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ROBERT GALLAGER: OK.
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00:00:22,840 --> 00:00:25,910
Today, I want to review
a little bit of
10
00:00:25,910 --> 00:00:27,920
what we did last time.
11
00:00:27,920 --> 00:00:32,860
I think with all the details,
some of you sort of lost the
12
00:00:32,860 --> 00:00:34,900
main pattern of what
was going on.
13
00:00:34,900 --> 00:00:37,920
So let me try to talk about
the main pattern.
14
00:00:37,920 --> 00:00:40,490
I don't want to talk about
the details anymore.
15
00:00:44,920 --> 00:00:48,920
I think, for the most part in
this course, the best way to
16
00:00:48,920 --> 00:00:52,200
understand the details of proofs
is to read the notes
17
00:00:52,200 --> 00:00:56,380
where you can read them at your
own rate, unless there's
18
00:00:56,380 --> 00:00:58,560
something wrong in the notes.
19
00:00:58,560 --> 00:01:02,240
I will typically avoid
that from now on.
20
00:01:02,240 --> 00:01:10,600
The main story that we want to
go through, first is the idea
21
00:01:10,600 --> 00:01:14,920
of what convergence with
probability 1 means.
22
00:01:14,920 --> 00:01:17,700
This is a very peculiar
concept.
23
00:01:17,700 --> 00:01:20,490
And I have to keep going through
it, have to keep
24
00:01:20,490 --> 00:01:23,800
talking about it with different
notation so that you
25
00:01:23,800 --> 00:01:30,970
can see it coming any
way you look at it.
26
00:01:30,970 --> 00:01:34,460
So what the theorem is-- and
it's a very useful theorem--
27
00:01:34,460 --> 00:01:39,580
it says let Y n be a set of
random variables, which
28
00:01:39,580 --> 00:01:44,970
satisfy the condition that the
sum of the expectations of the
29
00:01:44,970 --> 00:01:49,550
absolute value of each
random variable
30
00:01:49,550 --> 00:01:50,800
is less than infinity.
31
00:01:53,330 --> 00:01:55,570
First thing I want to point
out is what that means,
32
00:01:55,570 --> 00:01:59,015
because it's not entirely clear
what is the means to
33
00:01:59,015 --> 00:01:59,720
start with.
34
00:01:59,720 --> 00:02:06,990
When you talk about a limit
from n to infinity, from n
35
00:02:06,990 --> 00:02:17,750
equals 1 to infinity, what it
remains is this sum here
36
00:02:17,750 --> 00:02:20,010
really means the limit
as m goes to
37
00:02:20,010 --> 00:02:22,590
infinity of a finite sum.
38
00:02:22,590 --> 00:02:26,370
Anytime you talk about a limit
of a set of numbers, that's
39
00:02:26,370 --> 00:02:28,640
exactly what you mean by it.
40
00:02:28,640 --> 00:02:31,570
So if we're saying that this
quantity is less than
41
00:02:31,570 --> 00:02:35,210
infinity, it says two things.
42
00:02:35,210 --> 00:02:38,790
It says that these finite
sums are lesson
43
00:02:38,790 --> 00:02:41,390
infinity for all a m.
44
00:02:41,390 --> 00:02:44,970
And it also says that these
finite sums go to a limit.
45
00:02:44,970 --> 00:02:48,830
And the fact that these finite
sums are going to a limit as m
46
00:02:48,830 --> 00:02:55,180
gets big says what's a more
intuitive thing, which is that
47
00:02:55,180 --> 00:02:57,680
the limit as m goes
to infinity.
48
00:02:57,680 --> 00:03:01,470
And here, instead of going from
1 thing m, I'm going from
49
00:03:01,470 --> 00:03:05,020
m plus 1 to infinity, and it
doesn't make any difference
50
00:03:05,020 --> 00:03:09,050
whether I go from m plus 1 to
infinity or m to infinity.
51
00:03:09,050 --> 00:03:13,500
And what has to happen for this
limit to exist is the
52
00:03:13,500 --> 00:03:16,790
difference between this and
this, which is this,
53
00:03:16,790 --> 00:03:19,870
has to go to 0.
54
00:03:19,870 --> 00:03:23,230
So that what we're really saying
here is that the tail
55
00:03:23,230 --> 00:03:28,030
sum has to go to 0 here.
56
00:03:28,030 --> 00:03:32,370
Now, this is a much stronger
requirement than just saying
57
00:03:32,370 --> 00:03:36,080
that the expected value of the
magnitudes of these random
58
00:03:36,080 --> 00:03:39,370
variables has to go to 0.
59
00:03:39,370 --> 00:03:46,010
If, for example, the expected
value of y sub n is 1/n, then
60
00:03:46,010 --> 00:03:50,580
1/n, the limit of that as n
goes to infinity, is 0.
61
00:03:50,580 --> 00:03:52,350
So this is satisfied.
62
00:03:52,350 --> 00:03:58,760
But when you sum one/n here,
you don't get 0.
63
00:03:58,760 --> 00:04:00,010
And in fact, you get infinity.
64
00:04:05,000 --> 00:04:07,380
Yes, you get infinity.
65
00:04:07,380 --> 00:04:10,690
So this requires
more than this.
66
00:04:10,690 --> 00:04:14,280
This limit here, the requirement
that that equals
67
00:04:14,280 --> 00:04:22,540
0, implies that the sequence y
sub n actually converges to 0
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00:04:22,540 --> 00:04:26,580
in probability, rather than
with probability 1.
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00:04:26,580 --> 00:04:30,660
And the first problem in the
homework for this coming week
70
00:04:30,660 --> 00:04:32,690
is to actually show that.
71
00:04:32,690 --> 00:04:36,530
And when you show it, I hope
you find out that it is
72
00:04:36,530 --> 00:04:38,620
absolutely trivial to show it.
73
00:04:38,620 --> 00:04:41,340
It takes two lines to show
it, and that's really
74
00:04:41,340 --> 00:04:43,210
all you need here.
75
00:04:43,210 --> 00:04:47,370
The stronger requirement here,
let's us say something about
76
00:04:47,370 --> 00:04:48,880
the entire sample path.
77
00:04:48,880 --> 00:04:52,820
You see, this requirement really
is focusing on each
78
00:04:52,820 --> 00:04:55,110
individual value of
n, but it's not
79
00:04:55,110 --> 00:04:58,140
focusing on the sequences.
80
00:04:58,140 --> 00:05:02,250
This stronger quantity here is
really focusing on these
81
00:05:02,250 --> 00:05:05,270
entire sample paths.
82
00:05:05,270 --> 00:05:05,600
OK.
83
00:05:05,600 --> 00:05:08,550
So let's go on and review what
the strong law of large
84
00:05:08,550 --> 00:05:11,100
numbers says.
