trials, you can
39
00:01:43,430 --> 00:01:47,145
talk about successes
and failures.
40
00:01:47,145 --> 00:01:50,312
The language that we will be
using will be more the
41
00:01:50,312 --> 00:01:51,800
language of arrivals.
42
00:01:51,800 --> 00:01:56,020
That is, if in a given slot you
have a success, you say
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00:01:56,020 --> 00:01:57,930
that something arrived.
44
00:01:57,930 --> 00:02:00,470
If you have a failure,
nothing arrived.
45
00:02:00,470 --> 00:02:03,090
And that language is a little
more convenient and more
46
00:02:03,090 --> 00:02:06,820
natural, especially when we talk
about continuous time--
47
00:02:06,820 --> 00:02:10,250
to talk about arrivals
instead of successes.
48
00:02:10,250 --> 00:02:12,640
But in any case, for the
Bernoulli process let's keep,
49
00:02:12,640 --> 00:02:14,990
for a little bit, the language
of successes.
50
00:02:14,990 --> 00:02:18,530
Whereas working in discrete
time, we have time slots.
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00:02:18,530 --> 00:02:20,870
During each time slot,
we have an
52
00:02:20,870 --> 00:02:22,810
independent Bernoulli trial.
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00:02:22,810 --> 00:02:25,750
There is probability p
of having a success.
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00:02:25,750 --> 00:02:29,400
Different slots are independent
of each other.
55
00:02:29,400 --> 00:02:33,190
And this probability p is the
same for any given time slot.
56
00:02:33,190 --> 00:02:36,540
So for this process we will
discuss the one random
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00:02:36,540 --> 00:02:39,240
variable of interest, which
is the following.
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00:02:39,240 --> 00:02:43,060
If we have n time slots,
or n trials, how many
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00:02:43,060 --> 00:02:44,600
arrivals will there be?
60
00:02:44,600 --> 00:02:46,990
Or how many successes
will there be?
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00:02:46,990 --> 00:02:50,910
Well, this is just given
by the binomial PMF.
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00:02:50,910 --> 00:02:54,780
Number of successes in n trials
is a random variable
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00:02:54,780 --> 00:02:58,420
that has a binomial PMF, and
we know what this is.
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00:02:58,420 --> 00:03:01,150
Then we talked about
inter-arrival times.
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00:03:01,150 --> 00:03:04,490
The time until the first
arrival happens has a
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00:03:04,490 --> 00:03:07,890
geometric distribution.
67
00:03:07,890 --> 00:03:11,770
And we have seen that
from some time ago.
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00:03:11,770 --> 00:03:14,890
Now if you start thinking
about the time until k
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00:03:14,890 --> 00:03:20,040
arrivals happen, and we denote
that by Yk, this is the time
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00:03:20,040 --> 00:03:22,190
until the first arrival
happens.
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00:03:22,190 --> 00:03:24,860
And then after the first arrival
happens, you have to
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00:03:24,860 --> 00:03:27,000
wait some time until
the second arrival
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00:03:27,000 --> 00:03:28,730
happens, and so on.
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00:03:28,730 --> 00:03:32,650
And then the time from the
(k -1)th arrival, until
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00:03:32,650 --> 00:03:34,820
arrival number k.
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00:03:34,820 --> 00:03:37,680
The important thing to realize
here is that because the
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process has a memorylessness
property, once the first
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00:03:41,830 --> 00:03:45,330
arrival comes, it's as if we're
starting from scratch
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00:03:45,330 --> 00:03:48,030
and we will be flipping
our coins until the
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00:03:48,030 --> 00:03:49,560
next arrival comes.
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00:03:49,560 --> 00:03:52,350
So the time it will take until
the next arrival comes will
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00:03:52,350 --> 00:03:54,810
also be a geometric
random variable.
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00:03:54,810 --> 00:03:57,790
And because different slots
are independent, whatever
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00:03:57,790 --> 00:04:00,580
happens after the first arrival
is independent from
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00:04:00,580 --> 00:04:02,640
whatever happened before.
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00:04:02,640 --> 00:04:06,250
So T1 and T2 will be independent
random variables.
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00:04:06,250 --> 00:04:08,940
And similarly, all
the way up to Tk.
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00:04:08,940 --> 00:04:13,460
So the time until the k-th
arrival is a sum of
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00:04:13,460 --> 00:04:17,640
independent geometric random
variables, with the same
90
00:04:17,640 --> 00:04:19,459
parameter p.
91
00:04:19,459 --> 00:04:23,010
And we saw last time that we
can find the probability
92
00:04:23,010 --> 00:04:25,850
distribution of Yk.
93
00:04:25,850 --> 00:04:30,880
The probability that Yk takes
a value of t is equal to--
94
00:04:30,880 --> 00:04:36,110
there's this combinatorial
factor here, and then you get
95
00:04:36,110 --> 00:04:41,590
p to the k, (1-p) to the (t-k),
and this formula is
96
00:04:41,590 --> 00:04:48,020
true for t equal to
k, k+1, and so on.
97
00:04:48,020 --> 00:04:49,950
And this distribution
has a name.
98
00:04:49,950 --> 00:04:51,660
It's called the Pascal PMF.
99
00:04:51,660 --> 00:04:54,500
100
00:04:54,500 --> 00:04:57,080
So this is all there
is to know about
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00:04:57,080 --> 00:04:59,130
the Bernoulli process.
102
00:04:59,130 --> 00:05:02,880
One important comment is to
realize what exactly this
103
00:05:02,880 --> 00:05:05,540
memorylessness property
is saying.
104
00:05:05,540 --> 00:05:07,450
So I discussed it a little
bit last time.
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00:05:07,450 --> 00:05:10,490
Let me reiterate it.
106
00:05:10,490 --> 00:05:13,460
So we have a Bernoulli process,
which is a sequence
107
00:05:13,460 --> 00:05:15,230
of Bernoulli trials.
108
00:05:15,230 --> 00:05:17,990
And these are (0,1) random
variables that
109
00:05:17,990 --> 00:05:20,320
keep going on forever.
110
00:05:20,320 --> 00:05:27,400
So someone is watching this
movie of Bernoulli trials B_t.
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00:05:27,400 --> 00:05:31,540
And at some point, they say
they think, or something
112
00:05:31,540 --> 00:05:33,990
interesting has happened,
why don't you come
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00:05:33,990 --> 00:05:36,300
in and start watching?
114
00:05:36,300 --> 00:05:39,760
So at some time t, they
tell you to come
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00:05:39,760 --> 00:05:41,300
in and start watching.
116
00:05:41,300 --> 00:05:44,780
So what you will see once
you come in will
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00:05:44,780 --> 00:05:48,320
be this future trials.
118
00:05:48,320 --> 00:05:52,610
So actually what you will see
is a random process, whose
119
00:05:52,610 --> 00:05:57,170
first random variable is going
to be the first one that you
120
00:05:57,170 --> 00:05:59,340
see, B_(t +1).
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00:05:59,340 --> 00:06:03,040
The second one is going
to be this, and so on.
122
00:06:03,040 --> 00:06:06,850
So this is the process that's
seen by the person who's asked
123
00:06:06,850 --> 00:06:10,360
to come in and start watching
at that time.
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00:06:10,360 --> 00:06:15,220
And the claim is that this
process is itself a Bernoulli
125
00:06:15,220 --> 00:06:20,380
process, provided that the
person who calls you into the
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00:06:20,380 --> 00:06:23,740
room does not look
into the future.
127
00:06:23,740 --> 00:06:27,100
The person who calls you into
the room decides to call you
128
00:06:27,100 --> 00:06:31,180
in only on the basis of what
they have seen so far.
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00:06:31,180 --> 00:06:33,860
So for example, who calls you
into the room might have a
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00:06:33,860 --> 00:06:39,600
rule that says, as soon as I see
a sequence of 3 heads, I
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00:06:39,600 --> 00:06:43,730
ask the other person
to come in.
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00:06:43,730 --> 00:06:46,390
So if they use that particular
rule, it means that when
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00:06:46,390 --> 00:06:49,900
you're called in, the previous
3 were heads.
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00:06:49,900 --> 00:06:53,190
But this doesn't give you any
information about the future.
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00:06:53,190 --> 00:06:55,260
And so the future ones
will be just
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00:06:55,260 --> 00:06:57,160
independent Bernoulli trials.
137
00:06:57,160 --> 00:07:00,370
If on the other hand, the person
who calls you in has
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00:07:00,370 --> 00:07:04,180
seen the movie before and they
use a rule, such as, for
139
00:07:04,180 --> 00:07:09,710
example, I call you in just
before 3 heads show up for the
140
00:07:09,710 --> 00:07:10,820
first time.
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00:07:10,820 --> 00:07:13,850
So the person calls you in based
on knowledge that these
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00:07:13,850 --> 00:07:15,430
two would be three heads.
143
00:07:15,430 --> 00:07:17,490
If they have such foresight--
144
00:07:17,490 --> 00:07:19,820
if they can look into
the future--
145
00:07:19,820 --> 00:07:25,050
then X1, X2, X3, they're certain
to be three heads, so
146
00:07:25,050 --> 00:07:27,190
they do not correspond
to random
147
00:07:27,190 --> 00:07:29,680
independent Bernoulli trials.
148
00:07:29,680 --> 00:07:33,660
So to rephrase this, the
process is memoryless.
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00:07:33,660 --> 00:07:38,460
It does not matter what has
happened in the past.
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00:07:38,460 --> 00:07:42,090
And that's true even if you are
called into the room and
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00:07:42,090 --> 00:07:45,700
start watching at a random time,
as long as that random
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00:07:45,700 --> 00:07:50,950
time is determined in a causal
way on the basis of what has
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00:07:50,950 --> 00:07:52,340
happened so far.
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00:07:52,340 --> 00:07:55,660
So you are called into the room
in a causal manner, just
155
00:07:55,660 --> 00:07:57,630
based on what's happened
so far.
