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PROFESSOR: As the question said,
where are we as far as
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00:00:25,930 --> 00:00:26,800
the text goes.
10
00:00:26,800 --> 00:00:28,960
We're just going to
start Chapter 2
11
00:00:28,960 --> 00:00:31,290
today, Poisson processes.
12
00:00:31,290 --> 00:00:35,270
I want to spend about five
minutes reviewing a little bit
13
00:00:35,270 --> 00:00:39,130
about convergence, the things
we said last time,
14
00:00:39,130 --> 00:00:41,880
and then move on.
15
00:00:41,880 --> 00:00:46,240
There's a big break in the
course, at this point between
16
00:00:46,240 --> 00:00:51,570
Chapter 1 and Chapter 2 in the
sense that Chapter 1 is very
17
00:00:51,570 --> 00:00:54,940
abstract, a little
theoretical.
18
00:00:54,940 --> 00:01:00,750
It's dealing with probability
theory in general and the most
19
00:01:00,750 --> 00:01:04,239
general theorems in probability
stated in very
20
00:01:04,239 --> 00:01:06,270
simple and elementary form.
21
00:01:06,270 --> 00:01:11,490
But still, we're essentially
not dealing with
22
00:01:11,490 --> 00:01:12,660
applications at all.
23
00:01:12,660 --> 00:01:16,850
We're dealing with very, very
abstract things in Chapter 1.
24
00:01:16,850 --> 00:01:19,140
Chapter 2 is just the reverse.
25
00:01:19,140 --> 00:01:22,480
A Poisson process is the
most concrete thing
26
00:01:22,480 --> 00:01:24,420
you can think of.
27
00:01:24,420 --> 00:01:28,150
People use that as a model
for almost everything.
28
00:01:28,150 --> 00:01:29,710
Whether it's a reasonable
model or
29
00:01:29,710 --> 00:01:31,400
not is another question.
30
00:01:31,400 --> 00:01:34,300
But people use it as
a model constantly.
31
00:01:34,300 --> 00:01:37,570
And everything about
it is simple.
32
00:01:37,570 --> 00:01:41,570
For a Poisson process, you can
almost characterize it as
33
00:01:41,570 --> 00:01:44,310
saying everything you could
think of about a Poisson
34
00:01:44,310 --> 00:01:49,270
process is either true or
it's obviously false.
35
00:01:49,270 --> 00:01:50,830
And when you get to
that point, you
36
00:01:50,830 --> 00:01:52,670
understand Poisson processes.
37
00:01:52,670 --> 00:01:55,630
And you can go on to other
things and never have to
38
00:01:55,630 --> 00:01:57,530
really think about them
very hard again.
39
00:01:57,530 --> 00:02:00,820
Because at that point, you
really understand them.
40
00:02:00,820 --> 00:02:01,130
OK.
41
00:02:01,130 --> 00:02:06,130
So let's go on and review
things a little bit.
42
00:02:06,130 --> 00:02:11,060
What's convergence and how does
it affect sequences of
43
00:02:11,060 --> 00:02:12,550
IID random variables?
44
00:02:12,550 --> 00:02:14,550
Convergence is more
general than
45
00:02:14,550 --> 00:02:16,840
just IID random variables.
46
00:02:16,840 --> 00:02:21,700
But it applies to any sequence
of random variables.
47
00:02:21,700 --> 00:02:26,270
And the definition is that a
sequence of random variables
48
00:02:26,270 --> 00:02:30,970
converges in distribution to
another random variable z, if
49
00:02:30,970 --> 00:02:36,120
the limit, as n goes to
infinity, of the distribution
50
00:02:36,120 --> 00:02:39,200
function of zn converges to the
51
00:02:39,200 --> 00:02:41,700
distribution function of z.
52
00:02:41,700 --> 00:02:46,190
For this definition to make
sense, it doesn't matter what
53
00:02:46,190 --> 00:02:52,880
z is. z can be outside of
the sample space even.
54
00:02:52,880 --> 00:02:54,720
The only thing we're interested
in is this
55
00:02:54,720 --> 00:02:57,110
particular distribution
function.
56
00:02:57,110 --> 00:03:00,760
So what we're really saying is
a sequence of distributions
57
00:03:00,760 --> 00:03:05,670
converges to another
distribution function if in
58
00:03:05,670 --> 00:03:12,980
fact this limit occurs at every
point where f of z is
59
00:03:12,980 --> 00:03:13,450
continuous.
60
00:03:13,450 --> 00:03:17,590
In other words, if f of z is
discontinuous someplace, we
61
00:03:17,590 --> 00:03:20,440
had an example of that where
we're looking at the law of
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00:03:20,440 --> 00:03:24,490
large numbers and the
distribution function.
63
00:03:24,490 --> 00:03:27,330
Looked at in the right way
was a step function.
64
00:03:27,330 --> 00:03:30,010
It wasn't continuous
at the step.
65
00:03:30,010 --> 00:03:34,000
And therefore, you can't expect
anything to be said
66
00:03:34,000 --> 00:03:36,360
about that.
67
00:03:36,360 --> 00:03:40,710
So the typical example of
convergence in distribution is
68
00:03:40,710 --> 00:03:45,460
the central limit theorem which
says, if x1, x2 are IID,
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00:03:45,460 --> 00:03:47,900
they have a variance
sigma squared.
70
00:03:47,900 --> 00:03:51,370
And if s sub n the sum of these
random variables is a
71
00:03:51,370 --> 00:03:56,410
sum of x1 to xn, then zn
is the normalized sum.
72
00:03:56,410 --> 00:03:58,660
In other words, you
take this sum.
73
00:03:58,660 --> 00:04:01,530
You subtract off
the mean of it.
74
00:04:01,530 --> 00:04:06,430
And I think in the reproduction
of these slides
75
00:04:06,430 --> 00:04:16,232
that you have, I think that
n, right there was--
76
00:04:16,232 --> 00:04:17,492
here we go.
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00:04:17,492 --> 00:04:19,680
That's the other y.
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00:04:19,680 --> 00:04:24,280
I think that n was left off.
79
00:04:24,280 --> 00:04:27,880
If that n wasn't left off,
another n was left off.
80
00:04:27,880 --> 00:04:29,470
It's obviously needed there.
81
00:04:29,470 --> 00:04:32,640
This is a normalized random
variable because the variance
82
00:04:32,640 --> 00:04:42,120
of sn and of sn minus nx bar is
just sigma squared times n.
83
00:04:42,120 --> 00:04:44,340
Because they're any of these
random variables.
84
00:04:44,340 --> 00:04:49,020
So we're dividing by the
standard deviation.
85
00:04:49,020 --> 00:04:53,700
So this is a random variable
for each n which has a
86
00:04:53,700 --> 00:04:57,250
standard deviation
1 and mean 0.
87
00:04:57,250 --> 00:04:58,680
So it's normalized.
88
00:04:58,680 --> 00:05:01,610
And it converges in distribution
to a Gaussian
89
00:05:01,610 --> 00:05:06,470
random variable of mean 0 and
standard deviation 1.
90
00:05:06,470 --> 00:05:08,870
This notation here is
sort of standard.
91
00:05:08,870 --> 00:05:11,180
And we'll use it at
various times.
92
00:05:11,180 --> 00:05:15,780
It means a Gaussian distribution
with mean 0 and
93
00:05:15,780 --> 00:05:17,690
variance 1.
94
00:05:17,690 --> 00:05:22,270
So for an example of that, if
x1, x2, so forth, are IID with
95
00:05:22,270 --> 00:05:30,410
mean expected value of x and the
sum here, then sn over n
96
00:05:30,410 --> 00:05:33,670
converges in distribution to
the deterministic random
97
00:05:33,670 --> 00:05:36,230
variable x bar.
98
00:05:36,230 --> 00:05:37,730
That's a nice example of this.
99
00:05:37,730 --> 00:05:42,850
So we have two examples of
convergence and distribution.
100
00:05:42,850 --> 00:05:49,500
And that's what that says.
101
00:05:49,500 --> 00:05:53,290
So next, a sequence of random
variables converges in
102
00:05:53,290 --> 00:05:55,700
probability.
103
00:05:55,700 --> 00:05:59,400
When we start talking about
convergence in probability,
104
00:05:59,400 --> 00:06:04,360
there's another idea which we
are going to bring in, mostly
105
00:06:04,360 --> 00:06:07,510
in Chapter 4 when we get
to it, which is called
106
00:06:07,510 --> 00:06:09,840
convergence with
probability 1.
107
00:06:09,840 --> 00:06:11,950
Don't confuse those two
things because they're
108
00:06:11,950 --> 00:06:14,880
very different ideas.
109
00:06:14,880 --> 00:06:20,050
Because people confuse them,
many people call convergence
110
00:06:20,050 --> 00:06:24,490
with probability 1 almost sure
convergence or almost
111
00:06:24,490 --> 00:06:27,030
everywhere convergence.
112
00:06:27,030 --> 00:06:30,200
I don't like that notation.
113
00:06:30,200 --> 00:06:34,090
So I'll stay with the notation
with probability 1.
114
00:06:34,090 --> 00:06:36,950
But it means something very
different than converging in
115
00:06:36,950 --> 00:06:38,230
probability.
116
00:06:38,230 --> 00:06:42,120
So the definition is that a
set of random variables
117
00:06:42,120 --> 00:06:48,280
converges in probability to some
other random variable if
118
00:06:48,280 --> 00:06:50,000
this limit holds true.
119
00:06:50,000 --> 00:06:54,840
And you remember that diagram
we showed you last time.
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00:06:54,840 --> 00:06:57,890
Let me just quickly redraw it.
121
00:07:01,124 --> 00:07:04,420
Have this set of distribution
functions.
122
00:07:04,420 --> 00:07:09,370
Here's the mean here,
x bar, limits,
123
00:07:09,370 --> 00:07:12,840
plus and minus epsilon.
124
00:07:12,840 --> 00:07:18,000
And this sequence of random
variables has to come in down
125
00:07:18,000 --> 00:07:22,280
here and go out up there.
126
00:07:22,280 --> 00:07:27,930
This distance here and the
distance there gets very small
127
00:07:27,930 --> 00:07:30,570
and goes to 0 as n
gets very large.
128
00:07:30,570 --> 00:07:32,580
And that's the meaning
of what this says.
129
00:07:32,580 --> 00:07:35,860
So if you don't remember that
diagram, go look at it in the
130
00:07:35,860 --> 00:07:39,370
lecture notes last time or in
the text where it's explained
131
00:07:39,370 --> 00:07:42,830
in a lot more detail.
132
00:07:42,830 --> 00:07:47,590
So the typical example of that
is the weak law of large
133
00:07:47,590 --> 00:07:52,370
numbers of x1, blah, blah,
blah, are IID with mean,
134
00:07:52,370 --> 00:07:53,420
expected value of x.
135
00:07:53,420 --> 00:07:58,170
Remember now that we say that
a random variable has a mean
136
00:07:58,170 --> 00:08:04,370
if the expected value of the
absolute value of x is finite.
137
00:08:04,370 --> 00:08:09,510
It's not enough to have things
which have a distribution
138
00:08:09,510 --> 00:08:15,060
function which is badly behaved
for very big x, and
139
00:08:15,060 --> 00:08:17,770
badly behaved for very small
x, and the two of them
140
00:08:17,770 --> 00:08:18,510
cancelled out.
141
00:08:18,510 --> 00:08:19,800
That doesn't work.
142
00:08:19,800 --> 00:08:21,620
That doesn't mean
you have a mean.
143
00:08:21,620 --> 00:08:24,140
You need the expected value
as the absolute
144
00:08:24,140 --> 00:08:25,420
value of x to be finite.
145
00:08:28,240 --> 00:08:33,530
Now, the weak law of large
numbers says that the random
146
00:08:33,530 --> 00:08:38,929
variables sn over n, in other
words, the sample average, in
147
00:08:38,929 --> 00:08:42,870
fact converges to the
deterministic
148
00:08:42,870 --> 00:08:44,840
random variable x bar.
149
00:08:44,840 --> 00:08:48,220
And that convergence is
convergence in probability.
150
00:08:48,220 --> 00:08:50,530
Which means it's this kind
of convergence here.
151
00:08:50,530 --> 00:08:53,300
Which means that it's going
to a distribution
152
00:08:53,300 --> 00:08:55,260
which is a step function.
153
00:08:55,260 --> 00:08:59,010
There's a very big difference
between a distribution which
154
00:08:59,010 --> 00:09:01,870
is a step function and a
distribution which is
155
00:09:01,870 --> 00:09:04,510
something like a Gaussian
random variable.
156
00:09:04,510 --> 00:09:08,450
And what the big difference is
is that the random variables
157
00:09:08,450 --> 00:09:11,570
that are converging to each
other, if a bunch of random
158
00:09:11,570 --> 00:09:14,670
variables are all converging to
a constant, then they all
159
00:09:14,670 --> 00:09:16,730
have to be very close
to each other.
160
00:09:16,730 --> 00:09:21,930
And that's the property you're
really interested in in
161
00:09:21,930 --> 00:09:25,000
convergence in probability.
162
00:09:25,000 --> 00:09:29,430
So convergence in mean square,
finally, last definition which
163
00:09:29,430 --> 00:09:31,725
is easy to deal with.
164
00:09:31,725 --> 00:09:34,990
If a sequence of random
variables converges in the
165
00:09:34,990 --> 00:09:39,740
mean square to another random
variable, if this limit of the
166
00:09:39,740 --> 00:09:43,150
expected value, of the
difference between the two
167
00:09:43,150 --> 00:09:47,690
random variables squared, goes
to 0, this n gets big.