85
00:05:11,100 --> 00:05:16,460
And another note about the
theorem on convergence, how
86
00:05:16,460 --> 00:05:21,840
useful that theorem is depends
on how you choose these random
87
00:05:21,840 --> 00:05:24,745
variable, Y1, Y2,
and so forth.
88
00:05:27,430 --> 00:05:30,420
When we were proving the strong
law of large numbers in
89
00:05:30,420 --> 00:05:34,670
class last time, and in the
notes, we started off by
90
00:05:34,670 --> 00:05:38,150
assuming that the mean
of x is equal to 0.
91
00:05:38,150 --> 00:05:42,145
In fact, we'll see that that's
just for convenience.
92
00:05:44,690 --> 00:05:48,290
It's not something that has
anything to do with anything.
93
00:05:48,290 --> 00:05:51,310
It's just to get rid of a
lot of extra numbers.
94
00:05:51,310 --> 00:05:52,660
So we assume this.
95
00:05:52,660 --> 00:05:55,530
We also assume that the expected
value of x to the
96
00:05:55,530 --> 00:05:57,720
fourth was less than infinity.
97
00:05:57,720 --> 00:06:01,510
In other words, we assume that
this random variable that we
98
00:06:01,510 --> 00:06:05,220
were adding, these IID
random variables,
99
00:06:05,220 --> 00:06:07,720
had a fourth moment.
100
00:06:07,720 --> 00:06:09,540
Now, an awful lot of
random variables
101
00:06:09,540 --> 00:06:11,230
have a fourth moment.
102
00:06:11,230 --> 00:06:16,110
The real strong law of large
numbers, all that assumes is
103
00:06:16,110 --> 00:06:19,490
that the expected value
of the magnitude of x
104
00:06:19,490 --> 00:06:22,900
is less than infinity.
105
00:06:22,900 --> 00:06:26,740
So it has a much weaker
set of conditions.
106
00:06:26,740 --> 00:06:29,530
Most of the problems you run
into, doesn't make any
107
00:06:29,530 --> 00:06:33,830
difference whether you assume
this or you make the stronger
108
00:06:33,830 --> 00:06:37,070
assumption that the mean
is equal to 0.
109
00:06:37,070 --> 00:06:40,180
When you start applying this
theorem, it doesn't make any
110
00:06:40,180 --> 00:06:44,510
difference at all, because
there's no way you can tell,
111
00:06:44,510 --> 00:06:48,570
in a physical situation, whether
it is reasonable to
112
00:06:48,570 --> 00:06:57,970
assume that the fourth moment
is finite and the
113
00:06:57,970 --> 00:06:59,110
first moment isn't.
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00:06:59,110 --> 00:07:02,930
Because that question has to
do only with the very far
115
00:07:02,930 --> 00:07:05,030
tails of the distribution.
116
00:07:05,030 --> 00:07:10,630
I can take any distribution x
and I can truncate it at 10 to
117
00:07:10,630 --> 00:07:11,735
the google.
118
00:07:11,735 --> 00:07:15,750
And if I truncate it at 10 to
the google, it has a finite
119
00:07:15,750 --> 00:07:16,840
fourth moment.
120
00:07:16,840 --> 00:07:19,010
If I don't truncate
it, it might not
121
00:07:19,010 --> 00:07:20,900
have a fourth moment.
122
00:07:20,900 --> 00:07:24,240
There's no way you can tell
from looking at physical
123
00:07:24,240 --> 00:07:26,090
situations.
124
00:07:26,090 --> 00:07:30,190
So this question here is
primarily a question of
125
00:07:30,190 --> 00:07:34,020
modeling and what you're going
to do with the models.
126
00:07:34,020 --> 00:07:37,890
It's not something
which is crucial.
127
00:07:37,890 --> 00:07:42,710
But anyway, after we did that,
what we said was when we
128
00:07:42,710 --> 00:07:46,480
assume that this is less than
infinity, we can look at s sub
129
00:07:46,480 --> 00:07:54,000
n, which is x1, up to x n all
IID We then took this sum
130
00:07:54,000 --> 00:07:57,210
here, took to the fourth moment
of it, and we looked at
131
00:07:57,210 --> 00:07:59,630
all the cross terms and
what could happen.
132
00:07:59,630 --> 00:08:03,660
And we found that the only cross
terms that worked that
133
00:08:03,660 --> 00:08:09,250
were non-0 was where either
these quantities were paired
134
00:08:09,250 --> 00:08:15,080
together, x1 x1, x2, x2, or
where they were all the same.
135
00:08:15,080 --> 00:08:18,530
And then when we looked at that,
we very quickly realized
136
00:08:18,530 --> 00:08:22,880
that the expected value of s n
to the fourth was proportional
137
00:08:22,880 --> 00:08:25,020
to n squared.
138
00:08:25,020 --> 00:08:28,370
It was upper banded the by three
times n squared times
139
00:08:28,370 --> 00:08:30,880
the fourth moment of x.
140
00:08:30,880 --> 00:08:35,000
So when we look at s n to the
fourth divided by n to the
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00:08:35,000 --> 00:08:41,429
fourth, that quantity, summed
over n, goes as 1 over n
142
00:08:41,429 --> 00:08:44,810
squared, which has a
finite sum over n.
143
00:08:44,810 --> 00:08:48,420
And therefore, the probability
of the limit as n approaches
144
00:08:48,420 --> 00:08:53,150
infinity of s n to the fourth
over n fourth equals 0.
145
00:08:53,150 --> 00:08:56,560
The probability of the
sample path, the
146
00:08:56,560 --> 00:08:58,390
sample paths converge.
147
00:08:58,390 --> 00:09:03,200
The probability of that is equal
to 1, which says all
148
00:09:03,200 --> 00:09:07,580
sample paths converge
with probability 1.
149
00:09:07,580 --> 00:09:12,030
So this is enough, not quite
enough to prove the strong law
150
00:09:12,030 --> 00:09:13,210
of large numbers.
151
00:09:13,210 --> 00:09:16,690
Because what we're interested
in is not s n to the fourth
152
00:09:16,690 --> 00:09:17,730
over n fourth.
153
00:09:17,730 --> 00:09:21,810
We're interested in
s sub n over n.
154
00:09:21,810 --> 00:09:24,740
So we have to go one
step further.
155
00:09:24,740 --> 00:09:28,430
This is why it's tricky to
figure out what random
156
00:09:28,430 --> 00:09:33,950
variables you want to use when
you're trying to go from this
157
00:09:33,950 --> 00:09:37,390
theorem about convergence with
probability 1 to the strong
158
00:09:37,390 --> 00:09:42,480
law, or the strong law for
renewals, or any other kind of
159
00:09:42,480 --> 00:09:45,510
strong law doing
anything else.
160
00:09:45,510 --> 00:09:49,170
It's a fairly tricky matter to
choose what random variables
161
00:09:49,170 --> 00:09:50,940
you want to talk about there.