156
00:07:57,630 --> 00:08:00,030
What you're going to see
starting from that time will
157
00:08:00,030 --> 00:08:03,240
still be a sequence of
independent Bernoulli trials.
158
00:08:03,240 --> 00:08:06,560
And this is the argument that we
used here, essentially, to
159
00:08:06,560 --> 00:08:09,190
argue that this T2 is an
independent random
160
00:08:09,190 --> 00:08:11,030
variable from T1.
161
00:08:11,030 --> 00:08:14,530
So a person is watching the
movie, sees the first success.
162
00:08:14,530 --> 00:08:17,260
163
00:08:17,260 --> 00:08:19,810
And on the basis of what
they have seen--
164
00:08:19,810 --> 00:08:21,620
they have just seen the
first success--
165
00:08:21,620 --> 00:08:23,590
they ask you to come in.
166
00:08:23,590 --> 00:08:24,350
You come in.
167
00:08:24,350 --> 00:08:27,260
What you're going to see is a
sequence of Bernoulli trials.
168
00:08:27,260 --> 00:08:32,260
And you wait this long until
the next success comes in.
169
00:08:32,260 --> 00:08:35,390
What you see is a Bernoulli
process, as if the process was
170
00:08:35,390 --> 00:08:37,299
just starting right now.
171
00:08:37,299 --> 00:08:40,830
And that convinces us that this
should be a geometric
172
00:08:40,830 --> 00:08:43,200
random variable of the same
kind as this one, as
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00:08:43,200 --> 00:08:47,080
independent from what
happened before.
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00:08:47,080 --> 00:08:47,370
All right.
175
00:08:47,370 --> 00:08:49,610
So this is pretty much all there
is to know about the
176
00:08:49,610 --> 00:08:50,650
Bernoulli process.
177
00:08:50,650 --> 00:08:52,860
Plus the two things that we
did at the end of the last
178
00:08:52,860 --> 00:08:55,640
lecture where we merge two
independent Bernoulli
179
00:08:55,640 --> 00:08:58,070
processes, we get a
Bernoulli process.
180
00:08:58,070 --> 00:09:01,240
If we have a Bernoulli process
and we split it by flipping a
181
00:09:01,240 --> 00:09:05,450
coin and sending things one way
or the other, then we get
182
00:09:05,450 --> 00:09:07,790
two separate Bernoulli
processes.
183
00:09:07,790 --> 00:09:10,590
And we see that all of
these carry over to
184
00:09:10,590 --> 00:09:11,690
the continuous time.
185
00:09:11,690 --> 00:09:14,700
And our task for today is
basically to work these
186
00:09:14,700 --> 00:09:18,000
continuous time variations.
187
00:09:18,000 --> 00:09:21,440
So the Poisson process is a
continuous time version of the
188
00:09:21,440 --> 00:09:23,480
Bernoulli process.
189
00:09:23,480 --> 00:09:25,250
Here's the motivation
for considering
190
00:09:25,250 --> 00:09:26,930
it a Bernoulli process.
191
00:09:26,930 --> 00:09:29,850
So you have that person whose
job is to sit outside
192
00:09:29,850 --> 00:09:32,120
the door of a bank.
193
00:09:32,120 --> 00:09:38,280
And they have this long sheet,
and for every one second slot,
194
00:09:38,280 --> 00:09:42,560
they mark an X if a person
came in, or they mark
195
00:09:42,560 --> 00:09:45,760
something else if no one came
in during that slot.
196
00:09:45,760 --> 00:09:48,500
Now the bank manager is a really
scientifically trained
197
00:09:48,500 --> 00:09:50,530
person and wants very
accurate results.
198
00:09:50,530 --> 00:09:53,380
So they tell you, don't use
one second slots, use
199
00:09:53,380 --> 00:09:54,400
milliseconds slots.
200
00:09:54,400 --> 00:09:57,160
So you have all those slots
and you keep filling if
201
00:09:57,160 --> 00:09:59,950
someone arrived or not
during that slot.
202
00:09:59,950 --> 00:10:01,870
Well then you come
up with an idea.
203
00:10:01,870 --> 00:10:06,760
Why use millisecond slots and
keep putting crosses or zero's
204
00:10:06,760 --> 00:10:08,150
into each slot?
205
00:10:08,150 --> 00:10:12,380
It's much simpler if I just
record the exact times when
206
00:10:12,380 --> 00:10:14,010
people came in.
207
00:10:14,010 --> 00:10:16,380
So time is continuous.
208
00:10:16,380 --> 00:10:20,340
I don't keep doing something
at every time slot.
209
00:10:20,340 --> 00:10:24,440
But instead of the time axis,
I mark the times at which
210
00:10:24,440 --> 00:10:26,370
customers arrive.
211
00:10:26,370 --> 00:10:28,620
So there's no real
need for slots.
212
00:10:28,620 --> 00:10:32,370
The only information that you
want is when did we have
213
00:10:32,370 --> 00:10:34,130
arrivals of people.
214
00:10:34,130 --> 00:10:37,830
And we want to now model a
process of this kind happening
215
00:10:37,830 --> 00:10:41,880
in continuous time, that has the
same flavor, however, as
216
00:10:41,880 --> 00:10:44,210
the Bernoulli process.
217
00:10:44,210 --> 00:10:48,170
So that's the model we
want to develop.
218
00:10:48,170 --> 00:10:48,550
OK.
219
00:10:48,550 --> 00:10:52,880
So what are the properties
that we're going to have?
220
00:10:52,880 --> 00:10:57,190
First, we're going to assume
that intervals over the same
221
00:10:57,190 --> 00:11:01,340
length behave probabilistically
in an
222
00:11:01,340 --> 00:11:04,500
identical fashion.
223
00:11:04,500 --> 00:11:06,740
So what does that mean?
224
00:11:06,740 --> 00:11:09,640
Think of an interval of
some given length.
225
00:11:09,640 --> 00:11:12,450
During the interval of that
length, there's going to be a
226
00:11:12,450 --> 00:11:14,750
random number of arrivals.
227
00:11:14,750 --> 00:11:17,140
And that random number of
arrivals is going to have a
228
00:11:17,140 --> 00:11:19,050
probability distribution.
229
00:11:19,050 --> 00:11:20,630
So that probability
distribution--
230
00:11:20,630 --> 00:11:24,820
let's denote it by
this notation.
231
00:11:24,820 --> 00:11:29,410
We fix t, we fix the duration.
232
00:11:29,410 --> 00:11:31,650
So this is fixed.
233
00:11:31,650 --> 00:11:34,180
And we look at the
different k's.
234
00:11:34,180 --> 00:11:37,190
The probability of having 0
arrivals, the probability of 1
235
00:11:37,190 --> 00:11:40,100
arrival, the probability of
2 arrivals, and so on.
236
00:11:40,100 --> 00:11:42,630
So this thing is essentially
a PMF.
237
00:11:42,630 --> 00:11:46,920
So it should have the property
that the sum over all k's of
238
00:11:46,920 --> 00:11:49,890
this P_(k, tau) should
be equal to 1.
239
00:11:49,890 --> 00:11:52,620
240
00:11:52,620 --> 00:11:57,490
Now, hidden inside this notation
is an assumption of
241
00:11:57,490 --> 00:11:59,550
time homogeneity.
242
00:11:59,550 --> 00:12:03,520
That is, this probability
distribution for the number of
243
00:12:03,520 --> 00:12:09,020
arrivals only depends on the
length of the interval, but
244
00:12:09,020 --> 00:12:13,230
not the exact location of the
interval on the time axis.
245
00:12:13,230 --> 00:12:18,810
That is, if I take an interval
of length tau, and I ask about
246
00:12:18,810 --> 00:12:21,200
the number of arrivals
in this interval.
247
00:12:21,200 --> 00:12:24,990
And I take another interval of
length tau, and I ask about
248
00:12:24,990 --> 00:12:27,790
the number of arrivals
during that interval.
249
00:12:27,790 --> 00:12:31,040
Number of arrivals here, and
number of arrivals there have
250
00:12:31,040 --> 00:12:34,400
the same probability
distribution, which is
251
00:12:34,400 --> 00:12:36,990
denoted this way.
252
00:12:36,990 --> 00:12:41,650
So the statistical behavior of
arrivals here is the same as
253
00:12:41,650 --> 00:12:44,310
the statistical behavioral
of arrivals there.
254
00:12:44,310 --> 00:12:46,820
What's the relation with
the Bernoulli process?
255
00:12:46,820 --> 00:12:48,510
It's very much like
the assumption--
256
00:12:48,510 --> 00:12:49,710
the Bernoulli process--
257
00:12:49,710 --> 00:12:52,530
that in different slots,
we have the same
258
00:12:52,530 --> 00:12:54,450
probability of success.
259
00:12:54,450 --> 00:12:56,770
Every slot looks
probabilistically
260
00:12:56,770 --> 00:12:58,400
as any other slot.
261
00:12:58,400 --> 00:13:02,760
So similarly here, any interval
of length tau looks
262
00:13:02,760 --> 00:13:06,790
probabilistically as any other
interval of length tau.
263
00:13:06,790 --> 00:13:09,710
And the number of arrivals
during that interval is a
264
00:13:09,710 --> 00:13:12,350
random variable described
by these probabilities.
265
00:13:12,350 --> 00:13:15,570
Number of arrivals here is a
random variable described by
266
00:13:15,570 --> 00:13:18,270
these same probabilities.
267
00:13:18,270 --> 00:13:19,900
So that's our first
assumption.
268
00:13:19,900 --> 00:13:21,030
Then what else?
269
00:13:21,030 --> 00:13:23,340
In the Bernoulli process we
had the assumption that
270
00:13:23,340 --> 00:13:27,710
different time slots were
independent of each other.
271
00:13:27,710 --> 00:13:32,970
Here we do not have time slots,
but we can still think
272
00:13:32,970 --> 00:13:37,490
in a similar way and impose the
following assumption, that
273
00:13:37,490 --> 00:13:40,970
these joint time intervals are
statistically independent.