168
00:09:47,690 --> 00:09:51,500
That's what we had with the weak
law of large numbers if
169
00:09:51,500 --> 00:09:55,035
you assume that the random
variables each had a variance.
170
00:09:58,090 --> 00:10:00,350
So on to something new.
171
00:10:00,350 --> 00:10:02,890
On to Poisson processes.
172
00:10:02,890 --> 00:10:05,830
We first have to explain what
an arrival process is.
173
00:10:05,830 --> 00:10:08,760
And then we can get into
Poisson processes.
174
00:10:08,760 --> 00:10:13,540
Because arrival processes are
a very broad class of
175
00:10:13,540 --> 00:10:17,930
stochastic processes, in fact
discrete stochastic processes.
176
00:10:17,930 --> 00:10:21,750
But they have this property of
being characterized by things
177
00:10:21,750 --> 00:10:27,090
happening at various random
instance of time as opposed to
178
00:10:27,090 --> 00:10:30,710
a noise waveform or something
of that sort.
179
00:10:30,710 --> 00:10:34,390
So an arrival process is a
sequence of increasing random
180
00:10:34,390 --> 00:10:38,540
variables, 0 less than
s1, less than s2.
181
00:10:38,540 --> 00:10:42,850
What's it mean for a random
variable s1 to be less than a
182
00:10:42,850 --> 00:10:45,760
random variable s2?
183
00:10:45,760 --> 00:10:47,870
It means exactly the
same thing as it
184
00:10:47,870 --> 00:10:50,870
means for real numbers.
185
00:10:50,870 --> 00:10:56,850
s1 is less than s2 if the random
variable s2 minus s1 is
186
00:10:56,850 --> 00:11:00,710
a positive random variable,
namely if it only takes on
187
00:11:00,710 --> 00:11:06,520
non-negative values for all
omega in the sample space or
188
00:11:06,520 --> 00:11:13,060
for all omega except for some
peculiar set of probability 0.
189
00:11:13,060 --> 00:11:19,050
The differences in these arrival
epochs, why do I call
190
00:11:19,050 --> 00:11:20,580
them arrival epochs?
191
00:11:20,580 --> 00:11:23,200
Why do other people call
them arrival epochs?
192
00:11:23,200 --> 00:11:27,780
Because time is something which
gets used so often here
193
00:11:27,780 --> 00:11:29,000
that it gets confusing.
194
00:11:29,000 --> 00:11:32,680
So it's nice to call one thing
epochs instead of time.
195
00:11:32,680 --> 00:11:35,930
And then you know what you're
talking about a little better.
196
00:11:35,930 --> 00:11:42,070
The difference is s sub i minus
s sub i minus 1 for all
197
00:11:42,070 --> 00:11:49,040
i greater than or equal to 2
here, with x1 equal to s1.
198
00:11:49,040 --> 00:11:52,640
These are called interarrival
times and the si are called
199
00:11:52,640 --> 00:11:53,910
arrival epochs.
200
00:11:53,910 --> 00:11:58,170
The picture here really
shows it all.
201
00:11:58,170 --> 00:12:02,380
Here we have a sequence of
arrival instance, which is
202
00:12:02,380 --> 00:12:04,260
where these arrivals occur.
203
00:12:04,260 --> 00:12:09,970
By definition, x1 is the time at
which is the first arrival
204
00:12:09,970 --> 00:12:13,450
occurs, x2 is the difference
between the time when the
205
00:12:13,450 --> 00:12:16,030
second arrival occurs and
the first arrival
206
00:12:16,030 --> 00:12:18,080
occurs, and so forth.
207
00:12:18,080 --> 00:12:21,560
n of t is the number of arrivals
that have occurred up
208
00:12:21,560 --> 00:12:23,390
until time t.
209
00:12:23,390 --> 00:12:27,280
Which is, if we draw a staircase
function for each of
210
00:12:27,280 --> 00:12:30,920
these arrivals, n of t
is just the value of
211
00:12:30,920 --> 00:12:32,310
that staircase function.
212
00:12:32,310 --> 00:12:34,490
In other words, the counting
process, the
213
00:12:34,490 --> 00:12:36,360
arrival counting process--
214
00:12:36,360 --> 00:12:39,400
here's another typo in the
notes that you've got.
215
00:12:39,400 --> 00:12:41,050
It calls it Poisson
counting process.
216
00:12:41,050 --> 00:12:43,625
It should be arrival
counting process.
217
00:12:48,540 --> 00:12:52,290
What this staircase function
is is in fact
218
00:12:52,290 --> 00:12:54,120
the counting process.
219
00:12:54,120 --> 00:12:56,550
It says how many arrivals
there have been
220
00:12:56,550 --> 00:12:57,950
up until time t.
221
00:12:57,950 --> 00:13:00,320
And every once in a while,
that jumps up by 1.
222
00:13:00,320 --> 00:13:03,920
So it keeps increasing by
1 at various times.
223
00:13:03,920 --> 00:13:06,600
So that's the arrival
counting process.
224
00:13:06,600 --> 00:13:11,970
The important thing to get out
of this is if you understand
225
00:13:11,970 --> 00:13:15,760
everything about these random
variables, then you understand
226
00:13:15,760 --> 00:13:18,170
everything about these
random variables.
227
00:13:18,170 --> 00:13:20,580
And then you understand
everything about
228
00:13:20,580 --> 00:13:22,800
these random variables.
229
00:13:22,800 --> 00:13:25,950
There's a countable number of
these random variables.
230
00:13:25,950 --> 00:13:29,050
There's a countable number of
these random variables.
231
00:13:29,050 --> 00:13:31,230
There's an unaccountably
infinite number of these
232
00:13:31,230 --> 00:13:32,060
random variables.
233
00:13:32,060 --> 00:13:35,070
In other words, for every
t, n of t is a
234
00:13:35,070 --> 00:13:37,240
different random variable.
235
00:13:37,240 --> 00:13:42,020
I mean, it tends to be the
same for relatively large
236
00:13:42,020 --> 00:13:44,020
intervals of t sometimes.
237
00:13:44,020 --> 00:13:49,110
But this is a different random
variable for each value of t.
238
00:13:49,110 --> 00:13:53,600
So let's proceed with that.
239
00:13:53,600 --> 00:13:56,970
A sample path or sample function
of the process is a
240
00:13:56,970 --> 00:13:58,870
sequence of sample values.
241
00:13:58,870 --> 00:14:01,180
That's the same as we
have everywhere.
242
00:14:01,180 --> 00:14:05,450
You look at a sample point
of the process.
243
00:14:05,450 --> 00:14:13,160
Sample point of the whole
probability space maps into
244
00:14:13,160 --> 00:14:15,520
this sequence of random
variables, s1,
245
00:14:15,520 --> 00:14:18,990
s2, s3, and so forth.
246
00:14:18,990 --> 00:14:22,200
If you know what the sample
value is of each one of these
247
00:14:22,200 --> 00:14:27,570
random variables, then in fact,
you can draw this step
248
00:14:27,570 --> 00:14:28,890
function here.
249
00:14:28,890 --> 00:14:32,460
If you know the value, the
sample value, of each one of
250
00:14:32,460 --> 00:14:37,640
these, in the same way,
you can again
251
00:14:37,640 --> 00:14:39,890
draw this step function.
252
00:14:39,890 --> 00:14:42,330
And if you know what the
subfunction is, the step
253
00:14:42,330 --> 00:14:46,390
function in fact is the sample
value of n of t.
254
00:14:46,390 --> 00:14:51,380
Now, there's one thing a
little peculiar here.
255
00:14:51,380 --> 00:14:53,620
Each sample path corresponds
to a
256
00:14:53,620 --> 00:14:56,030
particular staircase function.
257
00:14:56,030 --> 00:14:58,880
And the process can be viewed
as the ensemble with joint
258
00:14:58,880 --> 00:15:03,110
probability distributions of
such staircase functions.
259
00:15:03,110 --> 00:15:07,280
Now, what does all that
gobbledygook mean?
260
00:15:07,280 --> 00:15:09,230
Very, very often in probability
261
00:15:09,230 --> 00:15:11,590
theory, we draw pictures.
262
00:15:11,590 --> 00:15:15,790
And these pictures are pictures
of what happens to
263
00:15:15,790 --> 00:15:17,490
random variables.
264
00:15:17,490 --> 00:15:19,840
And there's a cheat
in all of that.
265
00:15:19,840 --> 00:15:26,160
And the cheat here is that in
fact, this step function here
266
00:15:26,160 --> 00:15:28,910
is just a generic
step function.
267
00:15:28,910 --> 00:15:34,860
These points at which changes
occur are generic values at
268
00:15:34,860 --> 00:15:36,880
which changes occur.
269
00:15:36,880 --> 00:15:41,400
And we're representing those
values as random variables.
270
00:15:41,400 --> 00:15:44,380
When you represent these as
random variables, this whole
271
00:15:44,380 --> 00:15:48,010
function here, namely n
of t itself, becomes--
272
00:15:56,510 --> 00:15:59,680
if you have a particular set
of values for each one of
273
00:15:59,680 --> 00:16:03,140
these, then you have a
particular staircase function.
274
00:16:03,140 --> 00:16:06,350
With that particular staircase
function, you have a
275
00:16:06,350 --> 00:16:09,860
particular sample
path for n of t.
276
00:16:09,860 --> 00:16:13,350
In other words, a sample path
for any set of these random
277
00:16:13,350 --> 00:16:16,620
variables-- the arrival epochs,
or the interarrival
278
00:16:16,620 --> 00:16:19,740
intervals, or n of
t for each t--
279
00:16:19,740 --> 00:16:22,400
all of these are equivalent
to each other.
280
00:16:22,400 --> 00:16:29,130
For this reason, when we talk
about arrival processes, it's
281
00:16:29,130 --> 00:16:31,240
a little different than
what we usually do.
282
00:16:31,240 --> 00:16:36,470
Because usually, we say a random
process is a sequence
283
00:16:36,470 --> 00:16:39,740
or an uncountable number
of random variables.
284
00:16:39,740 --> 00:16:43,470
Here, just because we can
describe it in three different
285
00:16:43,470 --> 00:16:51,330
ways, this same stochastic
process gets described either
286
00:16:51,330 --> 00:16:56,700
as a sequence of interarrival
intervals, or as a sequence of
287
00:16:56,700 --> 00:17:01,660
arrival epochs, or as a
countable number these n of t
288
00:17:01,660 --> 00:17:03,010
random variables.
289
00:17:03,010 --> 00:17:06,839
And from now on, we make no
distinction between any of
290
00:17:06,839 --> 00:17:07,589
these things.
291
00:17:07,589 --> 00:17:11,560
We will, every once in while,
have to remind ourselves what
292
00:17:11,560 --> 00:17:15,300
these pictures mean because
they look very simple.
293
00:17:15,300 --> 00:17:17,180
They look like the pictures
of functions that
294
00:17:17,180 --> 00:17:18,550
you're used to drawing.
295
00:17:18,550 --> 00:17:20,980
But they don't really
mean the same thing.
296
00:17:20,980 --> 00:17:25,579
Because this picture is drawing
a generic sample path.
297
00:17:25,579 --> 00:17:29,380
For that generic sample path,
you have a set of sample
298
00:17:29,380 --> 00:17:37,580
values for the Xs, a sample path
for the arrival epochs, a
299
00:17:37,580 --> 00:17:41,750
sample set of values
for n of t.
300
00:17:41,750 --> 00:17:44,050
And when we draw the picture
calling these random
301
00:17:44,050 --> 00:17:46,120
variables, we really
mean the set of
302
00:17:46,120 --> 00:17:47,850
all such step functions.
303
00:17:47,850 --> 00:17:51,340
And we just automatically use
all those properties and those
304
00:17:51,340 --> 00:17:53,390
relationships.
305
00:17:53,390 --> 00:17:57,510
So it's not quite as simple as
what it appears to be, but
306
00:17:57,510 --> 00:17:58,760
it's almost as simple.
307
00:18:01,090 --> 00:18:06,280
You can also see that any sample
path can be specified
308
00:18:06,280 --> 00:18:10,370
by the sample values n of t for
all t, by si for all i, or
309
00:18:10,370 --> 00:18:13,050
by xi for all i.
310
00:18:13,050 --> 00:18:17,190
So that essentially, an arrival
process is specified
311
00:18:17,190 --> 00:18:18,310
by any one of these things.
312
00:18:18,310 --> 00:18:20,830
That's exactly what
I just said.
313
00:18:20,830 --> 00:18:25,720
The major relation we need to
relate the counting process to
314
00:18:25,720 --> 00:18:31,040
the arrival process, well,
there's one relationship here,
315
00:18:31,040 --> 00:18:34,260
which is perhaps the simplest
relationship.
316
00:18:34,260 --> 00:18:38,790
But this relationship is a nice
relationship to say what
317
00:18:38,790 --> 00:18:42,980
n of t is if you know
what s sub n is.
318
00:18:42,980 --> 00:18:48,430
It's not quite so nice if you
know what n of t is to figure
319
00:18:48,430 --> 00:18:50,820
what s sub n is.
320
00:18:50,820 --> 00:18:53,960
I mean, the information is
tucked into the statement, but
321
00:18:53,960 --> 00:18:56,800
it's tucked in a more convenient
way into this
322
00:18:56,800 --> 00:18:59,280
statement down here.
323
00:18:59,280 --> 00:19:04,520
This statement, I can see it
now after many years of
324
00:19:04,520 --> 00:19:06,910
dealing with it.
325
00:19:06,910 --> 00:19:10,400
I'm sure that you can
see it if you stare
326
00:19:10,400 --> 00:19:11,650
at it for five minutes.
327
00:19:14,700 --> 00:19:16,900
You will keep forgetting
the intuitive picture
328
00:19:16,900 --> 00:19:18,310
that goes with it.