162
00:09:50,940 --> 00:09:57,180
But here, it doesn't make any
difference, as we've said.
163
00:09:57,180 --> 00:10:03,970
If you let s n of omega over
n be a sub n to the 1/4.
164
00:10:03,970 --> 00:10:07,520
In other words, what I'm doing
now is I'm focusing on just
165
00:10:07,520 --> 00:10:09,510
one sample path.
166
00:10:09,510 --> 00:10:14,080
If I focus on one sample path,
omega, and then each one of
167
00:10:14,080 --> 00:10:23,690
these terms has some value, a
sub n, and then s sub n of
168
00:10:23,690 --> 00:10:29,220
omega over n is going to be
equal to a sub n to the 1/4
169
00:10:29,220 --> 00:10:33,910
power, the 1/4 power of this
quantity over here.
170
00:10:33,910 --> 00:10:38,900
Now, the question is if the
limit of these numbers--
171
00:10:38,900 --> 00:10:41,600
And now remember, now we're
talking about a single sample
172
00:10:41,600 --> 00:10:45,710
path, but all of these sample
paths behave the same way.
173
00:10:45,710 --> 00:10:51,090
So if this limit here for one
sample path is equal to 0,
174
00:10:51,090 --> 00:10:58,090
then the limit of a sub n to
the 1/4 is also equal to 0.
175
00:10:58,090 --> 00:10:59,960
Why is that true?
176
00:10:59,960 --> 00:11:04,290
That's just a result about
the real number system.
177
00:11:04,290 --> 00:11:07,500
It's a result about convergence
of real numbers.
178
00:11:07,500 --> 00:11:10,400
If you take a bunch of real
numbers, which are getting
179
00:11:10,400 --> 00:11:13,920
very, very small, and you take
the fourth root of those
180
00:11:13,920 --> 00:11:16,880
numbers, which are getting very
small, the fourth root is
181
00:11:16,880 --> 00:11:19,740
a lot bigger than the
number itself.
182
00:11:19,740 --> 00:11:23,980
But nonetheless, the fourth
root is being driven to 0
183
00:11:23,980 --> 00:11:27,530
also, or at least the absolute
value of the fourth root is
184
00:11:27,530 --> 00:11:30,720
being driven to 0 also.
185
00:11:30,720 --> 00:11:32,790
You can see this intuitively
without
186
00:11:32,790 --> 00:11:38,640
even proving any theorems.
187
00:11:38,640 --> 00:11:43,390
Except, it is a standard result,
just talking about the
188
00:11:43,390 --> 00:11:45,960
real number system.
189
00:11:45,960 --> 00:11:46,440
OK.
190
00:11:46,440 --> 00:11:50,770
So what that says is the
probability that the limit of
191
00:11:50,770 --> 00:11:55,680
s sub n over n equals 0.
192
00:11:55,680 --> 00:11:58,230
This is now is talking
about sample paths
193
00:11:58,230 --> 00:11:59,930
for each sample path.
194
00:11:59,930 --> 00:12:04,360
This limit s n over n either
exists or it doesn't exist.
195
00:12:04,360 --> 00:12:07,170
If it does exist, it's either
equal to 0 or equal to
196
00:12:07,170 --> 00:12:08,380
something else.
197
00:12:08,380 --> 00:12:11,550
It says that the probability
that it exists and that it's
198
00:12:11,550 --> 00:12:15,120
equal to 0 is equal to 1.
199
00:12:15,120 --> 00:12:15,360
OK.
200
00:12:15,360 --> 00:12:19,510
Now, remember last time we
talked about something a
201
00:12:19,510 --> 00:12:20,260
little bit funny.
202
00:12:20,260 --> 00:12:23,150
We talked about the
Bernoulli process.
203
00:12:23,150 --> 00:12:27,150
And we talked about the
Bernoulli process using one
204
00:12:27,150 --> 00:12:32,170
value of p for the probability
that x is equal to 1.
205
00:12:32,170 --> 00:12:38,670
And what we found is that the
probability of the sample path
206
00:12:38,670 --> 00:12:43,260
where s sub n over n approach to
p, the probability of that
207
00:12:43,260 --> 00:12:46,010
set was equal to 1.
208
00:12:46,010 --> 00:12:50,020
If we change the probability
that x is equal to 1 to some
209
00:12:50,020 --> 00:12:54,160
other value, that is still a
perfectly well-defined event.
210
00:12:58,020 --> 00:13:04,460
The event that a sample path,
that sub n over n for a sample
211
00:13:04,460 --> 00:13:05,880
path approaches p.
212
00:13:05,880 --> 00:13:11,820
But that sample path, that
event, becomes 0 as soon as
213
00:13:11,820 --> 00:13:14,190
you change p to some
other value.
214
00:13:14,190 --> 00:13:17,490
So what we're talking
about here is really
215
00:13:17,490 --> 00:13:19,510
a probability measure.
216
00:13:19,510 --> 00:13:23,450
We're not using any kind of
measure theory here, but you
217
00:13:23,450 --> 00:13:25,970
really have to be careful about
the fact that you're not
218
00:13:25,970 --> 00:13:35,390
talking about the number of
sequences for which this limit
219
00:13:35,390 --> 00:13:36,910
is equal to 0.
220
00:13:36,910 --> 00:13:39,790
You're really talking about
the probability of it.
221
00:13:39,790 --> 00:13:43,460
And you can't think of it in
terms of number of sequences.
222
00:13:43,460 --> 00:13:47,800
What's the most probable
sequence for a Bernoulli
223
00:13:47,800 --> 00:13:55,980
process where p is,
say, 0.317?
224
00:13:55,980 --> 00:13:57,360
Who knows what the
most probable
225
00:13:57,360 --> 00:14:00,965
sequence of length n is?
226
00:14:00,965 --> 00:14:02,378
What?
227
00:14:02,378 --> 00:14:03,520
There is none?
228
00:14:03,520 --> 00:14:06,170
Yes, there is.
229
00:14:06,170 --> 00:14:08,510
It's all 0's, yes.
230
00:14:08,510 --> 00:14:13,010
The all 0's sequence is more
likely than anything else.
231
00:14:13,010 --> 00:14:18,720
So why don't these sequences
converge to 0?
232
00:14:18,720 --> 00:14:23,940
I'm In this case, these
sequences actually converge to
233
00:14:23,940 --> 00:14:29,550
0.317, if that's
the value of p.
234
00:14:29,550 --> 00:14:31,010
So what's going on?
235
00:14:31,010 --> 00:14:34,910
What's going on is a trade-off
between the number of
236
00:14:34,910 --> 00:14:40,000
sequences and the probability
of those sequences.
237
00:14:40,000 --> 00:14:43,620
You look at a particular value
of n, there's only one
238
00:14:43,620 --> 00:14:46,890
sequence which is all 0's.
239
00:14:46,890 --> 00:14:50,270
It's much more likely than any
of the other sequences.