274
00:13:40,970 --> 00:13:42,640
What does that mean?
275
00:13:42,640 --> 00:13:45,530
Does a random number of arrivals
during this interval,
276
00:13:45,530 --> 00:13:48,270
and the random number of
arrivals during this interval,
277
00:13:48,270 --> 00:13:49,830
and the random number of
278
00:13:49,830 --> 00:13:51,730
arrivals during this interval--
279
00:13:51,730 --> 00:13:55,110
so these are three different
random variables--
280
00:13:55,110 --> 00:13:58,900
these three random variables are
independent of each other.
281
00:13:58,900 --> 00:14:02,170
How many arrivals we got here
is independent from how many
282
00:14:02,170 --> 00:14:04,040
arrivals we got there.
283
00:14:04,040 --> 00:14:07,410
So this is similar to saying
that different time slots were
284
00:14:07,410 --> 00:14:08,110
independent.
285
00:14:08,110 --> 00:14:10,020
That's what we did
in discrete time.
286
00:14:10,020 --> 00:14:13,190
The continuous time analog is
this independence assumption.
287
00:14:13,190 --> 00:14:16,400
So for example, in particular,
number of arrivals here is
288
00:14:16,400 --> 00:14:19,440
independent from the number
of arrivals there.
289
00:14:19,440 --> 00:14:22,980
So these are two basic
assumptions about the process.
290
00:14:22,980 --> 00:14:25,620
291
00:14:25,620 --> 00:14:30,270
Now in order to write down a
formula, eventually, about
292
00:14:30,270 --> 00:14:33,050
this probability
distribution--
293
00:14:33,050 --> 00:14:36,860
which is our next objective, we
would like to say something
294
00:14:36,860 --> 00:14:38,790
specific about this
distribution
295
00:14:38,790 --> 00:14:40,310
of number of arrivals--
296
00:14:40,310 --> 00:14:43,980
we need to add a little more
structure into the problem.
297
00:14:43,980 --> 00:14:47,380
And we're going to make the
following assumption.
298
00:14:47,380 --> 00:14:51,140
If we look at the time interval
of length delta--
299
00:14:51,140 --> 00:14:54,090
and delta now is supposed to
be a small number, so a
300
00:14:54,090 --> 00:14:55,900
picture like this--
301
00:14:55,900 --> 00:15:00,790
during a very small time
interval, there is a
302
00:15:00,790 --> 00:15:06,140
probability that we get exactly
one arrival, which is
303
00:15:06,140 --> 00:15:07,750
lambda times delta.
304
00:15:07,750 --> 00:15:10,630
Delta is the length of the
interval and lambda is a
305
00:15:10,630 --> 00:15:14,510
proportionality factor, which
is sort of the intensity of
306
00:15:14,510 --> 00:15:16,190
the arrival process.
307
00:15:16,190 --> 00:15:21,000
Bigger lambda means that a
little interval is more likely
308
00:15:21,000 --> 00:15:24,140
to get an arrival.
309
00:15:24,140 --> 00:15:25,595
So there's a probability lambda
310
00:15:25,595 --> 00:15:27,570
times delta of 1 arrival.
311
00:15:27,570 --> 00:15:31,740
The remaining probability
goes to 0 arrivals.
312
00:15:31,740 --> 00:15:35,560
And when delta is small, the
probability of 2 arrivals can
313
00:15:35,560 --> 00:15:39,660
be approximated by 0.
314
00:15:39,660 --> 00:15:42,460
So this is a description
of what happens during
315
00:15:42,460 --> 00:15:45,240
a small, tiny slot.
316
00:15:45,240 --> 00:15:48,200
Now this is something that's
supposed to be true in some
317
00:15:48,200 --> 00:15:51,530
limiting sense, when delta
is very small.
318
00:15:51,530 --> 00:15:56,070
So the exact version of this
statement would be that this
319
00:15:56,070 --> 00:16:02,050
is an equality, plus order
of delta squared terms.
320
00:16:02,050 --> 00:16:04,060
So this is an approximate
equality.
321
00:16:04,060 --> 00:16:07,850
And what approximation means is
that in the limit of small
322
00:16:07,850 --> 00:16:11,970
deltas, the dominant terms--
323
00:16:11,970 --> 00:16:15,900
the constant and the first order
term are given by this.
324
00:16:15,900 --> 00:16:19,760
Now when delta is very small,
second order terms in delta do
325
00:16:19,760 --> 00:16:21,130
not matter.
326
00:16:21,130 --> 00:16:24,380
They are small compared
to first order terms.
327
00:16:24,380 --> 00:16:26,190
So we ignore this.
328
00:16:26,190 --> 00:16:30,280
So you can either think in terms
of an exact relation,
329
00:16:30,280 --> 00:16:34,640
which is the probabilities are
given by this, plus delta
330
00:16:34,640 --> 00:16:35,990
squared terms.
331
00:16:35,990 --> 00:16:38,730
Or if you want to be a little
more loose, you just write
332
00:16:38,730 --> 00:16:41,000
here, as an approximate
equality.
333
00:16:41,000 --> 00:16:44,080
And the understanding is that
this equality holds--
334
00:16:44,080 --> 00:16:50,850
approximately becomes more
and more correct as
335
00:16:50,850 --> 00:16:53,410
delta goes to 0.
336
00:16:53,410 --> 00:16:57,250
So another version of that
statement would be that if you
337
00:16:57,250 --> 00:17:03,280
take the limit as delta goes to
0, of p, the probability of
338
00:17:03,280 --> 00:17:06,829
having 1 arrival in an interval
of length delta,
339
00:17:06,829 --> 00:17:10,300
divided by delta, this
is equal to lambda.
340
00:17:10,300 --> 00:17:16,010
So that would be one version of
an exact statement of what
341
00:17:16,010 --> 00:17:19,250
we are assuming here.
342
00:17:19,250 --> 00:17:22,750
So this lambda, we call it the
arrival rate, or the intensity
343
00:17:22,750 --> 00:17:23,930
of the process.
344
00:17:23,930 --> 00:17:27,349
And clearly, if you double
lambda, then a little interval
345
00:17:27,349 --> 00:17:29,340
is likely --
346
00:17:29,340 --> 00:17:31,630
you expect to get --
347
00:17:31,630 --> 00:17:34,200
the probability of obtaining
an arrival during that
348
00:17:34,200 --> 00:17:35,740
interval has doubled.
349
00:17:35,740 --> 00:17:40,400
So in some sense we have twice
as intense arrival process.
350
00:17:40,400 --> 00:17:46,470
If you look at the number
of arrivals during delta
351
00:17:46,470 --> 00:17:54,490
interval, what is the expected
value of that random variable?
352
00:17:54,490 --> 00:18:00,100
Well with probability lambda
delta we get 1 arrival.
353
00:18:00,100 --> 00:18:01,240
And with the remaining
354
00:18:01,240 --> 00:18:03,610
probability, we get 0 arrivals.
355
00:18:03,610 --> 00:18:06,680
So it's just lambda
times delta.
356
00:18:06,680 --> 00:18:10,640
So expected number of arrivals
during a little interval is
357
00:18:10,640 --> 00:18:12,170
lambda times delta.
358
00:18:12,170 --> 00:18:15,460
So expected number of arrivals
is proportional to lambda, and
359
00:18:15,460 --> 00:18:19,050
that's again why we call lambda
the arrival rate.
360
00:18:19,050 --> 00:18:22,820
If you send delta to the
denominator in this equality,
361
00:18:22,820 --> 00:18:26,380
it tells you that lambda is
the expected number of
362
00:18:26,380 --> 00:18:30,000
arrivals per unit time.
363
00:18:30,000 --> 00:18:37,010
So the arrival rate is expected
number of arrivals
364
00:18:37,010 --> 00:18:38,750
per unit time.
365
00:18:38,750 --> 00:18:42,580
And again, that justifies why
we call lambda the intensity
366
00:18:42,580 --> 00:18:43,830
of this process.
367
00:18:43,830 --> 00:18:46,316
368
00:18:46,316 --> 00:18:46,760
All right.
369
00:18:46,760 --> 00:18:49,680
So where are we now?
370
00:18:49,680 --> 00:18:53,740
For the Bernoulli process, the
number of arrivals during a
371
00:18:53,740 --> 00:19:00,210
given interval of length n had
the PMF that we knew it was
372
00:19:00,210 --> 00:19:01,545
the binomial PMF.
373
00:19:01,545 --> 00:19:04,190
374
00:19:04,190 --> 00:19:07,530
What is the formula for the
corresponding PMF for the
375
00:19:07,530 --> 00:19:09,190
continuous time process?
376
00:19:09,190 --> 00:19:12,570
Somehow we would like to use
our assumptions and come up
377
00:19:12,570 --> 00:19:16,100
with the formula for
this quantity.
378
00:19:16,100 --> 00:19:19,110
So this tells us about the
distribution of number of
379
00:19:19,110 --> 00:19:23,750
arrivals during an interval
of some general length.
380
00:19:23,750 --> 00:19:27,690
We have made assumptions about
the number of arrivals during
381
00:19:27,690 --> 00:19:30,410
an interval of small length.
382
00:19:30,410 --> 00:19:34,350
An interval of big length is
composed of many intervals of
383
00:19:34,350 --> 00:19:37,830
small length, so maybe this
is the way to go.
384
00:19:37,830 --> 00:19:43,120
Take a big interval, and split
it into many intervals of
385
00:19:43,120 --> 00:19:44,970
small length.
386
00:19:44,970 --> 00:19:48,410
So we have here our time axis.
387
00:19:48,410 --> 00:19:51,480
And we have an interval
of length tau.
388
00:19:51,480 --> 00:19:55,240
And I'm going to split it into
lots of little intervals of
389
00:19:55,240 --> 00:19:56,580
length delta.
390
00:19:56,580 --> 00:19:59,000
So how many intervals are
we going to have?
391
00:19:59,000 --> 00:20:03,060
The number of intervals is going
to be the total time,
392
00:20:03,060 --> 00:20:04,620
divided by delta.