329
00:19:18,310 --> 00:19:21,340
So I suggest that this is one
of the rare things in this
330
00:19:21,340 --> 00:19:24,780
course that you just
ought to remember.
331
00:19:24,780 --> 00:19:27,530
And then once you remember
it, you can always figure
332
00:19:27,530 --> 00:19:28,740
out why it's true.
333
00:19:28,740 --> 00:19:31,670
Here's the reason
why it's true.
334
00:19:31,670 --> 00:19:35,960
If s sub n is equal to tau for
some tau less than or equal to
335
00:19:35,960 --> 00:19:40,466
t, then n of tau has
to be equal to n.
336
00:19:40,466 --> 00:19:43,880
If s sub n is equal to tau,
here's the picture here,
337
00:19:43,880 --> 00:19:45,300
except there's not a tau here.
338
00:19:45,300 --> 00:19:52,180
If s sub 2 is equal to
tau, then n of 2--
339
00:19:52,180 --> 00:19:56,460
these are right continuous,
so n of 2 is equal to 2.
340
00:19:56,460 --> 00:20:01,100
And therefore, n of tau is less
than or equal to n of t.
341
00:20:01,100 --> 00:20:04,360
So that's the whole
reason down there.
342
00:20:04,360 --> 00:20:06,350
You can turn this
argument around.
343
00:20:06,350 --> 00:20:12,160
You can start out with n of t is
greater than or equal to n.
344
00:20:12,160 --> 00:20:15,220
It means n of t is equal
to some particular n.
345
00:20:15,220 --> 00:20:18,030
And turn the argument
upside down.
346
00:20:18,030 --> 00:20:19,490
And you get the same argument.
347
00:20:19,490 --> 00:20:25,800
So this tells you
what this is.
348
00:20:25,800 --> 00:20:28,160
This tells you what this is.
349
00:20:28,160 --> 00:20:32,290
If you do this for every n and
every t, then you do this for
350
00:20:32,290 --> 00:20:36,100
every n and every t.
351
00:20:36,100 --> 00:20:40,590
It's a very bizarre statement
because usually when you have
352
00:20:40,590 --> 00:20:44,820
relationships between functions,
you don't have the
353
00:20:44,820 --> 00:20:47,380
Ns and the Ts switching
around.
354
00:20:47,380 --> 00:20:50,330
And in this case, the
n is the subscript.
355
00:20:50,330 --> 00:20:52,620
That's the thing which says
which random variable you're
356
00:20:52,620 --> 00:20:53,970
talking about.
357
00:20:53,970 --> 00:20:57,670
And over here, t is the thing
which says what random
358
00:20:57,670 --> 00:20:59,690
variable you're talking about.
359
00:20:59,690 --> 00:21:01,860
So it's peculiar
in that sense.
360
00:21:01,860 --> 00:21:03,440
It's a statement which
requires a
361
00:21:03,440 --> 00:21:04,860
little bit of thought.
362
00:21:04,860 --> 00:21:07,910
I apologize for dwelling
on it because once you
363
00:21:07,910 --> 00:21:09,580
see it, it's obvious.
364
00:21:09,580 --> 00:21:12,940
But many of these obvious
things are not obvious.
365
00:21:19,030 --> 00:21:21,830
What we're going to do as we
move on is we're going to talk
366
00:21:21,830 --> 00:21:26,040
about these arrival processes in
any of these three ways we
367
00:21:26,040 --> 00:21:28,010
choose to talk about them.
368
00:21:28,010 --> 00:21:30,760
And we're going to go back
and forth between them.
369
00:21:30,760 --> 00:21:34,670
And with Poisson processes,
that's particularly easy.
370
00:21:34,670 --> 00:21:37,770
We can't do a whole lot more
with arrival processes.
371
00:21:37,770 --> 00:21:39,910
They're just too complicated.
372
00:21:39,910 --> 00:21:43,650
I mean, arrival processes
involve almost any kind of
373
00:21:43,650 --> 00:21:47,790
thing where things happen at
various points in time.
374
00:21:47,790 --> 00:21:51,960
So we simplify it to something
called a renewal process.
375
00:21:51,960 --> 00:21:55,750
Renewal processes are the
topic of Chapter 4.
376
00:21:55,750 --> 00:21:59,010
When you get to Chapter 4, you
will perhaps say that renewal
377
00:21:59,010 --> 00:22:03,320
processes are too complicated
to talk about also.
378
00:22:03,320 --> 00:22:06,730
I hope after we finish Chapter
4, you won't believe that it's
379
00:22:06,730 --> 00:22:08,990
too complicated to talk about.
380
00:22:08,990 --> 00:22:12,280
But these are fairly complicated
processes.
381
00:22:12,280 --> 00:22:16,370
But even here, it's an arrival
process where the interarrival
382
00:22:16,370 --> 00:22:21,060
intervals are independent and
identically distributed.
383
00:22:21,060 --> 00:22:25,730
Finally, a Poisson process is
a renewal process for which
384
00:22:25,730 --> 00:22:30,240
each x sub i has an exponential
distribution.
385
00:22:30,240 --> 00:22:35,070
Each interarrival has to have
the same distribution because
386
00:22:35,070 --> 00:22:39,110
since it's a renewal process,
these are all IID.
387
00:22:39,110 --> 00:22:44,040
And we let this distribution
function X be the generic
388
00:22:44,040 --> 00:22:45,330
random variable.
389
00:22:45,330 --> 00:22:47,350
And this is talking about
the distribution
390
00:22:47,350 --> 00:22:49,610
function of all of them.
391
00:22:49,610 --> 00:22:53,760
I don't know whether that 1
minus is in the slides I
392
00:22:53,760 --> 00:22:54,470
passed out.
393
00:22:54,470 --> 00:22:57,910
There's one kind of
error like that.
394
00:22:57,910 --> 00:22:59,960
And I'm not sure where it is.
395
00:22:59,960 --> 00:23:03,090
So anyway, lambda is some fixed
parameter called the
396
00:23:03,090 --> 00:23:05,180
rate of the Poisson process.
397
00:23:05,180 --> 00:23:09,770
So for each lambda greater than
0, you have a Poisson
398
00:23:09,770 --> 00:23:14,190
process where each of these
interarrival intervals are
399
00:23:14,190 --> 00:23:18,300
exponential random variables
of rate lambda.
400
00:23:18,300 --> 00:23:20,460
So that defines a
Poisson process.
401
00:23:20,460 --> 00:23:22,940
So we can all go home now
because we now know everything
402
00:23:22,940 --> 00:23:26,890
about Poisson processes
in principle.
403
00:23:26,890 --> 00:23:31,490
Everything we're going to say
from now on comes from this
404
00:23:31,490 --> 00:23:35,620
one simple statement here that
these interarrival intervals
405
00:23:35,620 --> 00:23:37,070
are exponential.
406
00:23:37,070 --> 00:23:39,910
There's something very, very
special about this exponential
407
00:23:39,910 --> 00:23:41,740
distribution.
408
00:23:41,740 --> 00:23:44,975
And that's what makes Poisson
processes so very special.
409
00:23:53,570 --> 00:23:58,070
And that special thing is this
memoryless property.
410
00:23:58,070 --> 00:24:01,760
A random variable is memoryless
if it's positive.
411
00:24:01,760 --> 00:24:06,530
And for all real t greater than
0 and x greater than 0,
412
00:24:06,530 --> 00:24:10,580
the probability that x is
greater than t plus x is equal
413
00:24:10,580 --> 00:24:13,580
to the probability that x is
greater than t times the
414
00:24:13,580 --> 00:24:16,260
probability that x is
greater than x.
415
00:24:16,260 --> 00:24:21,470
If you plug that in, then the
statement is the same whether
416
00:24:21,470 --> 00:24:24,620
you're dealing with densities,
or PMFs, or
417
00:24:24,620 --> 00:24:27,640
distribution function.
418
00:24:27,640 --> 00:24:30,980
You get the same product
relationship in each case.
419
00:24:30,980 --> 00:24:34,470
Since the interarrival interval
is exponential, the
420
00:24:34,470 --> 00:24:38,540
probability that a random
variable x is greater than
421
00:24:38,540 --> 00:24:43,570
some particular value x is equal
to e to the minus lambda
422
00:24:43,570 --> 00:24:46,330
x for x greater than zero.
423
00:24:46,330 --> 00:24:50,870
This you'll recognize, not as a
distribution function but as
424
00:24:50,870 --> 00:24:53,190
the complementary distribution
function.
425
00:24:53,190 --> 00:24:56,560
It's the probability that
X is greater than x.
426
00:24:56,560 --> 00:25:00,120
So it's the complementary
distribution function
427
00:25:00,120 --> 00:25:02,430
evaluated at the value of x.
428
00:25:02,430 --> 00:25:05,530
This is an exponential
which is going down.
429
00:25:05,530 --> 00:25:11,650
So these random variables
have a probability
430
00:25:11,650 --> 00:25:16,694
density which is this.
431
00:25:16,694 --> 00:25:25,110
This is f sub x of X. And they
have a distribution function
432
00:25:25,110 --> 00:25:28,150
which is this.
433
00:25:28,150 --> 00:25:30,420
And they have a complementary
distribution
434
00:25:30,420 --> 00:25:33,300
function which is this.
435
00:25:33,300 --> 00:25:35,940
Now, this is f of c.
436
00:25:35,940 --> 00:25:38,290
This is f.
437
00:25:38,290 --> 00:25:40,080
So there's nothing
much to them.
438
00:25:42,780 --> 00:25:47,770
And now there's a theorem
which says that a random
439
00:25:47,770 --> 00:25:52,200
variable is memoryless if and
only if it is exponential.
440
00:25:52,200 --> 00:25:55,520
We just showed here that an
exponential random variable is
441
00:25:55,520 --> 00:25:57,180
memoryless.
442
00:25:57,180 --> 00:26:02,700
To show it the other way
is almost obvious.
443
00:26:02,700 --> 00:26:04,810
You take this definition here.
444
00:26:04,810 --> 00:26:08,600
You take the logarithm of
each of these sides.
445
00:26:08,600 --> 00:26:12,850
When you get the logarithm of
this, it says the logarithm of
446
00:26:12,850 --> 00:26:18,070
the probability x is greater
than p plus x is the logarithm
447
00:26:18,070 --> 00:26:21,920
of this plus the logarithm
of this.
448
00:26:21,920 --> 00:26:25,600
What we have to show to get an
exponential is that this
449
00:26:25,600 --> 00:26:31,430
logarithm is linear
in its argument t.
450
00:26:31,430 --> 00:26:36,990
Now, if you have this is equal
to the sum of this and this
451
00:26:36,990 --> 00:26:41,220
for all t and x, it's sort
of says it's linear.
452
00:26:41,220 --> 00:26:47,235
There's an exercise, I think
it's Exercise 2.4, which shows
453
00:26:47,235 --> 00:26:49,660
that you have to be a
little bit careful.
454
00:26:49,660 --> 00:26:53,310
Or at least as it points
out, very, very picky
455
00:26:53,310 --> 00:26:57,050
mathematicians have to be a
little bit careful with that.
456
00:26:57,050 --> 00:27:01,980
And you can worry about that
or not as you choose.
457
00:27:01,980 --> 00:27:03,620
So that's the theorem.
458
00:27:03,620 --> 00:27:07,140
That's why Poisson processes
are special.
459
00:27:07,140 --> 00:27:08,900
And that's why we can
do all the things
460
00:27:08,900 --> 00:27:11,480
we can do with them.
461
00:27:11,480 --> 00:27:15,400
The reason why we call it
memoryless is more apparent if
462
00:27:15,400 --> 00:27:17,670
we use conditional
probabilities.
463
00:27:17,670 --> 00:27:20,210
With conditional probabilities,
the probability
464
00:27:20,210 --> 00:27:25,390
that the random variable X is
greater than t plus x, given
465
00:27:25,390 --> 00:27:29,340
that it's greater than t, is
equal to the probability that
466
00:27:29,340 --> 00:27:31,410
X is greater than x.
467
00:27:31,410 --> 00:27:35,530
If people in a checkout line
have exponential service times
468
00:27:35,530 --> 00:27:40,420
and you've waited 15 minutes for
the person in front, what
469
00:27:40,420 --> 00:27:44,130
is his or her remaining service
time, assuming the
470
00:27:44,130 --> 00:27:46,760
service time is exponential?
471
00:27:46,760 --> 00:27:47,310
What's the answer?
472
00:27:47,310 --> 00:27:48,890
You've waited 15 minutes.
473
00:27:48,890 --> 00:27:53,460
Your original service time is
exponential with rate lambda.
474
00:27:53,460 --> 00:27:55,990
What's the remaining
service time?
475
00:27:55,990 --> 00:27:58,560
Well, the answer is
it's exponential.
476
00:27:58,560 --> 00:28:00,700
That's this memoryless
property.
477
00:28:00,700 --> 00:28:05,880
It's called memoryless because
the random variable doesn't
478
00:28:05,880 --> 00:28:09,330
remember how long it
hasn't happened.
479
00:28:11,880 --> 00:28:15,190
So you can think of an
exponential random variable as
480
00:28:15,190 --> 00:28:17,520
something which takes
place in time.
481
00:28:17,520 --> 00:28:21,000
And in each instant of time, it
might or might not happen.
482
00:28:21,000 --> 00:28:24,110
And if it hasn't happened yet,
there's still the same
483
00:28:24,110 --> 00:28:27,210
probability in every remaining
increment that it's going to
484
00:28:27,210 --> 00:28:28,310
happen then.
485
00:28:28,310 --> 00:28:32,510
So you haven't gained anything
and you haven't lost anything
486
00:28:32,510 --> 00:28:34,280
by having to wait this long.