240
00:14:50,270 --> 00:14:55,070
There's an enormous number of
sequences where the relative
241
00:14:55,070 --> 00:14:59,560
frequency of them is
close to 0.317.
242
00:14:59,560 --> 00:15:02,500
They're very improbable, but
because of the very large
243
00:15:02,500 --> 00:15:06,040
number of them, those are the
ones that turn out to have all
244
00:15:06,040 --> 00:15:07,630
the probability here.
245
00:15:07,630 --> 00:15:10,880
So that's what's going
on in this strong
246
00:15:10,880 --> 00:15:11,880
law of large numbers.
247
00:15:11,880 --> 00:15:15,860
You have all of these effects
playing off against each
248
00:15:15,860 --> 00:15:19,080
other, and it's kind of
phenomenal that you wind up
249
00:15:19,080 --> 00:15:21,780
with an extraordinarily strong
theorem like this.
250
00:15:21,780 --> 00:15:25,320
When you call this the strong
law of large numbers, it, in
251
00:15:25,320 --> 00:15:30,710
fact, is an incredibly strong
theorem, which is not at all
252
00:15:30,710 --> 00:15:31,860
intuitively obvious.
253
00:15:31,860 --> 00:15:34,910
It's very, very far from
intuitively obvious.
254
00:15:34,910 --> 00:15:38,210
If you think it's intuitively
obvious, and you haven't
255
00:15:38,210 --> 00:15:42,040
studied it for a very long time,
go back and think about
256
00:15:42,040 --> 00:15:45,100
it again, because there's
something wrong in the way
257
00:15:45,100 --> 00:15:46,200
you're thinking about it.
258
00:15:46,200 --> 00:15:51,890
Because this is an absolutely
incredible theorem.
259
00:15:51,890 --> 00:15:52,500
OK.
260
00:15:52,500 --> 00:15:59,820
So I want to be a little more
general now and talk about
261
00:15:59,820 --> 00:16:03,490
sequences converging to
a constant alpha with
262
00:16:03,490 --> 00:16:05,200
probability 1.
263
00:16:05,200 --> 00:16:10,060
If the probability of the set of
omega such as the limit as
264
00:16:10,060 --> 00:16:12,160
n goes to infinity
is z n of omega.
265
00:16:12,160 --> 00:16:14,260
In other words, we will look
look at a sample path for a
266
00:16:14,260 --> 00:16:15,600
given omega.
267
00:16:15,600 --> 00:16:19,000
Let's look at the probability
that that's equal to alpha
268
00:16:19,000 --> 00:16:21,420
rather than equal to 0.
269
00:16:21,420 --> 00:16:25,400
That's the case of the
Bernoulli process.
270
00:16:25,400 --> 00:16:29,930
Bernoulli process with
probability p, if you're
271
00:16:29,930 --> 00:16:36,210
looking at s n of omega as s n
of omega over n, then this
272
00:16:36,210 --> 00:16:37,530
converges to alpha.
273
00:16:37,530 --> 00:16:41,030
You're looking at s n to the
fourth over n to the fourth,
274
00:16:41,030 --> 00:16:46,390
it converges to p to
the fourth power.
275
00:16:46,390 --> 00:16:48,420
And all those are equal to 1.
276
00:16:48,420 --> 00:16:54,300
Now, note that z n converges to
alpha if, and only if, z n
277
00:16:54,300 --> 00:16:58,290
minus alpha converges to 0.
278
00:16:58,290 --> 00:17:01,070
In other words, we're talking
about something that's
279
00:17:01,070 --> 00:17:03,030
relatively trivial here.
280
00:17:03,030 --> 00:17:05,450
It's not very important.
281
00:17:05,450 --> 00:17:08,609
Any time I have a sequence
of random variables that
282
00:17:08,609 --> 00:17:13,220
converges to some non-0
quantity, p, or alpha, or
283
00:17:13,220 --> 00:17:21,180
whatever, I can also talk
about z n minus alpha.
284
00:17:21,180 --> 00:17:24,180
And that's another sequence
of random variables.
285
00:17:24,180 --> 00:17:29,200
And if this converges to alpha,
this converges to 0.
286
00:17:29,200 --> 00:17:36,540
So all I was doing when I was
talking about this convergence
287
00:17:36,540 --> 00:17:39,940
theorem of everything converging
to 0, what I was
288
00:17:39,940 --> 00:17:44,800
doing was really taking these
random variables and looking
289
00:17:44,800 --> 00:17:47,830
at their variation around the
mean, at their fluctuation
290
00:17:47,830 --> 00:17:50,160
around the mean, rather
than the actual
291
00:17:50,160 --> 00:17:51,650
random variable itself.
292
00:17:51,650 --> 00:17:53,250
You can always do that.
293
00:17:53,250 --> 00:17:56,560
And by doing it, you need to
introduce a little more
294
00:17:56,560 --> 00:18:00,480
terminology, and you get rid of
a lot of mess, because then
295
00:18:00,480 --> 00:18:02,210
the mean doesn't
appear anymore.
296
00:18:02,210 --> 00:18:06,740
So when we start talking about
renewal processes, which we're
297
00:18:06,740 --> 00:18:09,350
going to do here, the
inter-renewal
298
00:18:09,350 --> 00:18:11,840
intervals are positive.
299
00:18:11,840 --> 00:18:16,820
It's important, the the fact
that they're positive, and
300
00:18:16,820 --> 00:18:18,690
that they never go negative.
301
00:18:18,690 --> 00:18:21,370
And because of that, we don't
really want to subtract the
302
00:18:21,370 --> 00:18:24,950
mean off them, because then we
would have a sequence of
303
00:18:24,950 --> 00:18:28,180
random variables that weren't
positive anymore.
304
00:18:28,180 --> 00:18:33,800
So instead of taking away the
mean to avoid a couple of
305
00:18:33,800 --> 00:18:39,710
extra symbols, we're going to
leave the mean in from now on,
306
00:18:39,710 --> 00:18:42,760
so these random variables are
going to converge to some
307
00:18:42,760 --> 00:18:47,300
constant, generally, rather
than converge to 0.
308
00:18:47,300 --> 00:18:49,850
And it doesn't make
any difference.
309
00:18:49,850 --> 00:18:50,340
OK.
310
00:18:50,340 --> 00:18:53,450
So now, the next thing I want to
do is talk about the strong
311
00:18:53,450 --> 00:18:56,230
law for renewal processes.
312
00:18:56,230 --> 00:18:59,640
In other words, I want to talk
about what happens when you
313
00:18:59,640 --> 00:19:05,130
have a renewal counting process
where n of t is the
314
00:19:05,130 --> 00:19:08,850
number of arrivals
up until time t.
315
00:19:08,850 --> 00:19:11,540
And we'd like to see if there's
any kind of law of
316
00:19:11,540 --> 00:19:16,530
large numbers about what happens
to n of t over t as t
317
00:19:16,530 --> 00:19:18,650
gets very large.