393
00:20:04,620 --> 00:20:07,520
394
00:20:07,520 --> 00:20:12,960
Now what happens during each one
of these little intervals?
395
00:20:12,960 --> 00:20:22,380
As long as the intervals are
small, what you have is that
396
00:20:22,380 --> 00:20:24,240
during an interval, you're
going to have
397
00:20:24,240 --> 00:20:27,220
either 0 or 1 arrival.
398
00:20:27,220 --> 00:20:29,940
The probability of more than
1 arrival during a little
399
00:20:29,940 --> 00:20:31,950
interval is negligible.
400
00:20:31,950 --> 00:20:35,380
So with this picture, you have
essentially a Bernoulli
401
00:20:35,380 --> 00:20:39,970
process that consists
of so many trials.
402
00:20:39,970 --> 00:20:43,160
And during each one of those
trials, we have a probability
403
00:20:43,160 --> 00:20:46,730
of success, which is
lambda times delta.
404
00:20:46,730 --> 00:20:51,845
405
00:20:51,845 --> 00:20:54,330
Different little intervals
here are
406
00:20:54,330 --> 00:20:56,140
independent of each other.
407
00:20:56,140 --> 00:20:58,670
That's one of our assumptions,
that these joint time
408
00:20:58,670 --> 00:21:00,380
intervals are independent.
409
00:21:00,380 --> 00:21:05,590
So approximately, what we have
is a Bernoulli process.
410
00:21:05,590 --> 00:21:06,980
We have independence.
411
00:21:06,980 --> 00:21:09,250
We have the number of
slots of interest.
412
00:21:09,250 --> 00:21:11,450
And during each one of the
slots we have a certain
413
00:21:11,450 --> 00:21:13,530
probability of success.
414
00:21:13,530 --> 00:21:17,300
So if we think of this as
another good approximation of
415
00:21:17,300 --> 00:21:18,870
the Poisson process--
416
00:21:18,870 --> 00:21:21,090
with the approximation becoming
more and more
417
00:21:21,090 --> 00:21:23,595
accurate as delta goes to 0 --
418
00:21:23,595 --> 00:21:28,150
what we should do would be to
take the formula for the PMF
419
00:21:28,150 --> 00:21:32,320
of number of arrivals in a
Bernoulli process, and then
420
00:21:32,320 --> 00:21:37,230
take the limit as
delta goes to 0.
421
00:21:37,230 --> 00:21:45,260
So in the Bernoulli process, the
probability of k arrivals
422
00:21:45,260 --> 00:21:52,730
is n choose k, and then
you have p to the k.
423
00:21:52,730 --> 00:21:57,610
Now in our case, we have here
lambda times delta, delta is
424
00:21:57,610 --> 00:21:59,340
tau over n.
425
00:21:59,340 --> 00:22:02,190
426
00:22:02,190 --> 00:22:08,410
Delta is tau over n, so p is
lambda times tau divided by n.
427
00:22:08,410 --> 00:22:11,010
So here's our p --
428
00:22:11,010 --> 00:22:13,690
Lambda tau over n --
429
00:22:13,690 --> 00:22:22,760
to the power k, and then times
one minus this-- this is our
430
00:22:22,760 --> 00:22:24,540
one minus p--
431
00:22:24,540 --> 00:22:25,790
to the power n-k.
432
00:22:25,790 --> 00:22:30,010
433
00:22:30,010 --> 00:22:35,730
So this is the exact formula
for the Bernoulli process.
434
00:22:35,730 --> 00:22:39,830
For the Poisson process, what we
do is we take that formula
435
00:22:39,830 --> 00:22:43,360
and we let delta go to 0.
436
00:22:43,360 --> 00:22:48,150
As delta goes to 0, n
goes to infinity.
437
00:22:48,150 --> 00:22:51,280
So that's the limit
that we're taking.
438
00:22:51,280 --> 00:22:55,580
On the other hand, this
expression lambda times tau--
439
00:22:55,580 --> 00:22:59,730
440
00:22:59,730 --> 00:23:03,740
lambda times tau, what
is it going to be?
441
00:23:03,740 --> 00:23:06,860
Lambda times tau is equal
to n times p.
442
00:23:06,860 --> 00:23:09,900
443
00:23:09,900 --> 00:23:11,990
n times p, is that
what I want?
444
00:23:11,990 --> 00:23:21,300
445
00:23:21,300 --> 00:23:22,550
No, let's see.
446
00:23:22,550 --> 00:23:26,600
447
00:23:26,600 --> 00:23:28,110
Lambda tau is np.
448
00:23:28,110 --> 00:23:29,990
Yeah.
449
00:23:29,990 --> 00:23:32,060
So lambda tau is np.
450
00:23:32,060 --> 00:23:53,370
451
00:23:53,370 --> 00:23:54,030
All right.
452
00:23:54,030 --> 00:23:59,320
So we have this relation,
lambda tau equals np.
453
00:23:59,320 --> 00:24:03,070
These two numbers being equal
kind of makes sense. np is the
454
00:24:03,070 --> 00:24:05,890
expected number of successes
you're going to get in the
455
00:24:05,890 --> 00:24:07,750
Bernoulli process.
456
00:24:07,750 --> 00:24:08,780
Lambda tau--
457
00:24:08,780 --> 00:24:11,800
since lambda is the arrival rate
and you have a total time
458
00:24:11,800 --> 00:24:15,710
of tau, lambda tau you can think
of it as the number of
459
00:24:15,710 --> 00:24:19,750
expected arrivals in the
Bernoulli process.
460
00:24:19,750 --> 00:24:22,000
We're doing a Bernoulli
approximation
461
00:24:22,000 --> 00:24:23,250
to the Poisson process.
462
00:24:23,250 --> 00:24:26,150
We take the formula for the
Bernoulli, and now take the
463
00:24:26,150 --> 00:24:30,060
limit as n goes to infinity.
464
00:24:30,060 --> 00:24:35,330
Now lambda tau over n is equal
to p, so it's clear what this
465
00:24:35,330 --> 00:24:37,040
term is going to give us.
466
00:24:37,040 --> 00:24:39,695
This is just p to the power k.
467
00:24:39,695 --> 00:24:48,820
468
00:24:48,820 --> 00:24:53,230
It will actually take a little
more work than that.
469
00:24:53,230 --> 00:24:58,210
Now I'm not going to do the
algebra, but I'm just telling
470
00:24:58,210 --> 00:25:03,390
you that one can take the limit
in this formula here, as
471
00:25:03,390 --> 00:25:05,000
n goes to infinity.
472
00:25:05,000 --> 00:25:08,860
And that will give you another
formula, the final formula for
473
00:25:08,860 --> 00:25:10,320
the Poisson PMF.
474
00:25:10,320 --> 00:25:13,400
One thing to notice is that here
you have something like 1
475
00:25:13,400 --> 00:25:17,460
minus a constant over
n, to the power n.
476
00:25:17,460 --> 00:25:21,630
And you may recall from calculus
a formula of this
477
00:25:21,630 --> 00:25:26,540
kind, that this converges
to e to the minus c.
478
00:25:26,540 --> 00:25:29,560
If you remember that formula
from calculus, then you will
479
00:25:29,560 --> 00:25:32,750
expect that here, in the limit,
you are going to get
480
00:25:32,750 --> 00:25:36,520
something like an e to
the minus lambda tau.
481
00:25:36,520 --> 00:25:39,180
So indeed, we will
get such a term.
482
00:25:39,180 --> 00:25:42,230
There is some work that needs
to be done to find the limit
483
00:25:42,230 --> 00:25:45,880
of this expression, times
that expression.
484
00:25:45,880 --> 00:25:48,690
The algebra is not hard,
it's in the text.
485
00:25:48,690 --> 00:25:51,340
Let's not spend more
time doing this.
486
00:25:51,340 --> 00:25:53,820
But let me just give you
the formula of what
487
00:25:53,820 --> 00:25:55,620
comes at the end.
488
00:25:55,620 --> 00:25:59,720
And the formula that comes at
the end is of this form.
489
00:25:59,720 --> 00:26:03,710
So what matters here is not so
much the specific algebra that
490
00:26:03,710 --> 00:26:07,690
you will do to go from this
formula to that one.
491
00:26:07,690 --> 00:26:09,370
It's kind of straightforward.
492
00:26:09,370 --> 00:26:14,535
What's important is the idea
that the Poisson process, by
493
00:26:14,535 --> 00:26:19,710
definition, can be approximated
by a Bernoulli
494
00:26:19,710 --> 00:26:25,040
process in which we have a very
large number of slots--
495
00:26:25,040 --> 00:26:27,970
n goes to infinity.
496
00:26:27,970 --> 00:26:32,600
Whereas we have a very small
probability of success during
497
00:26:32,600 --> 00:26:34,420
each time slot.
498
00:26:34,420 --> 00:26:38,640
So a large number of slots,
but tiny probability of
499
00:26:38,640 --> 00:26:40,480
success during each slot.
500
00:26:40,480 --> 00:26:42,370
And we take the limit
as the slots
501
00:26:42,370 --> 00:26:44,680
become smaller and smaller.
502
00:26:44,680 --> 00:26:47,170
So with this approximation
we end up with
503
00:26:47,170 --> 00:26:49,030
this particular formula.
504
00:26:49,030 --> 00:26:51,890
And this is the so-called
Poisson PMF.
505
00:26:51,890 --> 00:26:53,930
Now this function P here --
506
00:26:53,930 --> 00:26:55,190
has two arguments.
507
00:26:55,190 --> 00:26:58,320
The important thing to realize
is that when you think of this
508
00:26:58,320 --> 00:27:02,900
as a PMF, you fix t to tau.
509
00:27:02,900 --> 00:27:06,010
And for a fixed tau,
now this is a PMF.
510
00:27:06,010 --> 00:27:11,260
As I said before, the sum over
k has to be equal to 1.
511
00:27:11,260 --> 00:27:15,510
So for a given tau, these
probabilities add up to 1.