487
00:28:37,320 --> 00:28:41,120
Here's an interesting question
which you can tie yourself in
488
00:28:41,120 --> 00:28:42,800
knots for a little bit.
489
00:28:42,800 --> 00:28:44,835
Has your time waiting
been wasted?
490
00:28:48,360 --> 00:28:52,860
Namely the time you still have
to wait is exponential with
491
00:28:52,860 --> 00:28:56,700
the same rate as
it was before.
492
00:28:56,700 --> 00:29:00,070
So the expected amount of time
you have to wait is still the
493
00:29:00,070 --> 00:29:07,790
same as when you got into line
15 minutes ago with this one
494
00:29:07,790 --> 00:29:11,320
very slow person in
front of you.
495
00:29:11,320 --> 00:29:14,400
So have you wasting your time?
496
00:29:14,400 --> 00:29:15,700
Well, you haven't
gained anything.
497
00:29:19,010 --> 00:29:21,510
But you haven't really wasted
your time either.
498
00:29:21,510 --> 00:29:27,210
Because if you have to get
served in that line, then at
499
00:29:27,210 --> 00:29:31,500
some point, you're going to
have to go in that line.
500
00:29:31,500 --> 00:29:34,370
And you might look for a time
when the line is very short.
501
00:29:34,370 --> 00:29:36,910
You might be lucky and find
a time when the line is
502
00:29:36,910 --> 00:29:37,860
completely empty.
503
00:29:37,860 --> 00:29:40,360
And then you start getting
served right away.
504
00:29:40,360 --> 00:29:47,760
But if you ignore those issues,
then in fact, in a
505
00:29:47,760 --> 00:29:51,590
sense, you have wasted
your time.
506
00:29:51,590 --> 00:29:54,390
Another more interesting
question then is why do you
507
00:29:54,390 --> 00:29:58,410
move to another line if somebody
takes a long time?
508
00:29:58,410 --> 00:30:00,160
All of you have had
this experience.
509
00:30:00,160 --> 00:30:03,170
You're in a supermarket.
510
00:30:03,170 --> 00:30:08,130
Or you're at an airplane counter
or any of the places
511
00:30:08,130 --> 00:30:10,790
where you have to wait
for service.
512
00:30:10,790 --> 00:30:14,200
There's somebody, one person in
front of you, who has been
513
00:30:14,200 --> 00:30:16,150
there forever.
514
00:30:16,150 --> 00:30:18,990
And it seems as if they're going
to stay there forever.
515
00:30:18,990 --> 00:30:21,200
You notice another line
that only has one
516
00:30:21,200 --> 00:30:22,950
person being served.
517
00:30:22,950 --> 00:30:27,595
And most of us, especially very
impatient people like me,
518
00:30:27,595 --> 00:30:31,610
I'm going to walk over and
get into that other line.
519
00:30:31,610 --> 00:30:36,180
And the question is, is that
rational or isn't it rational?
520
00:30:36,180 --> 00:30:37,740
If the service times are
521
00:30:37,740 --> 00:30:40,750
exponential, it is not rational.
522
00:30:40,750 --> 00:30:44,030
It doesn't make any difference
whether I stay where I am or
523
00:30:44,030 --> 00:30:46,390
go to the other line.
524
00:30:46,390 --> 00:30:51,110
If the service times are fixed
duration, namely suppose every
525
00:30:51,110 --> 00:30:55,190
service time takes 10 minutes
and I've waited for a long
526
00:30:55,190 --> 00:30:59,410
time, is it rational for me
to move to the other line?
527
00:30:59,410 --> 00:31:03,840
Absolutely not because I'm
almost at the end of that 10
528
00:31:03,840 --> 00:31:05,190
minutes now.
529
00:31:05,190 --> 00:31:08,160
And I'm about to be served.
530
00:31:08,160 --> 00:31:09,470
So why do we move?
531
00:31:09,470 --> 00:31:14,020
Is it just psychology, that
we're very impatient?
532
00:31:14,020 --> 00:31:14,890
I don't think so.
533
00:31:14,890 --> 00:31:19,090
I think it's because we have all
seen that an awful lot of
534
00:31:19,090 --> 00:31:23,730
lines, particularly airline
reservation lines, and if your
535
00:31:23,730 --> 00:31:25,950
plane doesn't fly or something,
and you're trying
536
00:31:25,950 --> 00:31:31,960
to get rescheduled, or any of
these things, the service time
537
00:31:31,960 --> 00:31:36,860
is worse than Poisson in the
sense that if you've waited
538
00:31:36,860 --> 00:31:41,390
for 10 minutes, your expected
remaining waiting time is
539
00:31:41,390 --> 00:31:44,400
greater than it was before
you started waiting.
540
00:31:44,400 --> 00:31:48,190
The longer you wait, the longer
your expected remaining
541
00:31:48,190 --> 00:31:49,685
waiting time is.
542
00:31:49,685 --> 00:31:52,693
And that's called a heavy-tailed
distribution.
543
00:31:56,000 --> 00:31:59,500
What most of us have noticed, I
think, in our lives is that
544
00:31:59,500 --> 00:32:03,390
an awful lot of waiting lines
that human beings wait in are
545
00:32:03,390 --> 00:32:05,140
in fact heavy-tailed.
546
00:32:05,140 --> 00:32:09,920
So that in fact is part of the
reason why we move if somebody
547
00:32:09,920 --> 00:32:11,780
takes a long time.
548
00:32:11,780 --> 00:32:15,520
It's interesting to see
how the brain works.
549
00:32:15,520 --> 00:32:19,080
Because I'm sure that none
of you have ever really
550
00:32:19,080 --> 00:32:22,240
rationally analyzed this
question of why you move.
551
00:32:22,240 --> 00:32:23,390
Have you?
552
00:32:23,390 --> 00:32:25,390
I mean, I have because
I teach probability
553
00:32:25,390 --> 00:32:27,760
courses all the time.
554
00:32:27,760 --> 00:32:30,420
But I don't think anyone who
doesn't teach probability
555
00:32:30,420 --> 00:32:35,290
courses would be crazy enough
to waste their time on a
556
00:32:35,290 --> 00:32:37,240
question like this.
557
00:32:37,240 --> 00:32:40,730
But your brain automatically
figures that out.
558
00:32:40,730 --> 00:32:43,500
I mean, your brain is smart
enough to know that if you've
559
00:32:43,500 --> 00:32:46,300
waited for a long time, you're
probably going to have to wait
560
00:32:46,300 --> 00:32:48,510
for an even longer time.
561
00:32:48,510 --> 00:32:51,570
And it makes sense to move to
another line where your
562
00:32:51,570 --> 00:32:55,090
waiting time is probably
going to be shorter.
563
00:32:55,090 --> 00:32:59,460
So you're pretty smart if you
don't think about it too much.
564
00:33:01,980 --> 00:33:05,980
Here's an interesting theorem
now that makes use of this
565
00:33:05,980 --> 00:33:08,890
memoryless property.
566
00:33:08,890 --> 00:33:12,720
This is Theorem 2.2.1
in the text.
567
00:33:12,720 --> 00:33:14,990
It's not stated terribly
well there.
568
00:33:14,990 --> 00:33:17,660
And I'll tell you why
in a little bit.
569
00:33:17,660 --> 00:33:19,450
It's not stated too badly.
570
00:33:19,450 --> 00:33:20,680
I mean, it's stated correctly.
571
00:33:20,680 --> 00:33:23,620
But it's just a little hard to
understand what it says.
572
00:33:23,620 --> 00:33:27,460
If you have a Poisson process
of rate lambda and you're
573
00:33:27,460 --> 00:33:32,020
looking at any given time
t, here's t down here.
574
00:33:32,020 --> 00:33:35,380
You're looking at the
process of time t.
575
00:33:35,380 --> 00:33:37,690
The interval z--
576
00:33:37,690 --> 00:33:39,580
here's the interval z here--
577
00:33:39,580 --> 00:33:50,760
from t until the next arrival
has distribution e to the
578
00:33:50,760 --> 00:33:52,080
minus lambda z.
579
00:33:52,080 --> 00:33:57,930
And it has this distribution
for all real numbers
580
00:33:57,930 --> 00:33:59,480
greater than 0.
581
00:33:59,480 --> 00:34:04,880
The random variable Z is
independent of n of t.
582
00:34:04,880 --> 00:34:09,260
In other words, this random
variable here is independent
583
00:34:09,260 --> 00:34:14,080
of how many arrivals there
have been at time t.
584
00:34:14,080 --> 00:34:20,540
And given this, it's independent
of s sub n, which
585
00:34:20,540 --> 00:34:24,330
is the time at which the
last arrival occurred.
586
00:34:24,330 --> 00:34:27,429
Namely, here's n
of t equals 2.
587
00:34:27,429 --> 00:34:30,870
Here's s of 2 at time tau.
588
00:34:30,870 --> 00:34:37,630
So given both n of t and s sub
2 in this case, or s sub n of
589
00:34:37,630 --> 00:34:40,549
t as we might call it, and
that's what gets confusing.
590
00:34:40,549 --> 00:34:42,980
And I'll talk about
that later.
591
00:34:42,980 --> 00:34:48,380
Given those two things, the
number n of arrivals in 0t--
592
00:34:48,380 --> 00:34:51,900
well, I got off.
593
00:34:51,900 --> 00:34:54,960
The random variable Z is
independent of n of t.
594
00:34:54,960 --> 00:34:59,110
And given n of t, Z is
independent of all of these
595
00:34:59,110 --> 00:35:02,075
arrival epochs up
until time t.
596
00:35:02,075 --> 00:35:07,800
And it's also independent of
n of t for all values of
597
00:35:07,800 --> 00:35:10,660
tau up until t.
598
00:35:10,660 --> 00:35:12,260
That's what the theorem
states.
599
00:35:12,260 --> 00:35:16,120
What the theorem states is that
this memoryless property
600
00:35:16,120 --> 00:35:19,890
that we've just stated for
random variables is really a
601
00:35:19,890 --> 00:35:23,470
property of the Poisson
process.
602
00:35:23,470 --> 00:35:26,340
When we say that if a random
variable, it's a little hard
603
00:35:26,340 --> 00:35:30,080
to see why would anyone was
calling it memoryless.
604
00:35:30,080 --> 00:35:33,390
When you state it for a Poisson
process, it's very
605
00:35:33,390 --> 00:35:37,190
obvious why we want to
call it memoryless.
606
00:35:37,190 --> 00:35:41,850
It says that this time here from
t, from any arbitrary t,
607
00:35:41,850 --> 00:35:45,790
until the next arrival occurs,
that this is independent of
608
00:35:45,790 --> 00:35:52,380
all this junk that happens
before or up to time t.
609
00:35:52,380 --> 00:35:54,570
That's what the theorem says.
610
00:35:54,570 --> 00:35:57,600
Here's a sort of a
half proof of it.
611
00:35:57,600 --> 00:35:59,780
There's a careful proof
in the notes.
612
00:35:59,780 --> 00:36:01,800
The statement in the notes
is not that careful,
613
00:36:01,800 --> 00:36:02,790
but the proof is.
614
00:36:02,790 --> 00:36:09,010
And the proof is drawn
out perhaps too much.
615
00:36:09,010 --> 00:36:12,730
You can find your comfort level
between this and the
616
00:36:12,730 --> 00:36:14,960
much longer version
in the notes.
617
00:36:14,960 --> 00:36:17,910
You might understand
it well from this.
618
00:36:17,910 --> 00:36:21,430
Given n of t is equal to 2 in
this case, and in general,
619
00:36:21,430 --> 00:36:26,410
given that n of t is equal to
any constant n, and given that
620
00:36:26,410 --> 00:36:31,140
s sub 2 where this 2 is equal to
that 2, given that s sub 2
621
00:36:31,140 --> 00:36:38,010
is equal to tau, then x3, this
value here, the interarrival
622
00:36:38,010 --> 00:36:43,120
arrival time from this previous
arrival before t to
623
00:36:43,120 --> 00:36:48,470
the next arrival after t, namely
x3, is the thing which
624
00:36:48,470 --> 00:36:54,160
bridges across this time that
we selected, t. t is not a
625
00:36:54,160 --> 00:36:55,770
random thing.
626
00:36:55,770 --> 00:36:58,700
t is just something you're
interested in.
627
00:36:58,700 --> 00:37:02,550
I want to catch a plane at 7
o'clock tomorrow evening.
628
00:37:02,550 --> 00:37:05,520
t then is 7 o'clock
tomorrow evening.
629
00:37:05,520 --> 00:37:09,080
What's the time from the last
plane that went out to New
630
00:37:09,080 --> 00:37:12,460
York until the next plane that's
going out to New York?
631
00:37:12,460 --> 00:37:16,510
If the planes are so screwed
up that the schedule means
632
00:37:16,510 --> 00:37:18,990
nothing, then they're just
flying out whenever
633
00:37:18,990 --> 00:37:21,920
they can fly out.
634
00:37:21,920 --> 00:37:25,410
That's the meaning
of this x3 here.
635
00:37:25,410 --> 00:37:31,020
That says that x3, in fact,
has to be bigger
636
00:37:31,020 --> 00:37:32,680
than t minus tau.
637
00:37:32,680 --> 00:37:38,380
If we're given that n of t is
equal to 2 and that the time
638
00:37:38,380 --> 00:37:42,210
of the previous arrival is at
tau, we're given that there
639
00:37:42,210 --> 00:37:45,390
haven't been any arrivals
between the last arrival
640
00:37:45,390 --> 00:37:47,100
before t and t.
641
00:37:47,100 --> 00:37:48,470
That's what we're given.