318
00:19:18,650 --> 00:19:21,610
And there is such a law, and
that's the kind of thing that
319
00:19:21,610 --> 00:19:23,650
we want to focus on here.
320
00:19:23,650 --> 00:19:27,630
And that's what we're now
starting to talk about.
321
00:19:31,070 --> 00:19:35,790
What was it that made us want
to talk about the strong law
322
00:19:35,790 --> 00:19:39,390
of large numbers instead of the
weak law of large numbers?
323
00:19:39,390 --> 00:19:45,850
It was really the fact that
these sample paths converge.
324
00:19:45,850 --> 00:19:48,580
All these other kinds of
convergence, it's the
325
00:19:48,580 --> 00:19:52,030
distribution function that
converges, it's some
326
00:19:52,030 --> 00:19:54,020
probability of something
that converges.
327
00:19:54,020 --> 00:19:58,080
It's always something gross that
you can look at at every
328
00:19:58,080 --> 00:20:01,780
value of n, and then you can
find the limit of the
329
00:20:01,780 --> 00:20:04,940
distribution function, or find
the limit of the mean, or find
330
00:20:04,940 --> 00:20:07,820
the limit of the relative
frequency, or find the limit
331
00:20:07,820 --> 00:20:10,150
of something else.
332
00:20:10,150 --> 00:20:14,060
When we talk about the strong
law of large numbers, we are
333
00:20:14,060 --> 00:20:17,650
really talking about
these sample paths.
334
00:20:17,650 --> 00:20:22,440
And the fact that we could go
from a convergence theorem
335
00:20:22,440 --> 00:20:25,770
saying that s n to the fourth
over n to the fourth
336
00:20:25,770 --> 00:20:29,660
approached the limit, this was
the convergence theorem.
337
00:20:29,660 --> 00:20:34,300
And from that, we could show
the s sub n over n also
338
00:20:34,300 --> 00:20:37,900
approached the limit, that
really is the key to why the
339
00:20:37,900 --> 00:20:42,200
strong law of large numbers
gets used so much, and
340
00:20:42,200 --> 00:20:44,050
particularly gets used
when we were talking
341
00:20:44,050 --> 00:20:46,430
about renewal processes.
342
00:20:46,430 --> 00:20:48,650
What you will find when we
study we study renewal
343
00:20:48,650 --> 00:20:53,480
processes is that there's
a small part of renewal
344
00:20:53,480 --> 00:20:54,730
processes--
345
00:20:59,280 --> 00:21:02,930
there's a small part of the
theory which really says 80%
346
00:21:02,930 --> 00:21:04,970
of what's important.
347
00:21:04,970 --> 00:21:08,820
And it's almost trivially
simple, and it's built on the
348
00:21:08,820 --> 00:21:12,070
strong law for renewal
processes.
349
00:21:12,070 --> 00:21:15,230
Then there's a bunch of other
things which are not built on
350
00:21:15,230 --> 00:21:17,970
the strong law, they're built
on the weak law or something
351
00:21:17,970 --> 00:21:21,600
else, which are quite
tedious, and quite
352
00:21:21,600 --> 00:21:23,710
difficult, and quite messy.
353
00:21:23,710 --> 00:21:26,550
We go through them because
they get used in a lot of
354
00:21:26,550 --> 00:21:29,600
other places, and they let us
learn about a lot of things
355
00:21:29,600 --> 00:21:31,140
that are very important.
356
00:21:31,140 --> 00:21:35,380
But they still are
more difficult.
357
00:21:35,380 --> 00:21:37,860
And they're more difficult
because we're not talking
358
00:21:37,860 --> 00:21:39,730
about sample paths anymore.
359
00:21:39,730 --> 00:21:41,010
And you're going to see
that at the end
360
00:21:41,010 --> 00:21:42,630
of the lecture today.
361
00:21:42,630 --> 00:21:48,660
You'll see, I think, what is
probably the best illustration
362
00:21:48,660 --> 00:21:51,880
of why the strong law
of large numbers
363
00:21:51,880 --> 00:21:54,450
makes your life simple.
364
00:21:54,450 --> 00:21:55,560
OK.
365
00:21:55,560 --> 00:21:58,140
So we had this fact here.
366
00:21:58,140 --> 00:22:03,160
This theorem is what's going
to generalize this.
367
00:22:03,160 --> 00:22:07,450
Assume that z sub n and greater
than or equal to 1,
368
00:22:07,450 --> 00:22:10,440
this is a sequence of
random variables.
369
00:22:10,440 --> 00:22:13,710
And assume that this sequence of
random variables converges
370
00:22:13,710 --> 00:22:17,900
to some number, alpha,
with probability 1.
371
00:22:17,900 --> 00:22:22,570
In other words, you take sample
paths of this sequence
372
00:22:22,570 --> 00:22:24,320
of a random variables.
373
00:22:24,320 --> 00:22:29,480
And those sample paths, there
two sets of sample paths.
374
00:22:29,480 --> 00:22:33,810
One set of sample paths
converged to alpha, and that
375
00:22:33,810 --> 00:22:35,540
has probability 1.
376
00:22:35,540 --> 00:22:38,430
There's another set of sample
paths, some of them converge
377
00:22:38,430 --> 00:22:42,510
to something else, some of them
converge to nothing, some
378
00:22:42,510 --> 00:22:45,460
of them don't converge at all.
379
00:22:45,460 --> 00:22:46,870
Well, they converge
to nothing.
380
00:22:46,870 --> 00:22:49,320
They don't converge at all.
381
00:22:49,320 --> 00:22:53,230
But that set has probability 0,
so we don't worry about it.
382
00:22:53,230 --> 00:22:56,450
All we're worrying about is this
good set, which is the
383
00:22:56,450 --> 00:22:58,210
set which converges.
384
00:22:58,210 --> 00:23:02,570
And then what the theorem says
is if we have a function f of
385
00:23:02,570 --> 00:23:07,410
x, if it's a real-valued
function of a real variable,
386
00:23:07,410 --> 00:23:10,690
what does that mean?
387
00:23:10,690 --> 00:23:14,460
As an engineer, it means
it's a function.
388
00:23:14,460 --> 00:23:18,400
When you're an engineer and you
talk about functions, you
389
00:23:18,400 --> 00:23:24,080
don't talk about things that
aren't continuous.
390
00:23:24,080 --> 00:23:27,160
You talk about things
that are continuous.
391
00:23:27,160 --> 00:23:32,150
So all that's saying is it gives
us a nice, respectable
392
00:23:32,150 --> 00:23:35,710
function of a variable.
393
00:23:35,710 --> 00:23:39,860
It belongs to the national
academy of real variables that
394
00:23:39,860 --> 00:23:43,120
people like to use.