512
00:27:15,510 --> 00:27:20,590
The formula is moderately messy,
but not too messy.
513
00:27:20,590 --> 00:27:24,570
One can work with it without
too much pain.
514
00:27:24,570 --> 00:27:28,460
And what's the mean and
variance of this PMF?
515
00:27:28,460 --> 00:27:31,560
Well what's the expected
number of arrivals?
516
00:27:31,560 --> 00:27:35,680
If you think of this Bernoulli
analogy, we know that the
517
00:27:35,680 --> 00:27:37,940
expected number of arrivals
in the Bernoulli
518
00:27:37,940 --> 00:27:41,250
process is n times p.
519
00:27:41,250 --> 00:27:44,500
In the approximation that
we're using in these
520
00:27:44,500 --> 00:27:48,170
procedure, n times p is the
same as lambda tau.
521
00:27:48,170 --> 00:27:52,490
And that's why we get lambda tau
to be the expected number
522
00:27:52,490 --> 00:27:53,480
of arrivals.
523
00:27:53,480 --> 00:27:56,140
Here I'm using t
instead of tau.
524
00:27:56,140 --> 00:28:01,670
The expected number of
arrivals is lambda t.
525
00:28:01,670 --> 00:28:05,450
So if you double the time,
you expect to get
526
00:28:05,450 --> 00:28:07,290
twice as many arrivals.
527
00:28:07,290 --> 00:28:10,960
If you double the arrival rate,
you expect to get twice
528
00:28:10,960 --> 00:28:12,760
as many arrivals.
529
00:28:12,760 --> 00:28:15,290
How about the formula
for the variance?
530
00:28:15,290 --> 00:28:18,930
The variance of the Bernoulli
process is np,
531
00:28:18,930 --> 00:28:22,720
times one minus p.
532
00:28:22,720 --> 00:28:25,800
What does this go
to in the limit?
533
00:28:25,800 --> 00:28:31,170
In the limit that we're taking,
as delta goes to zero,
534
00:28:31,170 --> 00:28:33,550
then p also goes to zero.
535
00:28:33,550 --> 00:28:37,270
The probability of success in
any given slot goes to zero.
536
00:28:37,270 --> 00:28:39,700
So this term becomes
insignificant.
537
00:28:39,700 --> 00:28:45,980
So this becomes n times p, which
is again lambda t, or
538
00:28:45,980 --> 00:28:47,710
lambda tau.
539
00:28:47,710 --> 00:28:50,960
So the variance, instead of
having this more complicated
540
00:28:50,960 --> 00:28:54,290
formula of the variance is the
Bernoulli process, here it
541
00:28:54,290 --> 00:28:56,840
gets simplified and
it's lambda t.
542
00:28:56,840 --> 00:29:00,580
So interestingly, the variance
in the Poisson process is
543
00:29:00,580 --> 00:29:03,200
exactly the same as the
expected value.
544
00:29:03,200 --> 00:29:06,360
So you can look at this as
just some interesting
545
00:29:06,360 --> 00:29:07,780
coincidence.
546
00:29:07,780 --> 00:29:10,260
So now we're going to take
this formula and
547
00:29:10,260 --> 00:29:11,370
see how to use it.
548
00:29:11,370 --> 00:29:14,060
First we're going to do
a completely trivial,
549
00:29:14,060 --> 00:29:16,560
straightforward example.
550
00:29:16,560 --> 00:29:24,630
So 15 years ago when that
example was made, email was
551
00:29:24,630 --> 00:29:27,230
coming at a rate of five
messages per hour.
552
00:29:27,230 --> 00:29:30,660
I wish that was the
case today.
553
00:29:30,660 --> 00:29:38,450
And now emails that are coming
in, let's say during the day--
554
00:29:38,450 --> 00:29:41,750
the arrival rates of emails
are probably different in
555
00:29:41,750 --> 00:29:42,960
different times of the day.
556
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
557
00:29:46,840 --> 00:29:49,370
afternoon, there's probably
a constant rate.
558
00:29:49,370 --> 00:29:53,050
And email arrivals are
reasonably well modeled by a
559
00:29:53,050 --> 00:29:54,790
Poisson process.
560
00:29:54,790 --> 00:29:58,220
Speaking of modeling, it's
not just email arrivals.
561
00:29:58,220 --> 00:30:02,290
Whenever arrivals happen in a
completely random way, without
562
00:30:02,290 --> 00:30:05,370
any additional structure, the
Poisson process is a good
563
00:30:05,370 --> 00:30:07,010
model of these arrivals.
564
00:30:07,010 --> 00:30:10,200
So the times at which car
accidents will happen, that's
565
00:30:10,200 --> 00:30:11,450
a Poisson processes.
566
00:30:11,450 --> 00:30:15,530
567
00:30:15,530 --> 00:30:19,550
If you have a very, very weak
light source that's shooting
568
00:30:19,550 --> 00:30:24,290
out photons, just one at a time,
the times at which these
569
00:30:24,290 --> 00:30:27,240
photons will go out is
well modeled again
570
00:30:27,240 --> 00:30:28,670
by a Poisson process.
571
00:30:28,670 --> 00:30:30,540
So it's completely random.
572
00:30:30,540 --> 00:30:35,230
Or if you have a radioactive
material where one atom at a
573
00:30:35,230 --> 00:30:43,720
time changes at random times.
574
00:30:43,720 --> 00:30:45,920
So it's a very slow
radioactive decay.
575
00:30:45,920 --> 00:30:48,900
The time at which these alpha
particles, or whatever we get
576
00:30:48,900 --> 00:30:51,580
emitted, again is going
to be described
577
00:30:51,580 --> 00:30:53,200
by a Poisson process.
578
00:30:53,200 --> 00:30:58,220
So if you have arrivals, or
emissions, that happen at
579
00:30:58,220 --> 00:31:02,660
completely random times, and
once in a while you get an
580
00:31:02,660 --> 00:31:07,500
arrival or an event, then the
Poisson process is a very good
581
00:31:07,500 --> 00:31:10,070
model for these events.
582
00:31:10,070 --> 00:31:12,200
So back to emails.
583
00:31:12,200 --> 00:31:16,350
Get them at a rate of five
messages per day, per hour.
584
00:31:16,350 --> 00:31:19,420
In 30 minutes this
is half an hour.
585
00:31:19,420 --> 00:31:23,770
So what we have is that
lambda t, total
586
00:31:23,770 --> 00:31:26,520
number of arrivals is--
587
00:31:26,520 --> 00:31:29,020
the expected number
of arrivals is--
588
00:31:29,020 --> 00:31:33,810
lambda is five, t is one-half,
if we talk about hours.
589
00:31:33,810 --> 00:31:36,480
So lambda t is two to the 0.5.
590
00:31:36,480 --> 00:31:41,220
The probability of no new
messages is the probability of
591
00:31:41,220 --> 00:31:48,560
zero, in time interval of length
t, which, in our case,
592
00:31:48,560 --> 00:31:51,790
is one-half.
593
00:31:51,790 --> 00:31:55,510
And then we look back into the
formula from the previous
594
00:31:55,510 --> 00:31:59,550
slide, and the probability of
zero arrivals is lambda t to
595
00:31:59,550 --> 00:32:03,770
the power zero, divided by zero
factorial, and then an e
596
00:32:03,770 --> 00:32:05,450
to the lambda t.
597
00:32:05,450 --> 00:32:07,840
And you plug in the numbers
that we have.
598
00:32:07,840 --> 00:32:10,380
Lambda t to the zero
power is one.
599
00:32:10,380 --> 00:32:12,040
Zero factorial is one.
600
00:32:12,040 --> 00:32:15,500
So we're left with e
to the minus 2.5.
601
00:32:15,500 --> 00:32:18,860
And that number is 0.08.
602
00:32:18,860 --> 00:32:22,090
Similarly, you can ask for the
probability that you get
603
00:32:22,090 --> 00:32:24,850
exactly one message
in half an hour.
604
00:32:24,850 --> 00:32:27,420
And that would be-- the
probability of one message in
605
00:32:27,420 --> 00:32:28,680
one-half an hour--
606
00:32:28,680 --> 00:32:32,590
is going to be lambda t to the
first power, divided by 1
607
00:32:32,590 --> 00:32:38,230
factorial, e to the minus
lambda t, which--
608
00:32:38,230 --> 00:32:41,900
as we now get the extra lambda t
factor-- is going to be 2.5,
609
00:32:41,900 --> 00:32:43,650
e to the minus 2.5.
610
00:32:43,650 --> 00:32:46,930
And the numerical
answer is 0.20.
611
00:32:46,930 --> 00:32:50,450
So this is how you use the PMF
formula for the Poisson
612
00:32:50,450 --> 00:32:55,540
distribution that we had
in the previous slide.
613
00:32:55,540 --> 00:32:55,890
All right.
614
00:32:55,890 --> 00:33:00,010
So this was all about
the distribution of
615
00:33:00,010 --> 00:33:01,780
the number of arrivals.
616
00:33:01,780 --> 00:33:03,350
What else did we do last time?
617
00:33:03,350 --> 00:33:08,250
Last time we also talked about
the time it takes until the
618
00:33:08,250 --> 00:33:09,500
k-th arrival.
619
00:33:09,500 --> 00:33:12,390
620
00:33:12,390 --> 00:33:12,790
OK.
621
00:33:12,790 --> 00:33:16,020
So let's try to figure out
something about this
622
00:33:16,020 --> 00:33:18,260
particular distribution.
623
00:33:18,260 --> 00:33:21,180
We can derive the distribution
of the time of the k-th
624
00:33:21,180 --> 00:33:24,730
arrival by using the
exact same argument
625
00:33:24,730 --> 00:33:27,360
as we did last time.
626
00:33:27,360 --> 00:33:31,650
So now the time of the
k-th arrival is a
627
00:33:31,650 --> 00:33:33,830
continuous random variable.
628
00:33:33,830 --> 00:33:36,160
So it has a PDF.