642
00:37:48,470 --> 00:37:52,060
This was the last arrival before
t by the assumption
643
00:37:52,060 --> 00:37:52,940
we've made.
644
00:37:52,940 --> 00:37:56,450
So we're assuming there's
nothing in this interval.
645
00:37:56,450 --> 00:38:01,720
And then we're asking what is
the remaining time until x3 is
646
00:38:01,720 --> 00:38:02,840
all finished.
647
00:38:02,840 --> 00:38:06,850
And that's the random variable
that we call Z. So Z is x3
648
00:38:06,850 --> 00:38:09,270
minus t minus tau.
649
00:38:09,270 --> 00:38:14,260
The complementary distribution
function of Z conditional on
650
00:38:14,260 --> 00:38:20,640
both n and on s, this n here
and this s here is then
651
00:38:20,640 --> 00:38:24,580
exponential with e to
the minus lambda Z.
652
00:38:24,580 --> 00:38:29,740
Now, if I know that this is
exponential, what can I say
653
00:38:29,740 --> 00:38:31,800
about the random variable
Z itself?
654
00:38:34,510 --> 00:38:39,100
Well, there's an easy way to
find the distribution of Z
655
00:38:39,100 --> 00:38:42,315
when you know Z conditional
onto other things.
656
00:38:48,200 --> 00:38:51,910
You take what the distribution
is conditional on, each value
657
00:38:51,910 --> 00:38:53,600
of n and s.
658
00:38:53,600 --> 00:38:58,510
You then multiply that by the
probability that n and s have
659
00:38:58,510 --> 00:39:00,120
those particular values.
660
00:39:00,120 --> 00:39:03,120
And then you integrate.
661
00:39:03,120 --> 00:39:06,830
Now, we can look at this and
say we don't have to go
662
00:39:06,830 --> 00:39:08,260
through all of that.
663
00:39:08,260 --> 00:39:11,720
And in fact, we won't know what
the distribution of n is.
664
00:39:11,720 --> 00:39:14,670
And we certainly won't know what
the distribution of this
665
00:39:14,670 --> 00:39:18,760
previous arrival is for
quite a long time.
666
00:39:18,760 --> 00:39:21,310
Why don't we need
to know that?
667
00:39:21,310 --> 00:39:26,590
Well, because we know that
whatever n of t is and
668
00:39:26,590 --> 00:39:31,185
whatever s sub n of t is doesn't
make any difference.
669
00:39:31,185 --> 00:39:35,170
The distribution of Z is
still the same thing.
670
00:39:35,170 --> 00:39:39,620
So we know this has to be the
unconditional distribution
671
00:39:39,620 --> 00:39:43,880
function of Z also even without
knowing anything about
672
00:39:43,880 --> 00:39:47,020
n or knowing about s.
673
00:39:47,020 --> 00:39:51,930
And that means that the
complementary distribution
674
00:39:51,930 --> 00:39:57,790
function of Z is equal to e to
the minus lambda Z also.
675
00:39:57,790 --> 00:40:03,210
So that's sort of a proof if you
want to be really picky.
676
00:40:03,210 --> 00:40:05,870
And I would suggest you
try to be picky.
677
00:40:05,870 --> 00:40:09,640
When you read the notes, try to
understand why one has to
678
00:40:09,640 --> 00:40:12,260
say a little more than
one says here.
679
00:40:12,260 --> 00:40:13,620
Because that's the
way you really
680
00:40:13,620 --> 00:40:15,720
understand these things.
681
00:40:15,720 --> 00:40:18,630
But this really gives you
the idea of the proof.
682
00:40:18,630 --> 00:40:20,630
And it's pretty close
to a complete proof.
683
00:40:24,830 --> 00:40:26,430
This is saying what
we just said.
684
00:40:26,430 --> 00:40:30,380
The conditional distribution
of Z doesn't vary with the
685
00:40:30,380 --> 00:40:33,050
conditioning values.
686
00:40:33,050 --> 00:40:35,240
n of t equals n.
687
00:40:35,240 --> 00:40:37,150
And s sub n equals tau.
688
00:40:37,150 --> 00:40:42,040
So Z is statistically
independent of n of t and s
689
00:40:42,040 --> 00:40:43,550
sub n of t.
690
00:40:43,550 --> 00:40:47,890
You should look at the text
again, as I said, for more
691
00:40:47,890 --> 00:40:49,880
careful proof of that.
692
00:40:49,880 --> 00:40:53,940
What is this random variable
s sub n of t?
693
00:40:53,940 --> 00:40:57,640
It's clear from the picture
what it is.
694
00:40:57,640 --> 00:41:04,400
s sub n of t is the last
arrival before
695
00:41:04,400 --> 00:41:07,580
we're at time t.
696
00:41:07,580 --> 00:41:10,970
That's what it is in
the picture here.
697
00:41:10,970 --> 00:41:13,540
How do you define a random
variable like that?
698
00:41:17,080 --> 00:41:21,700
There's a temptation to
do it the following
699
00:41:21,700 --> 00:41:24,170
way which is incorrect.
700
00:41:24,170 --> 00:41:28,710
There's a temptation to say,
well, conditional on n of t,
701
00:41:28,710 --> 00:41:31,500
suppose n of t is equal to n.
702
00:41:31,500 --> 00:41:36,730
Let me then find the
distribution of s sub n.
703
00:41:36,730 --> 00:41:39,510
And that's not the right
way to do it.
704
00:41:39,510 --> 00:41:44,500
Because s sub n of t and n
of t are certainly not
705
00:41:44,500 --> 00:41:45,510
independent.
706
00:41:45,510 --> 00:41:48,590
n of t tells you what random
variable you want to look at.
707
00:41:52,110 --> 00:41:55,750
How do you define a random
variable in terms of a mapping
708
00:41:55,750 --> 00:42:02,520
from the sample space omega onto
the set of real numbers?
709
00:42:02,520 --> 00:42:07,040
So what you do here is you look
at a sample point omega.
710
00:42:07,040 --> 00:42:12,200
It maps into this random
variable n of t, the sample
711
00:42:12,200 --> 00:42:16,520
value of that at omega,
that's sum value n.
712
00:42:16,520 --> 00:42:22,210
And then you map that same
sample point into--
713
00:42:22,210 --> 00:42:24,520
now, you know which random
variable it is
714
00:42:24,520 --> 00:42:25,660
you're looking at.
715
00:42:25,660 --> 00:42:30,160
You take that same omega and
map it into sub time tau.
716
00:42:30,160 --> 00:42:34,740
So that's what we mean
by s sub n of t.
717
00:42:34,740 --> 00:42:38,800
If your mind glazes over at
that, don't worry about it.
718
00:42:38,800 --> 00:42:40,650
Think about it a
little bit now.
719
00:42:40,650 --> 00:42:42,900
Come back and think
about it later.
720
00:42:42,900 --> 00:42:46,220
Every time I don't think about
this for two weeks, my mind
721
00:42:46,220 --> 00:42:47,910
glazes over when I look at it.
722
00:42:47,910 --> 00:42:51,310
And I have to think very hard
about what this very peculiar
723
00:42:51,310 --> 00:42:53,610
looking random variable is.
724
00:42:53,610 --> 00:42:58,660
When I have a random variable
where I have a sequence of
725
00:42:58,660 --> 00:43:02,460
random variables, and I have a
random variable which is a
726
00:43:02,460 --> 00:43:06,270
random selection among those
random variables, it's a very
727
00:43:06,270 --> 00:43:08,180
complicated animal.
728
00:43:08,180 --> 00:43:09,977
And that's what this is.
729
00:43:09,977 --> 00:43:12,990
But we've just said
what it is.
730
00:43:12,990 --> 00:43:18,760
So you can think about
it as you go.
731
00:43:18,760 --> 00:43:20,600
The theorem essentially
extends the idea of
732
00:43:20,600 --> 00:43:23,340
memorylessness to the entire
Poisson process.
733
00:43:23,340 --> 00:43:26,390
In other words, this says that
a Poisson process is
734
00:43:26,390 --> 00:43:27,910
memoryless.
735
00:43:27,910 --> 00:43:30,440
You look at a particular
time t.
736
00:43:30,440 --> 00:43:33,980
And the time until the next
arrival is independent of
737
00:43:33,980 --> 00:43:35,780
everything that's going
before that.
738
00:43:40,490 --> 00:43:43,530
Starting at any time tau,
yeah, well, subsequent
739
00:43:43,530 --> 00:43:47,360
interrarrival times are
independent of Z
740
00:43:47,360 --> 00:43:50,150
and also of the past.
741
00:43:50,150 --> 00:43:53,420
I'm waving my hands
a little bit here.
742
00:43:53,420 --> 00:43:55,370
But in fact, what I'm
saying is right.
743
00:43:55,370 --> 00:43:58,930
We have these interarrival
intervals that we know are
744
00:43:58,930 --> 00:44:00,130
independent.
745
00:44:00,130 --> 00:44:04,090
The interarrival intervals which
have occurred completely
746
00:44:04,090 --> 00:44:10,400
before time t are independent of
this random variable Z. The
747
00:44:10,400 --> 00:44:14,760
next interarrival interval after
Z is independent of all
748
00:44:14,760 --> 00:44:17,370
the interarrival intervals
before that.
749
00:44:17,370 --> 00:44:25,550
And those interarrival intervals
before that are
750
00:44:25,550 --> 00:44:30,580
determined by the counting
process up until time t.
751
00:44:30,580 --> 00:44:33,230
So the counting process
corresponds to this
752
00:44:33,230 --> 00:44:37,020
corresponding interarrival
process.
753
00:44:37,020 --> 00:44:41,830
It's n of t prime minus n of t
for t prime greater than t.
754
00:44:41,830 --> 00:44:44,520
In other words, we now want to
look at a counting process
755
00:44:44,520 --> 00:44:49,460
which starts at time t and
follows whatever it has to
756
00:44:49,460 --> 00:44:52,960
follow from this original
counting process.
757
00:44:52,960 --> 00:44:56,800
And what we're saying is this
first arrival and this process
758
00:44:56,800 --> 00:45:00,740
starting at time t is
independent of everything that
759
00:45:00,740 --> 00:45:02,100
went before.
760
00:45:02,100 --> 00:45:05,960
And every subsequent
interarrival time after that
761
00:45:05,960 --> 00:45:10,070
is independent of everything
before time t.
762
00:45:10,070 --> 00:45:15,420
So this says that the process n
of t prime minus n of t as a
763
00:45:15,420 --> 00:45:17,270
process nt prime.
764
00:45:17,270 --> 00:45:21,560
This is a counting process nt
prime defined for t prime
765
00:45:21,560 --> 00:45:22,590
greater than t.
766
00:45:22,590 --> 00:45:27,870
So for fixed t, we now have
something which we can view
767
00:45:27,870 --> 00:45:31,670
over variable t prime as
a counting process.
768
00:45:31,670 --> 00:45:36,050
It's a Poisson process shifted
to start at time t, ie, for
769
00:45:36,050 --> 00:45:40,520
each t prime, n of t prime minus
the n of t has the same
770
00:45:40,520 --> 00:45:45,230
distribution as n of
t prime minus t.
771
00:45:45,230 --> 00:45:47,040
Same for joint distributions.
772
00:45:47,040 --> 00:45:49,740
In other words, this
random variable Z
773
00:45:49,740 --> 00:45:50,710
is exponential again.
774
00:45:50,710 --> 00:45:53,960
And all the future interarrival
times are
775
00:45:53,960 --> 00:45:55,000
exponential.
776
00:45:55,000 --> 00:45:58,940
So it's defined in exactly the
same way as the original
777
00:45:58,940 --> 00:46:01,100
random process is.
778
00:46:01,100 --> 00:46:03,155
So it's statistically
the same process.
779
00:46:05,770 --> 00:46:08,390
Which says two things
about it.
780
00:46:08,390 --> 00:46:09,810
Everything is the same.
781
00:46:09,810 --> 00:46:12,890
And everything is independent.
782
00:46:12,890 --> 00:46:15,560
We will call that stationary.
783
00:46:15,560 --> 00:46:17,140
Everything is the same.
784
00:46:17,140 --> 00:46:20,400
And independent, everything
is independent.
785
00:46:20,400 --> 00:46:23,310
And then we'll try to sort out
how things can be the same but
786
00:46:23,310 --> 00:46:25,750
also be independent.
787
00:46:25,750 --> 00:46:27,850
Oh, we already know that.
788
00:46:27,850 --> 00:46:33,010
We have two IID random
variables, x1 and x2.
789
00:46:33,010 --> 00:46:34,250
They're IID.
790
00:46:34,250 --> 00:46:36,890
They're independent and
identically distributed.
791
00:46:36,890 --> 00:46:39,330
Identity distributed means
that in one sense,
792
00:46:39,330 --> 00:46:40,870
they are the same.
793
00:46:40,870 --> 00:46:43,630
But they're also independent
of each other.
794
00:46:43,630 --> 00:46:48,080
So the random variables are
defined in the same way.
795
00:46:48,080 --> 00:46:50,180
And in that sense, they're
stationary.
796
00:46:50,180 --> 00:46:53,160
But they're independent of each
other by the definition
797
00:46:53,160 --> 00:46:55,390
of independence.
798
00:46:55,390 --> 00:47:00,570
So our new process is
independent of the old process
799
00:47:00,570 --> 00:47:08,950
in the interval 0 up to t.
800
00:47:08,950 --> 00:47:11,750
When we're talking about Poisson
processes and also
801
00:47:11,750 --> 00:47:17,350
arrival processes, we always
talk about intervals which are
802
00:47:17,350 --> 00:47:21,460
open on the left and closed
on the right.
803
00:47:21,460 --> 00:47:23,920
That's completely arbitrary.