395
00:23:43,120 --> 00:23:47,300
Then what the theorem says is
that the sequence of random
396
00:23:47,300 --> 00:23:50,160
variables f of z sub n--
397
00:23:50,160 --> 00:23:53,850
OK, we have a real-valued
function of a real variable.
398
00:23:53,850 --> 00:23:59,660
It maps, then, sample values
of z sub n into f of those
399
00:23:59,660 --> 00:24:01,010
sample values.
400
00:24:01,010 --> 00:24:04,380
And because of that, just as
we've done a dozen times
401
00:24:04,380 --> 00:24:07,550
already when you take a
real-valued function of a
402
00:24:07,550 --> 00:24:11,340
random variable, you have
a two-step mapping.
403
00:24:11,340 --> 00:24:17,480
You map from omega into
z n of omega.
404
00:24:17,480 --> 00:24:23,070
And then you map from z n of
omega into f of z n of omega.
405
00:24:23,070 --> 00:24:28,290
That's a simple-minded idea.
406
00:24:28,290 --> 00:24:28,890
OK.
407
00:24:28,890 --> 00:24:35,410
Example one of this, suppose
that f of x is x plus beta.
408
00:24:35,410 --> 00:24:38,000
All this is just a translation,
simple-minded
409
00:24:38,000 --> 00:24:42,970
function, president
of the academy.
410
00:24:42,970 --> 00:24:45,690
And supposes that the sequence
of random variables
411
00:24:45,690 --> 00:24:47,480
converges to alpha.
412
00:24:47,480 --> 00:24:51,690
Then this new set of random
variable u sub n equals z sub
413
00:24:51,690 --> 00:24:52,640
n plus beta.
414
00:24:52,640 --> 00:24:57,425
The translated version converges
to alpha plus beta.
415
00:24:57,425 --> 00:24:59,740
Well, you don't even need
a theorem to see that.
416
00:24:59,740 --> 00:25:03,270
I mean, you can just look at
it and say, of course.
417
00:25:03,270 --> 00:25:06,680
Example two is the one
we've already used.
418
00:25:06,680 --> 00:25:09,660
This one you do have to sweat
over a little bit, but we've
419
00:25:09,660 --> 00:25:12,110
already sweated over it, and
then we're not going to worry
420
00:25:12,110 --> 00:25:14,440
about it anymore.
421
00:25:14,440 --> 00:25:18,930
If f of x is equal to x to the
1/4 for x greater than or
422
00:25:18,930 --> 00:25:27,290
equal to 0, and z n, random
variable, the sequence of
423
00:25:27,290 --> 00:25:31,840
random variables, converges to 0
with probability 1 and these
424
00:25:31,840 --> 00:25:35,760
random variables are
non-negative, then f of z n
425
00:25:35,760 --> 00:25:38,690
converges to f of 0.
426
00:25:38,690 --> 00:25:41,730
That's the one that's a little
less obvious, because if you
427
00:25:41,730 --> 00:25:54,210
look at this function, when x is
very close to 0, but when x
428
00:25:54,210 --> 00:25:59,670
is equal to 0, it's 0,
it goes like this.
429
00:25:59,670 --> 00:26:03,070
When x is 1, you're
up to 1, I guess.
430
00:26:03,070 --> 00:26:06,390
But it really goes up with
an infinite slope here.
431
00:26:06,390 --> 00:26:10,710
It's still continuous at 0 if
you're looking only at the
432
00:26:10,710 --> 00:26:11,960
non-negative values.
433
00:26:14,740 --> 00:26:17,770
That's what we use to
prove the strong
434
00:26:17,770 --> 00:26:19,080
law of large numbers.
435
00:26:19,080 --> 00:26:22,490
None of you complained about
it last time, so you can't
436
00:26:22,490 --> 00:26:25,840
complain about it now.
437
00:26:25,840 --> 00:26:27,850
It's just part of what this
theorem is saying.
438
00:26:33,750 --> 00:26:35,370
This is what the theorem says.
439
00:26:38,870 --> 00:26:43,530
I'm just rewriting it
much more briefly.
440
00:26:43,530 --> 00:26:48,480
Here I'm going to give a "Pf."
For each omega such that limit
441
00:26:48,480 --> 00:26:52,500
of z n of omega equals alpha,
we use the result for a
442
00:26:52,500 --> 00:26:56,600
sequence of numbers that says
the limit of f of z n of
443
00:26:56,600 --> 00:27:01,740
omega, this limit of a set of
sequence of numbers, is equal
444
00:27:01,740 --> 00:27:05,390
to the function at
the limit value.
445
00:27:05,390 --> 00:27:08,560
Let me give you a little diagram
which shows you why
446
00:27:08,560 --> 00:27:09,810
that has to be true.
447
00:27:12,680 --> 00:27:14,795
Suppose you look at
this function f.
448
00:27:20,270 --> 00:27:22,820
This is f of x.
449
00:27:22,820 --> 00:27:30,460
And what we're doing now is
we're looking at a1 here.
450
00:27:30,460 --> 00:27:38,220
a1 is going to be f
of z1 of omega.
451
00:27:38,220 --> 00:27:42,620
a2, I'm just drawing random
numbers here.
452
00:27:42,620 --> 00:27:52,920
a3, a4, a5, a6, a7.
453
00:27:56,950 --> 00:27:58,360
And then I draw f of a1.
454
00:28:11,510 --> 00:28:15,940
So what I'm saying here, in
terms of real numbers, is this
455
00:28:15,940 --> 00:28:17,590
quite trivial thing.
456
00:28:17,590 --> 00:28:23,580
If this function is continuous
at this point, as n gets
457
00:28:23,580 --> 00:28:28,590
large, these numbers get
compressed into that limiting
458
00:28:28,590 --> 00:28:29,870
value there.
459
00:28:29,870 --> 00:28:32,990
And as these numbers get
compressed into that limiting
460
00:28:32,990 --> 00:28:38,080
value, these values up here
get compressed into that
461
00:28:38,080 --> 00:28:41,030
limiting value also.
462
00:28:41,030 --> 00:28:48,500
This is not a proof, but the way
I construct proofs is not
463
00:28:48,500 --> 00:28:52,430
to look for them someplace in
a book, but it's to draw a
464
00:28:52,430 --> 00:28:56,510
picture which shows me what the
idea of the proof is, then
465
00:28:56,510 --> 00:28:57,740
I prove it.
466
00:28:57,740 --> 00:29:01,330
This has an advantage in
research, because if you ever
467
00:29:01,330 --> 00:29:03,550
want to get a result in
research which you can
468
00:29:03,550 --> 00:29:05,800
publish, it has to be
something that you
469
00:29:05,800 --> 00:29:08,760
can't find in a book.
470
00:29:08,760 --> 00:29:12,110
If you do find it in a book
later, then in fact your
471
00:29:12,110 --> 00:29:14,740
result was not new And you're
not supposed to publish
472
00:29:14,740 --> 00:29:17,820
results that aren't new.