629
00:33:36,160 --> 00:33:38,430
Since we are in continuous
time, arrivals can
630
00:33:38,430 --> 00:33:39,900
happen at any time.
631
00:33:39,900 --> 00:33:42,310
So Yk is a continuous
random variable.
632
00:33:42,310 --> 00:33:45,200
633
00:33:45,200 --> 00:33:48,160
But now let's think of
a time interval of
634
00:33:48,160 --> 00:33:49,410
length little delta.
635
00:33:49,410 --> 00:33:52,370
636
00:33:52,370 --> 00:33:58,620
And use our usual interpretation
of PDFs.
637
00:33:58,620 --> 00:34:03,180
The PDF of a random variable
evaluated at a certain time
638
00:34:03,180 --> 00:34:08,010
times delta, this is the
probability that the Yk falls
639
00:34:08,010 --> 00:34:09,514
in this little interval.
640
00:34:09,514 --> 00:34:13,460
641
00:34:13,460 --> 00:34:16,639
So as I've said before, this
is the best way of thinking
642
00:34:16,639 --> 00:34:18,420
about PDFs.
643
00:34:18,420 --> 00:34:22,179
PDFs give you probabilities
of little intervals.
644
00:34:22,179 --> 00:34:25,540
So now let's try to calculate
this probability.
645
00:34:25,540 --> 00:34:29,880
For the k-th arrival to happen
inside this little interval,
646
00:34:29,880 --> 00:34:31,550
we need two things.
647
00:34:31,550 --> 00:34:35,790
We need an arrival to happen in
this interval, and we need
648
00:34:35,790 --> 00:34:41,530
k minus one arrivals to happen
during that interval.
649
00:34:41,530 --> 00:34:41,880
OK.
650
00:34:41,880 --> 00:34:45,469
You'll tell me, but it's
possible that we might have
651
00:34:45,469 --> 00:34:50,130
the k minus one arrival happen
here, and the k-th arrival to
652
00:34:50,130 --> 00:34:51,219
happen here.
653
00:34:51,219 --> 00:34:53,050
In principle, that's possible.
654
00:34:53,050 --> 00:34:56,139
But in the limit, when we take
delta very small, the
655
00:34:56,139 --> 00:34:59,850
probability of having two
arrivals in the same little
656
00:34:59,850 --> 00:35:01,830
slot is negligible.
657
00:35:01,830 --> 00:35:06,870
So assuming that no two arrivals
can happen in the
658
00:35:06,870 --> 00:35:10,940
same mini slot, then for the
k-th one to happen here, we
659
00:35:10,940 --> 00:35:15,710
must have k minus one during
this interval.
660
00:35:15,710 --> 00:35:20,210
Now because we have assumed that
these joint intervals are
661
00:35:20,210 --> 00:35:23,900
independent of each other,
this breaks down into the
662
00:35:23,900 --> 00:35:33,070
probability that we have exactly
k minus one arrivals,
663
00:35:33,070 --> 00:35:37,600
during the interval from zero to
t, times the probability of
664
00:35:37,600 --> 00:35:41,410
exactly one arrival during that
little interval, which is
665
00:35:41,410 --> 00:35:43,420
lambda delta.
666
00:35:43,420 --> 00:35:51,010
We do have a formula for this
from the previous slide, which
667
00:35:51,010 --> 00:35:59,340
is lambda t, to the k minus 1,
over k minus one factorial,
668
00:35:59,340 --> 00:36:07,190
times e to minus lambda t.
669
00:36:07,190 --> 00:36:09,070
And then lambda times delta.
670
00:36:09,070 --> 00:36:14,910
671
00:36:14,910 --> 00:36:16,160
Did I miss something?
672
00:36:16,160 --> 00:36:24,680
673
00:36:24,680 --> 00:36:26,310
Yeah, OK.
674
00:36:26,310 --> 00:36:26,970
All right.
675
00:36:26,970 --> 00:36:30,220
And now you cancel this
delta with that delta.
676
00:36:30,220 --> 00:36:36,820
And that gives us a formula for
the PDF of the time until
677
00:36:36,820 --> 00:36:39,170
the k-th arrival.
678
00:36:39,170 --> 00:36:43,290
This PDF, of course, depends
on the number k.
679
00:36:43,290 --> 00:36:46,850
The first arrival is going
to happen somewhere in
680
00:36:46,850 --> 00:36:48,040
this range of time.
681
00:36:48,040 --> 00:36:50,140
So this is the PDF
that it has.
682
00:36:50,140 --> 00:36:53,170
The second arrival, of course,
is going to happen later.
683
00:36:53,170 --> 00:36:54,860
And the PDF is this.
684
00:36:54,860 --> 00:36:57,880
So it's more likely to happen
around these times.
685
00:36:57,880 --> 00:37:01,410
The third arrival has this PDF,
so it's more likely to
686
00:37:01,410 --> 00:37:03,690
happen around those times.
687
00:37:03,690 --> 00:37:08,020
And if you were to take
k equal to 100,
688
00:37:08,020 --> 00:37:10,470
you might get a PDF--
689
00:37:10,470 --> 00:37:13,260
it's extremely unlikely that
the k-th arrival happens in
690
00:37:13,260 --> 00:37:18,060
the beginning, and it might
happen somewhere down there,
691
00:37:18,060 --> 00:37:20,010
far into the future.
692
00:37:20,010 --> 00:37:22,230
So depending on which particular
arrival we're
693
00:37:22,230 --> 00:37:25,510
talking about, it has a
different probability
694
00:37:25,510 --> 00:37:26,350
distribution.
695
00:37:26,350 --> 00:37:30,340
The time of the 100th arrival,
of course, is expected to be a
696
00:37:30,340 --> 00:37:34,100
lot larger than the time
of the first arrival.
697
00:37:34,100 --> 00:37:38,550
Incidentally, the time of the
first arrival has a PDF whose
698
00:37:38,550 --> 00:37:40,160
form is quite simple.
699
00:37:40,160 --> 00:37:43,850
If you let k equal to one here,
this term disappears.
700
00:37:43,850 --> 00:37:46,310
That term becomes a one.
701
00:37:46,310 --> 00:37:49,880
You're left with just lambda,
e to the minus lambda.
702
00:37:49,880 --> 00:37:53,210
And you recognize it, it's the
exponential distribution.
703
00:37:53,210 --> 00:37:57,210
So the time until the first
arrival in a Poisson process
704
00:37:57,210 --> 00:38:00,160
is an exponential
distribution.
705
00:38:00,160 --> 00:38:02,150
What was the time of the
first arrival in
706
00:38:02,150 --> 00:38:03,970
the Bernoulli process?
707
00:38:03,970 --> 00:38:07,060
It was a geometric
distribution.
708
00:38:07,060 --> 00:38:11,170
Well, not coincidentally, these
two look quite a bit
709
00:38:11,170 --> 00:38:13,030
like the other.
710
00:38:13,030 --> 00:38:17,980
A geometric distribution
has this kind of shape.
711
00:38:17,980 --> 00:38:21,900
The exponential distribution
has that kind of shape.
712
00:38:21,900 --> 00:38:25,560
The geometric is just a discrete
version of the
713
00:38:25,560 --> 00:38:27,090
exponential.
714
00:38:27,090 --> 00:38:29,860
In the Bernoulli case, we
are in discrete time.
715
00:38:29,860 --> 00:38:32,540
We have a PMF for the
time of the first
716
00:38:32,540 --> 00:38:35,080
arrival, which is geometric.
717
00:38:35,080 --> 00:38:38,540
In the Poisson case, what we
get is the limit of the
718
00:38:38,540 --> 00:38:41,560
geometric as you let those
lines become closer and
719
00:38:41,560 --> 00:38:46,480
closer, which gives you the
exponential distribution.
720
00:38:46,480 --> 00:38:50,430
Now the Poisson process shares
all the memorylessness
721
00:38:50,430 --> 00:38:52,870
properties of the Bernoulli
process.
722
00:38:52,870 --> 00:38:56,750
And the way one can argue is
just in terms of this picture.
723
00:38:56,750 --> 00:39:00,250
Since the Poisson process is
the limit of Bernoulli
724
00:39:00,250 --> 00:39:03,570
processes, whatever qualitative
processes you have
725
00:39:03,570 --> 00:39:07,340
in the Bernoulli process
remain valid
726
00:39:07,340 --> 00:39:08,360
for the Poisson process.
727
00:39:08,360 --> 00:39:11,470
In particular we have this
memorylessness property.
728
00:39:11,470 --> 00:39:15,120
You let the Poisson process run
for some time, and then
729
00:39:15,120 --> 00:39:16,600
you start watching it.
730
00:39:16,600 --> 00:39:18,520
What ever happened in
the past has no
731
00:39:18,520 --> 00:39:20,220
bearing about the future.
732
00:39:20,220 --> 00:39:23,150
Starting from right now, what's
going to happen in the
733
00:39:23,150 --> 00:39:27,330
future is described again by a
Poisson process, in the sense
734
00:39:27,330 --> 00:39:30,530
that during every little slot of
length delta, there's going
735
00:39:30,530 --> 00:39:33,790
to be a probability of lambda
delta of having an arrival.
736
00:39:33,790 --> 00:39:36,590
And that probably lambda
delta is the same-- is
737
00:39:36,590 --> 00:39:38,070
always lambda delta--
738
00:39:38,070 --> 00:39:41,270
no matter what happened in
the past of the process.
739
00:39:41,270 --> 00:39:47,040
And in particular, we could use
this argument to say that
740
00:39:47,040 --> 00:39:50,460
the time until the k-th arrival
is the time that it
741
00:39:50,460 --> 00:39:53,720
takes for the first
arrival to happen.
742
00:39:53,720 --> 00:39:56,380
OK, let me do it for
k equal to two.
743
00:39:56,380 --> 00:39:59,630
And then after the first arrival
happens, you wait a
744
00:39:59,630 --> 00:40:02,600
certain amount of time until
the second arrival happens.