804
00:47:23,920 --> 00:47:27,250
But if you don't make one
convention or the other, you
805
00:47:27,250 --> 00:47:31,940
could make them closed on the
left and open on the right,
806
00:47:31,940 --> 00:47:34,830
and that would be
consistent also.
807
00:47:34,830 --> 00:47:35,675
But nobody does.
808
00:47:35,675 --> 00:47:38,210
And it would be much
more confusing.
809
00:47:38,210 --> 00:47:42,870
So it's much easier to make
things closed on the right.
810
00:47:47,280 --> 00:47:50,820
So we're up to stationary and
independent increments.
811
00:47:50,820 --> 00:47:52,990
Well, we're not up to there.
812
00:47:52,990 --> 00:47:54,960
We're almost finished
with that.
813
00:47:54,960 --> 00:47:59,340
We've virtually already said
that increments are stationary
814
00:47:59,340 --> 00:48:00,000
and independent.
815
00:48:00,000 --> 00:48:04,560
And an increment is just a piece
of a Poisson process.
816
00:48:04,560 --> 00:48:06,220
That's an increment,
a piece of it.
817
00:48:09,620 --> 00:48:16,070
So a counting process has the
stationary increment property
818
00:48:16,070 --> 00:48:21,493
if n of t prime minus n of t has
the same distribution as n
819
00:48:21,493 --> 00:48:26,130
of t prime minus t for all
t prime greater than t
820
00:48:26,130 --> 00:48:28,060
greater than 0.
821
00:48:28,060 --> 00:48:30,950
In other words, you look at
this counting process.
822
00:48:30,950 --> 00:48:32,740
Goes up.
823
00:48:32,740 --> 00:48:35,970
Then you start at some
particular value of t.
824
00:48:35,970 --> 00:48:38,300
Let me draw a picture of that.
825
00:48:38,300 --> 00:48:39,550
Make it a little clearer.
826
00:49:00,330 --> 00:49:05,960
And the new Poisson process
starts at this value and goes
827
00:49:05,960 --> 00:49:08,360
up from there.
828
00:49:08,360 --> 00:49:13,970
So this thing here is
what we call n of t
829
00:49:13,970 --> 00:49:18,000
prime minus n of t.
830
00:49:18,000 --> 00:49:19,520
Because here's n of t.
831
00:49:22,190 --> 00:49:26,560
Here's t prime out here for
any value out here.
832
00:49:26,560 --> 00:49:29,880
And we're looking at the number
of arrivals up until
833
00:49:29,880 --> 00:49:31,410
time t prime.
834
00:49:31,410 --> 00:49:34,442
And what we're talking about,
when we're talking about n of
835
00:49:34,442 --> 00:49:40,110
t prime minus n of t, we're
talking about what happens in
836
00:49:40,110 --> 00:49:43,210
this region here.
837
00:49:43,210 --> 00:49:48,010
And we're saying that this is
a Poisson process again.
838
00:49:48,010 --> 00:49:50,530
And now in a minute, we're going
to say that this Poisson
839
00:49:50,530 --> 00:49:57,720
process is independent of what
happened up until time t.
840
00:49:57,720 --> 00:49:59,460
But Poisson processes have this
841
00:49:59,460 --> 00:50:02,540
stationary increment property.
842
00:50:02,540 --> 00:50:08,930
And a counting process has the
independent increment property
843
00:50:08,930 --> 00:50:14,970
if for every sequence of times,
t1, t2, up to t sub n.
844
00:50:14,970 --> 00:50:22,830
The random variables n of t1 and
tilde of t1, t2, we didn't
845
00:50:22,830 --> 00:50:23,830
talk about that.
846
00:50:23,830 --> 00:50:27,750
But I think it's defined
on one of those slides.
847
00:50:27,750 --> 00:50:41,480
n of t and t prime is defined as
n of t prime minus n of t.
848
00:50:44,830 --> 00:50:49,740
So n of t and t prime is really
the number of arrivals
849
00:50:49,740 --> 00:50:55,360
that have occurred from
t up until t prime--
850
00:50:55,360 --> 00:50:59,370
open on t, closed on t prime.
851
00:50:59,370 --> 00:51:07,410
So a counting process has the
independent increment property
852
00:51:07,410 --> 00:51:11,310
if for every finite set of
times, these random variables
853
00:51:11,310 --> 00:51:13,920
here are independent.
854
00:51:13,920 --> 00:51:16,860
The number of arrivals in the
first increment, number of
855
00:51:16,860 --> 00:51:19,860
arrivals in the second
increment, number of arrivals
856
00:51:19,860 --> 00:51:23,320
in the third increment, no
matter how you choose t1, t2,
857
00:51:23,320 --> 00:51:27,490
up to t sub n, what happens here
is independent of what
858
00:51:27,490 --> 00:51:29,600
happens here, is independent
of what happens
859
00:51:29,600 --> 00:51:31,790
here, and so forth.
860
00:51:31,790 --> 00:51:33,270
It's not only that
what happens in
861
00:51:33,270 --> 00:51:35,540
Las Vegas stays there.
862
00:51:35,540 --> 00:51:38,200
It's that what happens in
Boston stays there, what
863
00:51:38,200 --> 00:51:41,050
happens in Philadelphia stays
there, and so forth.
864
00:51:41,050 --> 00:51:43,740
What happens anywhere
stays anywhere.
865
00:51:43,740 --> 00:51:45,850
It never gets out of there.
866
00:51:45,850 --> 00:51:48,170
That's what we mean by
independence in this case.
867
00:51:48,170 --> 00:51:51,710
So it's a strong statement.
868
00:51:51,710 --> 00:51:55,000
But we've essentially said that
Poisson processes have
869
00:51:55,000 --> 00:51:57,410
that property.
870
00:51:57,410 --> 00:52:00,540
So this property implies is the
number of arrivals in each
871
00:52:00,540 --> 00:52:04,640
of the set of non-overlapping
intervals are independent
872
00:52:04,640 --> 00:52:06,270
random variables.
873
00:52:06,270 --> 00:52:18,010
For a Poisson process, we've
seen that the number of
874
00:52:18,010 --> 00:52:23,450
arrivals in t sub i minus 1 to t
sub i is independent of this
875
00:52:23,450 --> 00:52:26,960
whole set of random
variables here.
876
00:52:26,960 --> 00:52:30,520
Now, remember that when we're
talking about multiple random
877
00:52:30,520 --> 00:52:34,010
variables, we say that multiple
random variables are
878
00:52:34,010 --> 00:52:35,400
independent.
879
00:52:35,400 --> 00:52:37,740
It's not enough to be pairwise
independent.
880
00:52:37,740 --> 00:52:40,080
They all have to
be independent.
881
00:52:40,080 --> 00:52:44,260
But this thing we've just said
says that this is independent
882
00:52:44,260 --> 00:52:46,560
of all of these things.
883
00:52:46,560 --> 00:52:50,260
If this is independent of all of
these things, and then the
884
00:52:50,260 --> 00:52:55,610
next interval n of ti, ti plus
1, is independent of
885
00:52:55,610 --> 00:52:59,290
everything in the past, and so
forth all the way up, then all
886
00:52:59,290 --> 00:53:02,870
of those random variables are
statistically independent of
887
00:53:02,870 --> 00:53:03,710
each other.
888
00:53:03,710 --> 00:53:08,130
So in fact, we're saying more
than pairwise statistical
889
00:53:08,130 --> 00:53:10,790
independence.
890
00:53:10,790 --> 00:53:14,970
If you're panicking about
these minor differences
891
00:53:14,970 --> 00:53:19,860
between pairwise independence
and real independence, don't
892
00:53:19,860 --> 00:53:21,240
worry about it too much.
893
00:53:21,240 --> 00:53:24,430
Because the situations
where that happens
894
00:53:24,430 --> 00:53:26,240
are relatively rare.
895
00:53:26,240 --> 00:53:28,030
They don't happen
all the time.
896
00:53:28,030 --> 00:53:29,720
But they do happen
occasionally.
897
00:53:29,720 --> 00:53:33,157
So you should be aware of it.
898
00:53:33,157 --> 00:53:35,100
You shouldn't get in
a panic about it.
899
00:53:35,100 --> 00:53:38,200
Because normally, you don't
have to worry about it.
900
00:53:38,200 --> 00:53:42,240
In other words, when you're
taking a quiz, don't worry
901
00:53:42,240 --> 00:53:45,520
about any of the fine points.
902
00:53:45,520 --> 00:53:49,060
Figure out roughly how
to do the problems.
903
00:53:49,060 --> 00:53:51,620
Do them more or less.
904
00:53:51,620 --> 00:53:56,010
And then come back and deal with
the fine points later.
905
00:53:56,010 --> 00:53:59,430
Don't spend the whole quiz time
wrapped up on one little
906
00:53:59,430 --> 00:54:03,370
fine point and not get
to anything else.
907
00:54:03,370 --> 00:54:06,340
One of the important things to
learn in understanding a
908
00:54:06,340 --> 00:54:10,380
subject like this is to figure
out what are the fine points,
909
00:54:10,380 --> 00:54:12,370
what are the important points.
910
00:54:12,370 --> 00:54:14,740
How do you tell whether
something is important in a
911
00:54:14,740 --> 00:54:16,000
particular context.
912
00:54:16,000 --> 00:54:19,870
And that just takes intuition.
913
00:54:19,870 --> 00:54:23,730
That takes some intuition from
working with these processes.
914
00:54:23,730 --> 00:54:26,970
And you pick that
up as you go.
915
00:54:26,970 --> 00:54:30,840
But anyway, we wind up now
with the statement that
916
00:54:30,840 --> 00:54:34,440
Poisson processes have
stationary and independent
917
00:54:34,440 --> 00:54:35,070
increments.
918
00:54:35,070 --> 00:54:37,730
Which means that what happens
in each interval is
919
00:54:37,730 --> 00:54:42,800
independent of what happens
in each other interval.
920
00:54:42,800 --> 00:54:47,390
So we're done with that until
we get to alternate
921
00:54:47,390 --> 00:54:50,045
definitions of a Poisson
process.
922
00:54:50,045 --> 00:54:54,590
And we now want to deal with
the Erlang and the Poisson
923
00:54:54,590 --> 00:55:00,400
distributions, which are just
very plug and chug kinds of
924
00:55:00,400 --> 00:55:03,700
things to a certain extent.
925
00:55:03,700 --> 00:55:09,695
For a Poisson process of rate
lambda, the density function
926
00:55:09,695 --> 00:55:17,130
of arrival epoch s2, s2 is
the sum of x1 plus x2.
927
00:55:17,130 --> 00:55:20,730
x1 is an exponential random
variable of rate lambda.
928
00:55:20,730 --> 00:55:26,420
x2 is an independent random
variable of rate lambda.
929
00:55:26,420 --> 00:55:31,470
How do you find the probability
density function
930
00:55:31,470 --> 00:55:34,470
as a sum of two independent
random variables, which both
931
00:55:34,470 --> 00:55:35,810
have a density?
932
00:55:35,810 --> 00:55:37,930
You convolve them.
933
00:55:37,930 --> 00:55:43,610
That's something that you've
known ever since you studied
934
00:55:43,610 --> 00:55:47,170
any kind of linear systems, or
from any probability, or
935
00:55:47,170 --> 00:55:47,970
anything else.
936
00:55:47,970 --> 00:55:51,050
Convolution is the way to
solve this problem.
937
00:55:51,050 --> 00:55:54,670
When you convolve these
two random variables,
938
00:55:54,670 --> 00:55:56,400
here I've done it.
939
00:55:56,400 --> 00:56:00,322
You get lambda squared t times
e to the minus lambda t.
940
00:56:03,040 --> 00:56:07,400
This kind of form here with an
e to the minus lambda t, and
941
00:56:07,400 --> 00:56:11,430
with a t, or t squared, or so
forth, is a particularly easy
942
00:56:11,430 --> 00:56:13,500
form to integrate.
943
00:56:13,500 --> 00:56:16,790
So we just do this
again and again.
944
00:56:16,790 --> 00:56:19,400
And when we do it again and
again, we find out that the
945
00:56:19,400 --> 00:56:25,010
density function as a sum of n
of these random variables, you
946
00:56:25,010 --> 00:56:27,840
keep picking up an extra lambda
every time you convolve
947
00:56:27,840 --> 00:56:31,860
in another exponential
random variable.
948
00:56:31,860 --> 00:56:36,450
You pick up an extra factor of
t whenever you do this again.
949
00:56:36,450 --> 00:56:39,770
This stays the same
as it does here.
950
00:56:39,770 --> 00:56:45,480
And strangely enough, this n
minus 1 factorial appears down
951
00:56:45,480 --> 00:56:51,770
here when you start integrating
something with
952
00:56:51,770 --> 00:56:54,315
some power of t in it.
953
00:56:54,315 --> 00:56:57,030
So when you integrate this,
this is what you get.
954
00:56:57,030 --> 00:57:01,030
And it's called the
Erlang density.
955
00:57:01,030 --> 00:57:02,230
Any questions about this?
956
00:57:02,230 --> 00:57:04,620
Or any questions
about anything?
957
00:57:08,650 --> 00:57:09,420
I'm getting hoarse.
958
00:57:09,420 --> 00:57:10,310
I need questions.
959
00:57:10,310 --> 00:57:14,860
[LAUGHS]
960
00:57:14,860 --> 00:57:19,380
There's nothing much to
worry about there.
961
00:57:19,380 --> 00:57:21,900
But now, we want to stop and
smell the roses while doing
962
00:57:21,900 --> 00:57:24,710
all this computation.
963
00:57:24,710 --> 00:57:27,640
Let's do this a slightly
different way.