473
00:29:17,820 --> 00:29:20,990
So the idea of drawing a picture
and then proving it
474
00:29:20,990 --> 00:29:24,610
from the picture is really
a very valuable
475
00:29:24,610 --> 00:29:26,960
aid in doing research.
476
00:29:26,960 --> 00:29:29,750
And if you draw this picture,
then you can easily construct
477
00:29:29,750 --> 00:29:32,410
a proof of the picture.
478
00:29:32,410 --> 00:29:35,590
But I'm not going
to do it here.
479
00:29:35,590 --> 00:29:39,410
Now, let's go onto renewal
processes.
480
00:29:39,410 --> 00:29:40,600
Each and inter-renewal
481
00:29:40,600 --> 00:29:44,610
interval, x sub i, is positive.
482
00:29:44,610 --> 00:29:47,070
That was what we said
in starting to talk
483
00:29:47,070 --> 00:29:50,370
about renewal processes.
484
00:29:50,370 --> 00:29:53,740
Assuming that the expected
value of x exists, the
485
00:29:53,740 --> 00:29:59,080
expected value of x is then
strictly greater than 0.
486
00:29:59,080 --> 00:30:02,262
You're going to prove that in
the homework this time, too.
487
00:30:02,262 --> 00:30:06,460
When I was trying to get this
lecture ready, I didn't want
488
00:30:06,460 --> 00:30:10,870
to prove anything in detail,
so I had to follow the
489
00:30:10,870 --> 00:30:15,200
strategy of assigning problems,
and the problem set
490
00:30:15,200 --> 00:30:17,460
where you would, in fact, prove
these things, which are
491
00:30:17,460 --> 00:30:20,500
not difficult but
which require a
492
00:30:20,500 --> 00:30:23,010
little bit of thought.
493
00:30:23,010 --> 00:30:25,920
And since the expected value of
x is greater than or equal
494
00:30:25,920 --> 00:30:31,350
to 0, the expected value of s1,
which is expected value of
495
00:30:31,350 --> 00:30:35,920
x, the expected value of x1 plus
x2, which is s2, and so
496
00:30:35,920 --> 00:30:40,760
forth, all of these quantities
are greater than 0 also.
497
00:30:40,760 --> 00:30:44,470
And for each finite n, the
expected value of s sub n over
498
00:30:44,470 --> 00:30:47,550
n is greater than 0 also. so
we're talking about a whole
499
00:30:47,550 --> 00:30:51,640
bunch of positive quantities.
500
00:30:51,640 --> 00:30:54,250
So this strong law of large
numbers is then
501
00:30:54,250 --> 00:30:56,390
going to apply here.
502
00:30:56,390 --> 00:31:00,780
The probability of the sample
paths, s n of omega it over n,
503
00:31:00,780 --> 00:31:05,600
the probability that that sample
path converges to the
504
00:31:05,600 --> 00:31:09,430
mean of x, that's just 1.
505
00:31:09,430 --> 00:31:13,896
Then I use the theorem on
the last page about--
506
00:31:13,896 --> 00:31:16,330
It's this theorem.
507
00:31:16,330 --> 00:31:21,350
When I use f of x equals 1/x
that's continuous at this
508
00:31:21,350 --> 00:31:22,280
positive value.
509
00:31:22,280 --> 00:31:24,960
That's why it's important to
have a positive value for the
510
00:31:24,960 --> 00:31:26,430
expected value of x.
511
00:31:26,430 --> 00:31:30,640
The expected value of x is
equal to 0, 1/x is not
512
00:31:30,640 --> 00:31:35,770
continuous at x equals 0, and
you're in deep trouble.
513
00:31:35,770 --> 00:31:39,030
That's one of the reasons why
you really want to assume
514
00:31:39,030 --> 00:31:45,070
renewal theory and not allow
any inter-renewal intervals
515
00:31:45,070 --> 00:31:46,640
that are equal to 0.
516
00:31:46,640 --> 00:31:49,810
It just louses up the whole
theory, makes things much more
517
00:31:49,810 --> 00:31:53,660
difficult, and you gain
nothing by it.
518
00:31:53,660 --> 00:31:58,760
So we get this statement then,
the probability of sample
519
00:31:58,760 --> 00:32:04,650
points such that the limit of
n over s n of omega is equal
520
00:32:04,650 --> 00:32:05,970
to 1 over x bar.
521
00:32:05,970 --> 00:32:07,750
That limit is equal to 1.
522
00:32:07,750 --> 00:32:08,800
What does that mean?
523
00:32:08,800 --> 00:32:11,570
I look at that and it doesn't
mean anything to me.
524
00:32:11,570 --> 00:32:16,060
I can't see what it means
until I draw a picture.
525
00:32:16,060 --> 00:32:19,760
I'm really into pictures
today.
526
00:32:19,760 --> 00:32:22,920
This was the statement that we
said the probability of this
527
00:32:22,920 --> 00:32:27,300
set of omega such that the limit
of n over s n of omega
528
00:32:27,300 --> 00:32:30,690
is equal to 1 over x
bar is equal to 1.
529
00:32:30,690 --> 00:32:35,690
This is valid whenever you have
a renewal process for
530
00:32:35,690 --> 00:32:42,380
which x bar exists, namely,
the expected value of the
531
00:32:42,380 --> 00:32:44,690
magnitude of x exists.
532
00:32:44,690 --> 00:32:48,670
That was one of the assumptions
we had in here.
533
00:32:48,670 --> 00:32:50,520
And now, this is going
to imply the
534
00:32:50,520 --> 00:32:53,320
strong law renewal processes.
535
00:32:53,320 --> 00:32:57,160
Here's the picture, which lets
us interpret what this means
536
00:32:57,160 --> 00:32:59,860
and let's just go
further with it.
537
00:32:59,860 --> 00:33:06,420
The picture now is you have this
counting process, which
538
00:33:06,420 --> 00:33:10,580
also amounts to a picture of
any set of inter-arrival
539
00:33:10,580 --> 00:33:18,070
instance, x1, x2, x3, x4, and
so forth, and any set of
540
00:33:18,070 --> 00:33:22,110
arrival epochs, s1,
s2, and so forth.
541
00:33:22,110 --> 00:33:26,070
We look at a particular
value of t.
542
00:33:26,070 --> 00:33:33,650
And what I'm interested
in is n of t over t.
543
00:33:33,650 --> 00:33:37,300
I have a theorem about
n over s n of omega.
544
00:33:37,300 --> 00:33:38,790
That's not what I'm
interested in.
545
00:33:38,790 --> 00:33:41,490
I'm interested in
n of t over t.
546
00:33:41,490 --> 00:33:45,480
And this picture shows me what
the relationship is.
547
00:33:45,480 --> 00:33:48,630
So I start out with a
given value of t.