745
00:40:02,600 --> 00:40:06,410
Now once the first arrival
happened, that's in the past.
746
00:40:06,410 --> 00:40:07,400
You start watching.
747
00:40:07,400 --> 00:40:10,690
From now on you have mini slots
of length delta, each
748
00:40:10,690 --> 00:40:13,230
one having a probability of
success lambda delta.
749
00:40:13,230 --> 00:40:16,230
It's as if we started the
Poisson process from scratch.
750
00:40:16,230 --> 00:40:19,310
So starting from that time,
the time until the next
751
00:40:19,310 --> 00:40:22,840
arrival is going to be again an
exponential distribution,
752
00:40:22,840 --> 00:40:26,010
which doesn't care about what
happened in the past, how long
753
00:40:26,010 --> 00:40:28,000
it took you for the
first arrival.
754
00:40:28,000 --> 00:40:33,410
So these two random variables
are going to be independent
755
00:40:33,410 --> 00:40:38,140
and exponential, with the
same parameter lambda.
756
00:40:38,140 --> 00:40:42,570
So among other things, what we
have done here is we have
757
00:40:42,570 --> 00:40:48,130
essentially derived the PDF of
the sum of k independent
758
00:40:48,130 --> 00:40:49,320
exponentials.
759
00:40:49,320 --> 00:40:53,990
The time of the k-th arrival
is the sum of k
760
00:40:53,990 --> 00:40:56,230
inter-arrival times.
761
00:40:56,230 --> 00:40:59,380
The inter-arrival times are all
independent of each other
762
00:40:59,380 --> 00:41:01,450
because of memorylessness.
763
00:41:01,450 --> 00:41:04,245
And they all have the same
exponential distribution.
764
00:41:04,245 --> 00:41:07,130
765
00:41:07,130 --> 00:41:08,980
And by the way, this
gives you a way to
766
00:41:08,980 --> 00:41:11,080
simulate the Poisson process.
767
00:41:11,080 --> 00:41:14,070
If you wanted to simulate it
on your computer, you would
768
00:41:14,070 --> 00:41:20,140
have one option to break time
into tiny, tiny slots.
769
00:41:20,140 --> 00:41:24,030
And for every tiny slot, use
your random number generator
770
00:41:24,030 --> 00:41:27,520
to decide whether there
was an arrival or not.
771
00:41:27,520 --> 00:41:29,810
To get it very accurate,
you would have to
772
00:41:29,810 --> 00:41:32,090
use tiny, tiny slots.
773
00:41:32,090 --> 00:41:35,280
So that would be a lot
of computation.
774
00:41:35,280 --> 00:41:38,530
The more clever way of
simulating the Poisson process
775
00:41:38,530 --> 00:41:42,320
is you use your random number
generator to generate a sample
776
00:41:42,320 --> 00:41:45,280
from an exponential distribution
and call that
777
00:41:45,280 --> 00:41:47,240
your first arrival time.
778
00:41:47,240 --> 00:41:50,050
Then go back to the random
number generator, generate
779
00:41:50,050 --> 00:41:53,040
another independent sample,
again from the same
780
00:41:53,040 --> 00:41:54,780
exponential distribution.
781
00:41:54,780 --> 00:41:58,490
That's the time between the
first and the second arrival,
782
00:41:58,490 --> 00:42:01,390
and you keep going that way.
783
00:42:01,390 --> 00:42:03,260
So as a sort of a
quick summary,
784
00:42:03,260 --> 00:42:04,910
this is the big picture.
785
00:42:04,910 --> 00:42:08,630
This table doesn't tell
you anything new.
786
00:42:08,630 --> 00:42:12,230
But it's good to have it as a
reference, and to look at it,
787
00:42:12,230 --> 00:42:14,740
and to make sure you understand
what all the
788
00:42:14,740 --> 00:42:16,300
different boxes are.
789
00:42:16,300 --> 00:42:18,930
Basically the Bernoulli process
runs in discrete time.
790
00:42:18,930 --> 00:42:20,960
The Poisson process runs
in continuous time.
791
00:42:20,960 --> 00:42:25,140
There's an analogy of arrival
rates, p per trial, or
792
00:42:25,140 --> 00:42:27,270
intensity per unit time.
793
00:42:27,270 --> 00:42:32,190
We did derive, or sketched the
derivation for the PMF of the
794
00:42:32,190 --> 00:42:33,610
number of arrivals.
795
00:42:33,610 --> 00:42:37,810
And the Poisson distribution,
which is the distribution that
796
00:42:37,810 --> 00:42:40,450
we get, this Pk of t.
797
00:42:40,450 --> 00:42:44,220
Pk and t is the limit of the
binomial when we take the
798
00:42:44,220 --> 00:42:49,710
limit in this particular way,
as delta goes to zero, and n
799
00:42:49,710 --> 00:42:51,220
goes to infinity.
800
00:42:51,220 --> 00:42:54,270
The geometric becomes an
exponential in the limit.
801
00:42:54,270 --> 00:42:56,960
And the distribution of the
time of the k-th arrival--
802
00:42:56,960 --> 00:42:59,600
we had a closed form formula
last time for
803
00:42:59,600 --> 00:43:01,050
the Bernoulli process.
804
00:43:01,050 --> 00:43:03,930
We got the closed form
formula this time
805
00:43:03,930 --> 00:43:05,230
for the Poisson process.
806
00:43:05,230 --> 00:43:08,940
And we actually used exactly the
same argument to get these
807
00:43:08,940 --> 00:43:12,320
two closed form formulas.
808
00:43:12,320 --> 00:43:12,650
All right.
809
00:43:12,650 --> 00:43:18,280
So now let's talk about adding
or merging Poisson processes.
810
00:43:18,280 --> 00:43:21,060
And there's two statements
that we can make here.
811
00:43:21,060 --> 00:43:25,970
One has to do with adding
Poisson random variables, just
812
00:43:25,970 --> 00:43:26,820
random variables.
813
00:43:26,820 --> 00:43:28,290
There's another statement about
814
00:43:28,290 --> 00:43:30,770
adding Poisson processes.
815
00:43:30,770 --> 00:43:34,540
And the second is a bigger
statement than the first.
816
00:43:34,540 --> 00:43:36,140
But this is a warm up.
817
00:43:36,140 --> 00:43:39,140
Let's work with the
first statement.
818
00:43:39,140 --> 00:43:42,460
So the claim is that the sum of
independent Poisson random
819
00:43:42,460 --> 00:43:45,340
variables is Poisson.
820
00:43:45,340 --> 00:43:45,990
OK.
821
00:43:45,990 --> 00:43:50,490
So suppose that we have a
Poisson process with rate--
822
00:43:50,490 --> 00:43:51,760
just for simplicity--
823
00:43:51,760 --> 00:43:53,170
lambda one.
824
00:43:53,170 --> 00:43:56,240
And I take the interval
from zero to two.
825
00:43:56,240 --> 00:44:00,620
And that take then the interval
from two until five.
826
00:44:00,620 --> 00:44:03,720
The number of arrivals during
this interval--
827
00:44:03,720 --> 00:44:06,730
let's call it n from
zero to two--
828
00:44:06,730 --> 00:44:13,920
is going to be a Poisson
random variable, with
829
00:44:13,920 --> 00:44:18,240
parameter, or with mean, two.
830
00:44:18,240 --> 00:44:24,340
The number of arrivals during
this interval is n from time
831
00:44:24,340 --> 00:44:26,340
two until five.
832
00:44:26,340 --> 00:44:31,120
This is again a Poisson random
variable with mean equal to
833
00:44:31,120 --> 00:44:34,690
three, because the arrival rate
is 1 and the duration of
834
00:44:34,690 --> 00:44:36,990
the interval is three.
835
00:44:36,990 --> 00:44:41,320
These two random variables
are independent.
836
00:44:41,320 --> 00:44:43,760
They obey the Poisson
distribution
837
00:44:43,760 --> 00:44:45,640
that we derived before.
838
00:44:45,640 --> 00:44:50,930
If you add them, what you get
is the number of arrivals
839
00:44:50,930 --> 00:44:53,850
during the interval
from zero to five.
840
00:44:53,850 --> 00:44:56,290
Now what kind of distribution
does this
841
00:44:56,290 --> 00:44:57,910
random variable have?
842
00:44:57,910 --> 00:45:00,760
Well this is the number of
arrivals over an interval of a
843
00:45:00,760 --> 00:45:03,600
certain length in a
Poisson process.
844
00:45:03,600 --> 00:45:08,580
Therefore, this is also Poisson
with mean five.
845
00:45:08,580 --> 00:45:16,520
846
00:45:16,520 --> 00:45:19,040
Because for the Poisson process
we know that this
847
00:45:19,040 --> 00:45:23,300
number of arrivals is Poisson,
this is Poisson, but also the
848
00:45:23,300 --> 00:45:26,610
number of overall arrivals
is also Poisson.
849
00:45:26,610 --> 00:45:30,040
This establishes that the sum
of a Poisson plus a Poisson
850
00:45:30,040 --> 00:45:32,200
random variable gives
us another
851
00:45:32,200 --> 00:45:33,630
Poisson random variable.
852
00:45:33,630 --> 00:45:37,110
So adding Poisson random
variables gives us a Poisson
853
00:45:37,110 --> 00:45:38,720
random variable.
854
00:45:38,720 --> 00:45:42,660
But now I'm going to make a more
general statement that
855
00:45:42,660 --> 00:45:44,940
it's not just number
of arrivals during
856
00:45:44,940 --> 00:45:46,415
a fixed time interval--
857
00:45:46,415 --> 00:45:50,420
858
00:45:50,420 --> 00:45:53,240
it's not just numbers of
arrivals for given time
859
00:45:53,240 --> 00:45:54,260
intervals--
860
00:45:54,260 --> 00:45:57,770
but rather if you take two
different Poisson processes
861
00:45:57,770 --> 00:46:02,330
and add them up, the process
itself is Poisson in the sense
862
00:46:02,330 --> 00:46:05,930
that this process is going to
satisfy all the assumptions of
863
00:46:05,930 --> 00:46:07,510
a Poisson process.