964
00:57:27,640 --> 00:57:40,100
The joint density of x1 up to
x sub n is lambda x1 times e
965
00:57:40,100 --> 00:57:44,850
to the minus lambda x1, times
lambda x2, times e to the
966
00:57:44,850 --> 00:57:48,270
minus lambda x2, and so forth.
967
00:57:48,270 --> 00:57:51,180
So excuse me.
968
00:57:51,180 --> 00:57:54,880
The probability density of an
exponential random variable is
969
00:57:54,880 --> 00:57:57,950
lambda times e to the
minus lambda x.
970
00:57:57,950 --> 00:58:06,900
So the joint density is lambda
e to the minus lambda x1.
971
00:58:06,900 --> 00:58:08,270
I told you I was
getting hoarse.
972
00:58:08,270 --> 00:58:09,580
And my mind is getting hoarse.
973
00:58:09,580 --> 00:58:14,280
So you better start asking
some questions or I will
974
00:58:14,280 --> 00:58:16,480
evolve into meaningless
chatter.
975
00:58:20,490 --> 00:58:26,770
And this is just lambda to the
n times e to the minus lambda
976
00:58:26,770 --> 00:58:34,840
times the summation of x sub
i from i equals 1 to n.
977
00:58:34,840 --> 00:58:37,160
Now, that's sort of interesting
because this joint
978
00:58:37,160 --> 00:58:43,710
density is just this
simple-minded thing.
979
00:58:43,710 --> 00:58:47,010
You can write it as lambda to
the n times e to the minus
980
00:58:47,010 --> 00:58:50,730
lambda s sub n, where
s sub n is the
981
00:58:50,730 --> 00:58:53,610
time of the n-th arrival.
982
00:58:53,610 --> 00:58:57,910
This says that the joint
distribution of all of these
983
00:58:57,910 --> 00:59:01,610
interarrival times only
depends on when the
984
00:59:01,610 --> 00:59:04,310
last one comes in.
985
00:59:04,310 --> 00:59:09,750
And you can transform that to a
joint density on each of the
986
00:59:09,750 --> 00:59:14,900
arrival epochs as lambda to
the n times e to the minus
987
00:59:14,900 --> 00:59:17,290
lambda s sub n.
988
00:59:17,290 --> 00:59:18,540
Is this obvious to everyone?
989
00:59:21,620 --> 00:59:22,990
You're lying.
990
00:59:22,990 --> 00:59:26,700
If you're not shaking your
head, you're lying.
991
00:59:26,700 --> 00:59:29,350
Because it's not
obvious at all.
992
00:59:29,350 --> 00:59:35,110
What we're doing here, it's sort
of obvious if you look at
993
00:59:35,110 --> 00:59:37,040
the picture.
994
00:59:37,040 --> 00:59:40,420
It's not obvious when you
do the mathematics.
995
00:59:40,420 --> 00:59:44,280
What the picture says
is-- let me see if I
996
00:59:44,280 --> 00:59:45,530
find the picture again.
997
00:59:52,930 --> 00:59:53,720
OK.
998
00:59:53,720 --> 00:59:55,800
Here's the picture up here.
999
00:59:55,800 --> 00:59:58,953
We're looking at these
interarrival intervals.
1000
01:00:01,890 --> 01:00:04,450
I think it'll be clearer if
I draw it a different way.
1001
01:00:07,820 --> 01:00:09,070
There we go.
1002
01:00:14,360 --> 01:00:17,410
Let's just draw this
in a line.
1003
01:00:17,410 --> 01:00:19,590
Here's 0.
1004
01:00:19,590 --> 01:00:21,970
Here's s1.
1005
01:00:21,970 --> 01:00:24,360
Here's s2.
1006
01:00:24,360 --> 01:00:26,826
Here's s3.
1007
01:00:26,826 --> 01:00:28,076
And here's s4.
1008
01:00:30,790 --> 01:00:33,970
And here's x1.
1009
01:00:33,970 --> 01:00:35,220
Here's x2.
1010
01:00:41,130 --> 01:00:42,380
Here's x3.
1011
01:00:44,726 --> 01:00:45,976
And here's x4.
1012
01:00:49,930 --> 01:00:53,370
Now, what we're talking about,
we can go from the density of
1013
01:00:53,370 --> 01:01:01,150
each of these intervals to the
density of each of these sums
1014
01:01:01,150 --> 01:01:02,980
in a fairly straightforward
way.
1015
01:01:02,980 --> 01:01:12,530
If you write this all out as a
density, what you find is that
1016
01:01:12,530 --> 01:01:15,780
in making a transformation
from the density of these
1017
01:01:15,780 --> 01:01:21,230
interarrival intervals to the
density of these, what you're
1018
01:01:21,230 --> 01:01:25,170
essentially doing is taking this
density and multiplying
1019
01:01:25,170 --> 01:01:27,900
it by a matrix.
1020
01:01:27,900 --> 01:01:33,770
And the matrix is a diagonal
matrix, is an
1021
01:01:33,770 --> 01:01:35,840
upper triangular matrix.
1022
01:01:35,840 --> 01:01:38,490
Because this depends
only on this.
1023
01:01:38,490 --> 01:01:40,680
This depends only on
this and this.
1024
01:01:40,680 --> 01:01:42,930
This depends only on
this and this.
1025
01:01:42,930 --> 01:01:45,620
This depends only on
each of these.
1026
01:01:45,620 --> 01:01:50,670
So it's a triangular matrix with
terms on the diagonal.
1027
01:01:50,670 --> 01:01:53,300
And when you look at a matrix
like that, the terms on the
1028
01:01:53,300 --> 01:01:57,720
diagonal are 1s because what's
getting added each time is 1
1029
01:01:57,720 --> 01:01:58,900
times a new variable.
1030
01:01:58,900 --> 01:02:03,430
So we have a matrix with 1s on
the main diagonal and other
1031
01:02:03,430 --> 01:02:06,250
stuff above that.
1032
01:02:06,250 --> 01:02:09,000
And what that means is that
when you make this
1033
01:02:09,000 --> 01:02:11,770
transformation in densities,
the determinant of
1034
01:02:11,770 --> 01:02:13,550
that matrix is 1.
1035
01:02:13,550 --> 01:02:17,400
And the value that you then
get when you go from the
1036
01:02:17,400 --> 01:02:24,420
density of these to the density
of these, it's a
1037
01:02:24,420 --> 01:02:26,160
uniform density again.
1038
01:02:26,160 --> 01:02:29,760
So in fact, it has to
look like what we
1039
01:02:29,760 --> 01:02:32,530
said it looks like.
1040
01:02:32,530 --> 01:02:34,300
So I was kidding you there.
1041
01:02:34,300 --> 01:02:39,310
It's not so obvious how to
do that although it looks
1042
01:02:39,310 --> 01:02:40,560
reasonable.
1043
01:02:50,195 --> 01:02:51,180
AUDIENCE: [INAUDIBLE].
1044
01:02:51,180 --> 01:02:51,870
PROFESSOR: Yeah.
1045
01:02:51,870 --> 01:02:53,380
AUDIENCE: I'm sorry.
1046
01:02:53,380 --> 01:02:58,140
Is it also valid to make an
argument based on symmetry?
1047
01:02:58,140 --> 01:03:01,700
PROFESSOR: It will be later.
1048
01:03:01,700 --> 01:03:04,670
The symmetry is not
clear here yet.
1049
01:03:04,670 --> 01:03:06,570
I mean, the symmetry isn't
clear because you're
1050
01:03:06,570 --> 01:03:08,996
starting at time 0.
1051
01:03:08,996 --> 01:03:14,050
And because you're starting
at time 0, you don't have
1052
01:03:14,050 --> 01:03:16,530
symmetry here yet.
1053
01:03:16,530 --> 01:03:20,750
If we started at time 0 and we
ended at some time t, we could
1054
01:03:20,750 --> 01:03:23,120
try to claim there is some
kind of symmetry between
1055
01:03:23,120 --> 01:03:25,010
everything that happened
in the middle.
1056
01:03:25,010 --> 01:03:27,880
And we'll try to
do that later.
1057
01:03:27,880 --> 01:03:33,370
But at the moment, we would get
into even more trouble if
1058
01:03:33,370 --> 01:03:34,620
we try to do it by symmetry.
1059
01:03:41,220 --> 01:03:44,440
But anyway, what this is saying
is that this joint
1060
01:03:44,440 --> 01:03:48,310
density is really--
1061
01:03:48,310 --> 01:03:51,740
if you know where this point is,
the joint density of all
1062
01:03:51,740 --> 01:03:55,570
of these things remains the same
no matter how you move
1063
01:03:55,570 --> 01:03:57,450
these things around.
1064
01:03:57,450 --> 01:04:01,210
If I move s1 around a little
bit, it means that x1 gets a
1065
01:04:01,210 --> 01:04:04,500
little smaller, x2 gets
a little bit bigger.
1066
01:04:04,500 --> 01:04:07,200
And if you look at the joint
density there, the joint
1067
01:04:07,200 --> 01:04:11,120
density stays absolutely the
same because you have e to the
1068
01:04:11,120 --> 01:04:15,240
minus lambda x1 times e to
the minus lambda x2.
1069
01:04:15,240 --> 01:04:17,980
And the sum of the two for a
fixed value here is the same
1070
01:04:17,980 --> 01:04:19,370
as it was before.
1071
01:04:19,370 --> 01:04:22,120
So you can move all of these
things around in any
1072
01:04:22,120 --> 01:04:23,610
way you want to.
1073
01:04:23,610 --> 01:04:28,760
And the joint density depends
only on the last one.
1074
01:04:28,760 --> 01:04:30,600
And that's a very strange
property and it's a very
1075
01:04:30,600 --> 01:04:32,910
interesting property.
1076
01:04:32,910 --> 01:04:36,200
And it sort of is the same as
this independent increment
1077
01:04:36,200 --> 01:04:37,960
property that we've been
talking about.
1078
01:04:37,960 --> 01:04:41,790
But we'll see why that
is in just a minute.
1079
01:04:41,790 --> 01:04:46,280
But anyway, once we have that
property, we can then
1080
01:04:46,280 --> 01:04:55,480
integrate this over the volume
of s1, s2, s3, and s4, over
1081
01:04:55,480 --> 01:05:01,310
that volume which has the
property that it stops at that
1082
01:05:01,310 --> 01:05:03,480
one particular point there.
1083
01:05:03,480 --> 01:05:07,490
And we do that integration
subject to the fact that s3
1084
01:05:07,490 --> 01:05:11,490
has to be less than or equal to
s4, s2 has to be less than
1085
01:05:11,490 --> 01:05:14,790
or equal to s3, and
so forth down.
1086
01:05:14,790 --> 01:05:18,360
When you do that integration,
you get exactly the same thing
1087
01:05:18,360 --> 01:05:20,540
as you got before when you
did the integration.
1088
01:05:20,540 --> 01:05:23,420
The integration that you did
before was essentially doing
1089
01:05:23,420 --> 01:05:26,180
this, if you look at what
you did before.
1090
01:05:29,700 --> 01:05:35,700
You were taking lambda times e
to the minus lambda x times
1091
01:05:35,700 --> 01:05:38,650
lambda times t minus x.
1092
01:05:38,650 --> 01:05:41,390
And the x doesn't make
any difference here.
1093
01:05:41,390 --> 01:05:43,340
The x cancels out.
1094
01:05:43,340 --> 01:05:45,410
That's exactly what's
going on.
1095
01:05:45,410 --> 01:05:50,185
And if you do it in terms of s1
and s2, the s1 cancels out.
1096
01:05:50,185 --> 01:05:52,330
The s1 is the same as x here.
1097
01:05:52,330 --> 01:05:54,810
So there is that cancellation
here.
1098
01:05:54,810 --> 01:05:59,090
And therefore, this Erlang
density is just a marginal
1099
01:05:59,090 --> 01:06:02,100
distribution of a very
interesting joint
1100
01:06:02,100 --> 01:06:04,670
distribution, which depends
only on the last term.
1101
01:06:09,040 --> 01:06:14,760
So next, we have a theorem
which says for a Poisson
1102
01:06:14,760 --> 01:06:21,250
process, the PMF for n of t, the
Probability Mass Function,
1103
01:06:21,250 --> 01:06:23,360
is the Poisson PMF.
1104
01:06:23,360 --> 01:06:26,710
It sounds like I'm not really
saying anything because what
1105
01:06:26,710 --> 01:06:28,640
else would it be?
1106
01:06:28,640 --> 01:06:35,320
Because you've always heard that
the Poisson PMF is this
1107
01:06:35,320 --> 01:06:37,750
particular function here.
1108
01:06:37,750 --> 01:06:40,960
Well, in fact, there's
some reason for that.
1109
01:06:40,960 --> 01:06:44,180
And in fact, if we want to say
that a Poisson process is
1110
01:06:44,180 --> 01:06:49,000
defined in terms of these
exponential interarrival
1111
01:06:49,000 --> 01:06:50,960
times, then we have to
show that this is
1112
01:06:50,960 --> 01:06:54,320
consistent with that.
1113
01:06:54,320 --> 01:06:58,370
The way I'll prove that
here, this is a
1114
01:06:58,370 --> 01:07:01,330
little more than a PF.
1115
01:07:01,330 --> 01:07:05,445
Maybe we should say it's a
P-R-O-F. Leave out the double
1116
01:07:05,445 --> 01:07:08,980
O because it's not
quite complete.
1117
01:07:08,980 --> 01:07:13,350
But what we want to do is to
calculate the probability that
1118
01:07:13,350 --> 01:07:18,770
the n plus first arrival occurs
sometime between t and
1119
01:07:18,770 --> 01:07:21,770
t plus delta.
1120
01:07:21,770 --> 01:07:24,160
And we'll do it in two
different ways.
1121
01:07:24,160 --> 01:07:28,220
And one way involves the
probability mass function for
1122
01:07:28,220 --> 01:07:29,440
the Poisson.
1123
01:07:29,440 --> 01:07:33,730
The other way involves
the Erlang density.
1124
01:07:33,730 --> 01:07:36,630
And since we already know the
Erlang density, we can use
1125
01:07:36,630 --> 01:07:39,850
that to get the PMF
for n of t.
1126
01:07:42,480 --> 01:07:47,930
So using the Erlang density,
the probability that the n
1127
01:07:47,930 --> 01:07:51,730
plus first arrival falls in
this little tiny interval
1128
01:07:51,730 --> 01:07:53,950
here, we're thinking of
delta as being small.
1129
01:07:53,950 --> 01:07:56,740
And we're going to let
delta approach 0.
1130
01:07:56,740 --> 01:08:02,520
It's going to be the density
of the n plus first arrival
1131
01:08:02,520 --> 01:08:05,690
times delta plus o of delta.
1132
01:08:05,690 --> 01:08:10,490
o of delta is something that
goes to 0 as delta increases
1133
01:08:10,490 --> 01:08:12,110
faster than delta does.
1134
01:08:12,110 --> 01:08:17,170
It's something which has the
property that o of delta
1135
01:08:17,170 --> 01:08:20,670
divided by delta goes to
0 as delta gets large.
1136
01:08:20,670 --> 01:08:24,310
So this is just saying that this
is approximately equal to
1137
01:08:24,310 --> 01:08:30,540
the density of the n plus
first arrival times this
1138
01:08:30,540 --> 01:08:31,100
[INAUDIBLE]
1139
01:08:31,100 --> 01:08:31,479
here.
1140
01:08:31,479 --> 01:08:34,609
The density stays essentially
constant over
1141
01:08:34,609 --> 01:08:36,029
a very small delta.
1142
01:08:36,029 --> 01:08:38,609
It's a continuous density.
1143
01:08:38,609 --> 01:08:42,479
Next, we use the independent
increment property, which says
1144
01:08:42,479 --> 01:08:46,729
that the probability that t is
less than sn plus 1, is less
1145
01:08:46,729 --> 01:08:52,930
than or equal to t plus delta,
is the PMF that n of t is
1146
01:08:52,930 --> 01:08:59,649
equal to n at the beginning is
the interval, and then that in
1147
01:08:59,649 --> 01:09:05,279
the middle of the interval,
there's exactly one arrival.
1148
01:09:05,279 --> 01:09:09,970
And the probabilities of exactly
one arrival, is just
1149
01:09:09,970 --> 01:09:12,830
lambda delta plus o of delta.
1150
01:09:15,500 --> 01:09:16,580
Namely, that's because of the
1151
01:09:16,580 --> 01:09:19,830
independent increment property.
1152
01:09:19,830 --> 01:09:22,990
What's this o of delta
doing out here?
1153
01:09:22,990 --> 01:09:26,609
Why isn't this exactly
equal to this?
1154
01:09:26,609 --> 01:09:28,179
And why do I need
something else?
1155
01:09:31,460 --> 01:09:35,279
What am I leaving out
of this equation?
1156
01:09:35,279 --> 01:09:37,710
The probability that
our arrival comes--
1157
01:09:45,720 --> 01:09:47,789
here's t.
1158
01:09:47,789 --> 01:09:51,755
Here's t plus delta.
1159
01:09:51,755 --> 01:09:54,880
I'm talking about something
happening here.
1160
01:09:54,880 --> 01:09:56,990
At this point, n of t is here.
1161
01:09:59,710 --> 01:10:04,290
And I'm finding the probability
that n of t plus
1162
01:10:04,290 --> 01:10:10,850
delta is equal to n of
t plus 1 essentially.
1163
01:10:10,850 --> 01:10:12,500
I'm looking for the probability
of there being one
1164
01:10:12,500 --> 01:10:15,070
arrival in this interval here.
1165
01:10:18,980 --> 01:10:20,515
So what's the matter
with that equation?
1166
01:10:28,870 --> 01:10:33,520
This is the probability that
the n plus first arrival
1167
01:10:33,520 --> 01:10:37,220
occurs somewhere in this
interval here.
1168
01:10:37,220 --> 01:10:37,940
Yeah.
1169
01:10:37,940 --> 01:10:41,144
AUDIENCE: Is that last term
then the probability that
1170
01:10:41,144 --> 01:10:44,360
there's not anymore other
parameter standards as well?
1171
01:10:44,360 --> 01:10:46,380
PROFESSOR: It doesn't
include-- yes.
1172
01:10:46,380 --> 01:10:52,560
This last term which I had to
add is in fact the negligible
1173
01:10:52,560 --> 01:11:01,440
term that at time n of t, there
is less than n arrivals.
1174
01:11:01,440 --> 01:11:04,530
And then I get 2 arrivals in
this little interval delta.
1175
01:11:07,150 --> 01:11:09,300
So that's why I need
that extra term.
1176
01:11:09,300 --> 01:11:13,390
But anyway, when I relate these
two terms, I get the
1177
01:11:13,390 --> 01:11:18,050
probability mass function of n
of t is equal to the Erlang
1178
01:11:18,050 --> 01:11:22,820
density at t, where
the n plus first
1179
01:11:22,820 --> 01:11:25,140
arrival divided by lambda.
1180
01:11:25,140 --> 01:11:28,130
And that's what that
term is there.
1181
01:11:34,440 --> 01:11:37,940
So that gives us the
Poisson PMF.
1182
01:11:37,940 --> 01:11:42,500
Interesting observation about
this, it's a function
1183
01:11:42,500 --> 01:11:43,840
only of lambda t.
1184
01:11:43,840 --> 01:11:47,340
It's not a function of lambda
or t separately.
1185
01:11:47,340 --> 01:11:50,210
It's a function only of the
two of them together.
1186
01:11:50,210 --> 01:11:52,860
It has to be that.
1187
01:11:52,860 --> 01:11:56,180
Because you can use scaling
arguments on this.
1188
01:11:56,180 --> 01:12:00,530
If you have a Poisson process
of rate lambda and I measure
1189
01:12:00,530 --> 01:12:03,520
things in millimeters instead
of centimeters,
1190
01:12:03,520 --> 01:12:05,400
what's going to happen?
1191
01:12:05,400 --> 01:12:09,250
My rate is going to change
by a factor of 10.
1192
01:12:09,250 --> 01:12:13,060
My values of t are going to
change by a factor of 10.
1193
01:12:15,615 --> 01:12:17,630
This is a probability
mass function.
1194
01:12:17,630 --> 01:12:20,240
That has to stay the same.
1195
01:12:20,240 --> 01:12:24,860
So this has to be a function
only of the product lambda t
1196
01:12:24,860 --> 01:12:26,910
because of scaling
argument here.
1197
01:12:30,020 --> 01:12:35,290
Now, the other thing here, and
this is interesting because if
1198
01:12:35,290 --> 01:12:40,830
you look at n of t, the number
of arrivals up until time t is
1199
01:12:40,830 --> 01:12:45,500
the sum of the number of
arrivals up until some shorter
1200
01:12:45,500 --> 01:12:52,800
time t1 plus the number of
arrivals between t1 and t.
1201
01:12:52,800 --> 01:12:54,650
We know that the number
of arrivals up
1202
01:12:54,650 --> 01:12:57,100
until time t1 is Poisson.
1203
01:12:57,100 --> 01:13:01,280
The number of arrivals between
t1 and t is Poisson.
1204
01:13:01,280 --> 01:13:04,830
Those two values are independent
of each other.
1205
01:13:04,830 --> 01:13:07,580
I can choose t1 in the middle
to be anything I
1206
01:13:07,580 --> 01:13:09,190
want to make it.
1207
01:13:09,190 --> 01:13:13,080
And this says that the sum of
two Poisson random variables
1208
01:13:13,080 --> 01:13:16,180
has to be Poisson.
1209
01:13:16,180 --> 01:13:17,330
Now, I'm very lazy.
1210
01:13:17,330 --> 01:13:23,800
And I've gone through life
without ever convolving this
1211
01:13:23,800 --> 01:13:28,530
PMF to find out that in fact
the sum of 2 Poisson random
1212
01:13:28,530 --> 01:13:32,920
variables is in fact
Poisson itself.
1213
01:13:32,920 --> 01:13:35,700
Because I actually believe the
argument I just went through.
1214
01:13:40,510 --> 01:13:44,600
If you're skeptical, you will
probably want to actually do
1215
01:13:44,600 --> 01:13:48,420
the digital convolution to
show that the sum of two
1216
01:13:48,420 --> 01:13:54,540
independent Poisson random
variables is in fact Poisson.
1217
01:13:54,540 --> 01:13:56,860
And it extends to any [? tay ?]
disjoint interval.
1218
01:13:56,860 --> 01:14:01,050
So the same argument says that
any sum of Poisson random
1219
01:14:01,050 --> 01:14:04,210
variables is Poisson.
1220
01:14:04,210 --> 01:14:07,650
I do want to get through any
alternate definitions of a
1221
01:14:07,650 --> 01:14:12,240
Poisson process because
that makes a natural
1222
01:14:12,240 --> 01:14:14,630
stopping point here.
1223
01:14:14,630 --> 01:14:18,810
Question-- is it true that any
arrival process for which n of
1224
01:14:18,810 --> 01:14:22,680
t has a Poisson probability
mass function for a given
1225
01:14:22,680 --> 01:14:26,070
lambda and for all
t is a Poisson
1226
01:14:26,070 --> 01:14:27,330
process of rate lambda?
1227
01:14:30,150 --> 01:14:33,695
In other words, that's a
pretty strong property.
1228
01:14:33,695 --> 01:14:36,620
It says I found the probability
mass functions for
1229
01:14:36,620 --> 01:14:38,915
n of t at every value of t.
1230
01:14:38,915 --> 01:14:42,690
Does that describe a process?
1231
01:14:42,690 --> 01:14:44,525
Well, you see the
answer there.
1232
01:14:44,525 --> 01:14:50,950
As usual, marginal PMFs,
distribution functions don't
1233
01:14:50,950 --> 01:14:55,480
specify a process because they
don't specify the joint
1234
01:14:55,480 --> 01:14:57,180
probabilities.
1235
01:14:57,180 --> 01:15:00,080
But here, we've just pointed
out that these joint
1236
01:15:00,080 --> 01:15:01,690
probabilities are
all independent.
1237
01:15:01,690 --> 01:15:05,590
You can take a set of
probability mass functions for
1238
01:15:05,590 --> 01:15:08,840
this interval, this interval,
this interval, this interval,
1239
01:15:08,840 --> 01:15:10,020
and so forth.
1240
01:15:10,020 --> 01:15:18,090
And for any set of t1, t2, and
so forth up, we know that the
1241
01:15:18,090 --> 01:15:22,820
number of arrivals in zero to
t1, the number of arrivals in
1242
01:15:22,820 --> 01:15:26,850
t1 to t2, and so forth all the
way up are all independent
1243
01:15:26,850 --> 01:15:28,110
random variables.
1244
01:15:28,110 --> 01:15:32,830
And therefore, when we know the
Poisson probability mass
1245
01:15:32,830 --> 01:15:40,770
function, we really also know,
and we've also shown, that
1246
01:15:40,770 --> 01:15:43,990
these random variables are
independent of each other.
1247
01:15:43,990 --> 01:15:49,010
We have the joint PMF for any
sum of these random variables.
1248
01:15:49,010 --> 01:15:59,690
So in fact, in this particular
case, it's enough to know what
1249
01:15:59,690 --> 01:16:03,790
the probability mass function
is at each time t plus the
1250
01:16:03,790 --> 01:16:05,820
fact that we have this
1251
01:16:05,820 --> 01:16:08,020
independent increment property.
1252
01:16:08,020 --> 01:16:10,610
And we need the stationary
increment property, too, to
1253
01:16:10,610 --> 01:16:14,030
know that these values are
the same at each t.
1254
01:16:14,030 --> 01:16:17,300
So the theorem is that if an
arrival process has the
1255
01:16:17,300 --> 01:16:21,420
stationary and independent
increment properties, and if n
1256
01:16:21,420 --> 01:16:26,840
of t has the Poisson PMF for
given lambda and all t greater
1257
01:16:26,840 --> 01:16:32,090
than 0, then the process itself
has to be Poisson.
1258
01:16:32,090 --> 01:16:36,890
VHW stands for Violently
Hand Waving.
1259
01:16:39,690 --> 01:16:43,150
So that's even a little
worse than a PF.
1260
01:16:43,150 --> 01:16:45,220
Says the stationary and
independent increment
1261
01:16:45,220 --> 01:16:48,580
properties show that the joint
distribution of arrivals over
1262
01:16:48,580 --> 01:16:51,580
any given set of disjoint
intervals is that
1263
01:16:51,580 --> 01:16:53,410
of a Poisson process.
1264
01:16:53,410 --> 01:16:55,390
And clearly that's enough.
1265
01:16:55,390 --> 01:16:56,660
And it almost is.
1266
01:16:56,660 --> 01:16:59,470
And you should read the proof in
the notes which does just a
1267
01:16:59,470 --> 01:17:03,240
little more than that to make
this an actual proof.
1268
01:17:03,240 --> 01:17:03,550
OK.
1269
01:17:03,550 --> 01:17:05,330
I think I'll stop there.
1270
01:17:05,330 --> 01:17:08,020
And we will talk a little
bit about the Bernoulli
1271
01:17:08,020 --> 01:17:10,060
process next time.
1272
01:17:10,060 --> 01:17:11,310
Thank you.