864
00:46:07,510 --> 00:46:11,060
So the story is that you have
a red bulb that flashes at
865
00:46:11,060 --> 00:46:13,350
random times at the rate
of lambda one.
866
00:46:13,350 --> 00:46:14,980
It's a Poisson process.
867
00:46:14,980 --> 00:46:19,080
You have an independent process
where a green bulb
868
00:46:19,080 --> 00:46:21,230
flashes at random times.
869
00:46:21,230 --> 00:46:24,800
And you happen to be color
blind, so you just see when
870
00:46:24,800 --> 00:46:26,630
something is flashing.
871
00:46:26,630 --> 00:46:29,920
So these two are assumed to be
independent Poisson processes.
872
00:46:29,920 --> 00:46:34,968
What can we say about the
process that you observe?
873
00:46:34,968 --> 00:46:40,250
So in the processes that you
observe, if you take a typical
874
00:46:40,250 --> 00:46:45,170
time interval of length little
delta, what can happen during
875
00:46:45,170 --> 00:46:48,380
that little time interval?
876
00:46:48,380 --> 00:46:55,280
The red process may have
something flashing.
877
00:46:55,280 --> 00:46:56,815
So red flashes.
878
00:46:56,815 --> 00:46:59,850
879
00:46:59,850 --> 00:47:01,580
Or the red does not.
880
00:47:01,580 --> 00:47:06,610
881
00:47:06,610 --> 00:47:12,170
And for the other bulb, the
green bulb, there's two
882
00:47:12,170 --> 00:47:13,020
possibilities.
883
00:47:13,020 --> 00:47:17,910
The green one flashes.
884
00:47:17,910 --> 00:47:20,900
And the other possibility is
that the green does not.
885
00:47:20,900 --> 00:47:24,990
886
00:47:24,990 --> 00:47:25,330
OK.
887
00:47:25,330 --> 00:47:29,070
So there's four possibilities
about what can happen during a
888
00:47:29,070 --> 00:47:31,170
little slot.
889
00:47:31,170 --> 00:47:36,080
The probability that the red one
flashes and the green one
890
00:47:36,080 --> 00:47:39,750
flashes, what is this
probability?
891
00:47:39,750 --> 00:47:43,510
It's lambda one delta that the
first one flashes, and lambda
892
00:47:43,510 --> 00:47:47,290
two delta that the
second one does.
893
00:47:47,290 --> 00:47:50,280
I'm multiplying probabilities
here because I'm making the
894
00:47:50,280 --> 00:47:52,645
assumption that the two
processes are independent.
895
00:47:52,645 --> 00:47:55,330
896
00:47:55,330 --> 00:47:57,330
OK.
897
00:47:57,330 --> 00:48:00,130
Now the probability that
the red one flashes
898
00:48:00,130 --> 00:48:01,440
is lambda one delta.
899
00:48:01,440 --> 00:48:08,210
But the green one doesn't is
one, minus lambda two delta.
900
00:48:08,210 --> 00:48:12,840
Here the probability would be
that the red one does not,
901
00:48:12,840 --> 00:48:16,400
times the probability that
the green one does.
902
00:48:16,400 --> 00:48:20,750
And then here we have the
probability that none of them
903
00:48:20,750 --> 00:48:26,790
flash, which is whatever
is left.
904
00:48:26,790 --> 00:48:29,600
But it's one minus lambda
one delta, times one
905
00:48:29,600 --> 00:48:33,160
minus lambda two delta.
906
00:48:33,160 --> 00:48:36,920
Now we're thinking about
delta as small.
907
00:48:36,920 --> 00:48:43,260
So think of the case where delta
goes to zero, but in a
908
00:48:43,260 --> 00:48:49,840
way that we keep the
first order terms.
909
00:48:49,840 --> 00:48:54,070
We keep the delta terms, but
we throw away the delta
910
00:48:54,070 --> 00:48:55,020
squared terms.
911
00:48:55,020 --> 00:48:58,170
Delta squared terms are much
smaller than the delta terms
912
00:48:58,170 --> 00:49:00,260
when delta becomes small.
913
00:49:00,260 --> 00:49:01,920
If we do that--
914
00:49:01,920 --> 00:49:05,650
if we only keep the order
of delta terms--
915
00:49:05,650 --> 00:49:07,940
this term effectively
disappears.
916
00:49:07,940 --> 00:49:09,110
This is delta squared.
917
00:49:09,110 --> 00:49:11,550
So we make it zero.
918
00:49:11,550 --> 00:49:14,550
So the probability of having
simultaneously a red and a
919
00:49:14,550 --> 00:49:17,940
green flash during a little
interval is negligible.
920
00:49:17,940 --> 00:49:20,150
What do we get here?
921
00:49:20,150 --> 00:49:23,200
Lambda delta times
one survives, but
922
00:49:23,200 --> 00:49:24,910
this times that doesn't.
923
00:49:24,910 --> 00:49:28,820
So we can throw that away.
924
00:49:28,820 --> 00:49:32,190
So the approximation that we
get is lambda one delta.
925
00:49:32,190 --> 00:49:34,010
Similarly here, this
goes away.
926
00:49:34,010 --> 00:49:36,420
We're left with a lambda
two delta.
927
00:49:36,420 --> 00:49:42,140
And this is whatever remains,
whatever is left.
928
00:49:42,140 --> 00:49:45,000
So what do we have?
929
00:49:45,000 --> 00:49:51,400
That there is a probability of
seeing a flash, either a red
930
00:49:51,400 --> 00:49:54,360
or a green, which is
lambda one delta,
931
00:49:54,360 --> 00:49:57,020
plus lambda two delta.
932
00:49:57,020 --> 00:50:03,590
So if we take a little interval
of length delta here,
933
00:50:03,590 --> 00:50:11,780
it's going to see an arrival
with probability approximately
934
00:50:11,780 --> 00:50:15,100
lambda one, plus lambda
two, delta.
935
00:50:15,100 --> 00:50:20,940
So every slot in this merged
process has an arrival
936
00:50:20,940 --> 00:50:25,780
probability with a rate which
is the sum of the rates of
937
00:50:25,780 --> 00:50:27,600
these two processes.
938
00:50:27,600 --> 00:50:29,640
So this is one part
of the definition
939
00:50:29,640 --> 00:50:31,680
of the Poisson process.
940
00:50:31,680 --> 00:50:34,890
There's a few more things that
one would need to verify.
941
00:50:34,890 --> 00:50:37,980
Namely, that intervals of the
same length have the same
942
00:50:37,980 --> 00:50:41,000
probability distribution and
that different slots are
943
00:50:41,000 --> 00:50:42,710
independent of each other.
944
00:50:42,710 --> 00:50:50,780
This can be argued by starting
from here because different
945
00:50:50,780 --> 00:50:53,620
intervals in this process are
independent from each other.
946
00:50:53,620 --> 00:50:56,900
Different intervals here are
independent from each other.
947
00:50:56,900 --> 00:50:59,900
It's not hard to argue that
different intervals in the
948
00:50:59,900 --> 00:51:03,580
merged process will also be
independent of each other.
949
00:51:03,580 --> 00:51:06,480
So the conclusion that comes
at the end is that this
950
00:51:06,480 --> 00:51:10,130
process is a Poisson process,
with a total rate which is
951
00:51:10,130 --> 00:51:13,210
equal to the sum of the rate
of the two processes.
952
00:51:13,210 --> 00:51:17,010
And now if I tell you that an
arrival happened in the merged
953
00:51:17,010 --> 00:51:20,530
process at a certain time,
how likely is it that
954
00:51:20,530 --> 00:51:23,470
it came from here?
955
00:51:23,470 --> 00:51:24,950
How likely is it?
956
00:51:24,950 --> 00:51:26,980
We go to this picture.
957
00:51:26,980 --> 00:51:30,140
Given that an arrival
occurred--
958
00:51:30,140 --> 00:51:36,050
which is the event that this
or that happened--
959
00:51:36,050 --> 00:51:39,330
what is the probability that
it came from the first
960
00:51:39,330 --> 00:51:42,060
process, the red one?
961
00:51:42,060 --> 00:51:45,190
Well it's the probability
of this divided by the
962
00:51:45,190 --> 00:51:48,030
probability of this,
times that.
963
00:51:48,030 --> 00:51:52,760
Given that this event occurred,
you want to find the
964
00:51:52,760 --> 00:51:56,560
conditional probability
of that sub event.
965
00:51:56,560 --> 00:51:58,960
So we're asking the question,
out of the total probability
966
00:51:58,960 --> 00:52:00,660
of these two, what
fraction of that
967
00:52:00,660 --> 00:52:02,790
probability is assigned here?
968
00:52:02,790 --> 00:52:05,300
And this is lambda one
delta, after we
969
00:52:05,300 --> 00:52:07,040
ignore the other terms.
970
00:52:07,040 --> 00:52:09,170
This is lambda two delta.
971
00:52:09,170 --> 00:52:15,040
So that fraction is going to be
lambda one, over lambda one
972
00:52:15,040 --> 00:52:16,770
plus lambda two.
973
00:52:16,770 --> 00:52:17,640
What does this tell you?
974
00:52:17,640 --> 00:52:21,820
If lambda one and lambda two are
equal, given that I saw an
975
00:52:21,820 --> 00:52:25,580
arrival here, it's equally
likely to be red or green.
976
00:52:25,580 --> 00:52:29,716
But if the reds have a much
higher arrival rate, when I
977
00:52:29,716 --> 00:52:32,700
see an arrival here, it's
more likely this
978
00:52:32,700 --> 00:52:34,050
number will be large.
979
00:52:34,050 --> 00:52:38,390
So it's more likely to have
come from the red process.
980
00:52:38,390 --> 00:52:40,830
OK so we'll continue with
this story and do some
981
00:52:40,830 --> 00:52:42,080
applications next time.
982
00:52:42,080 --> 00:52:43,330