548
00:33:48,630 --> 00:33:52,300
For a given value of t, there's
a well-defined random
549
00:33:52,300 --> 00:33:56,100
variable, which is the number
of arrivals up to and
550
00:33:56,100 --> 00:33:58,250
including time t.
551
00:33:58,250 --> 00:34:02,670
From n of t, I get a
well-defined random variable,
552
00:34:02,670 --> 00:34:13,750
which is the arrival epoch of
the latest arrival less than
553
00:34:13,750 --> 00:34:15,790
or equal to time t.
554
00:34:15,790 --> 00:34:19,570
Now, this is a very funny
kind of random variable.
555
00:34:19,570 --> 00:34:22,210
I mean, we've talked about
random variables which are
556
00:34:22,210 --> 00:34:25,840
functions of other
random variables.
557
00:34:25,840 --> 00:34:28,620
And in a sense, that's
what this.
558
00:34:28,620 --> 00:34:31,929
But it's a little more
awful than that.
559
00:34:31,929 --> 00:34:35,280
Because here we have this
well-defined set of arrival
560
00:34:35,280 --> 00:34:41,760
epochs, and now we're taking a
particular arrival, which is
561
00:34:41,760 --> 00:34:44,260
determined by this t
we're looking at.
562
00:34:44,260 --> 00:34:49,770
So t defines n of t, and n of t
defines s sub n of t, if we
563
00:34:49,770 --> 00:34:53,070
have this entire sample
function.
564
00:34:53,070 --> 00:34:55,520
So this is well-defined.
565
00:34:55,520 --> 00:34:59,960
We will find as we proceed with
this that this random
566
00:34:59,960 --> 00:35:06,030
variable, the time of the
arrival most recently before
567
00:35:06,030 --> 00:35:09,900
t, it's in fact a very, very
strange random variable.
568
00:35:09,900 --> 00:35:12,250
There are strange things
associated with it.
569
00:35:12,250 --> 00:35:17,120
When I look at t minus s sub n
of t, or when I look at the
570
00:35:17,120 --> 00:35:23,670
arrival after t, s sub n of t
plus 1 minus t, those random
571
00:35:23,670 --> 00:35:26,700
variables are peculiar.
572
00:35:26,700 --> 00:35:30,850
And we're going to explain why
they are peculiar and use the
573
00:35:30,850 --> 00:35:34,650
strong law for renewal processes
to look at them in a
574
00:35:34,650 --> 00:35:36,150
kind of a simple way.
575
00:35:36,150 --> 00:35:40,070
But now, the thing we have
here is if we look at the
576
00:35:40,070 --> 00:35:47,260
slope of b of t, the slope of
this line here at each value
577
00:35:47,260 --> 00:35:53,220
of t, this slope is n
of t divided by t.
578
00:35:53,220 --> 00:35:55,050
That's this slope.
579
00:35:55,050 --> 00:36:02,390
This slope here is n of
t over s sub n of t.
580
00:36:02,390 --> 00:36:07,970
Namely, this is the slope up
to the point of the arrival
581
00:36:07,970 --> 00:36:10,960
right before t.
582
00:36:10,960 --> 00:36:14,600
This slope is then going
to decrease as
583
00:36:14,600 --> 00:36:16,800
we move across here.
584
00:36:16,800 --> 00:36:20,020
And at this value here, it's
going to pop up again.
585
00:36:20,020 --> 00:36:25,800
So we have a family of slopes,
which is going to look like--
586
00:36:33,220 --> 00:36:35,350
What's it going to do?
587
00:36:35,350 --> 00:36:38,050
I don't know where it's going to
start out, so I won't even
588
00:36:38,050 --> 00:36:39,650
worry about that.
589
00:36:39,650 --> 00:36:42,080
I'll just start someplace
here.
590
00:36:42,080 --> 00:36:44,702
It's going to be decreasing.
591
00:36:44,702 --> 00:36:45,952
Then there's going
to be an arrival.
592
00:36:49,570 --> 00:36:52,430
At that point, it's going to
increase a little bit.
593
00:36:52,430 --> 00:36:54,330
It's going to be decreasing.
594
00:36:54,330 --> 00:36:55,890
There's another arrival.
595
00:36:55,890 --> 00:37:01,550
So this is s sub n, s sub
n plus 1, and so forth.
596
00:37:04,180 --> 00:37:08,390
So the slope is slowly
decreasing.
597
00:37:08,390 --> 00:37:11,580
And then it changes
discontinuously every time you
598
00:37:11,580 --> 00:37:12,570
have an arrival.
599
00:37:12,570 --> 00:37:15,190
That's the way this behaves.
600
00:37:15,190 --> 00:37:19,400
You start out here, it decreases
slowly, it jumps up,
601
00:37:19,400 --> 00:37:21,490
then it decreases slowly
until the next
602
00:37:21,490 --> 00:37:23,890
arrival, and so forth.
603
00:37:23,890 --> 00:37:27,070
So that's the kind of thing
we're looking at.
604
00:37:27,070 --> 00:37:32,830
But one thing we know is that
n of t over t, that's the
605
00:37:32,830 --> 00:37:37,460
slope in the middle here, is
less than or equal to n of t
606
00:37:37,460 --> 00:37:38,800
over s sub n of t.
607
00:37:38,800 --> 00:37:39,770
Why is that?
608
00:37:39,770 --> 00:37:46,850
Well, n of t is equal to n of
t, but t is greater than or
609
00:37:46,850 --> 00:37:49,520
equal to the time of the
most recent arrival.
610
00:37:52,790 --> 00:37:56,510
So we have n of t over t is less
than or equal to n of t
611
00:37:56,510 --> 00:37:59,250
over s sub n of t.
612
00:37:59,250 --> 00:38:03,180
The other thing that's important
to observe is that
613
00:38:03,180 --> 00:38:05,780
now we want to look at
what happens as t
614
00:38:05,780 --> 00:38:07,930
gets larger and larger.
615
00:38:07,930 --> 00:38:15,626
And what happens to this
ratio, n of t over t?
616
00:38:15,626 --> 00:38:19,010
Well, this ratio n of t over t
is this thing we were looking
617
00:38:19,010 --> 00:38:21,690
at here, which is
kind of a mess.
618
00:38:21,690 --> 00:38:24,550
It jumps up, it goes down a
little bit, jumps up, goes
619
00:38:24,550 --> 00:38:26,690
down a little bit, jumps up.
620
00:38:26,690 --> 00:38:34,550
But the set of values that
it goes through--
621
00:38:34,550 --> 00:38:38,570
the set of values that this goes
through, namely, the set
622
00:38:38,570 --> 00:38:41,410
right before each
of these jumps--
623
00:38:41,410 --> 00:38:46,860
is the same set of values
as n over s sub n.
624
00:38:46,860 --> 00:38:51,810
As I look through this sequence,
I look at this n of
625
00:38:51,810 --> 00:38:55,670
t over s sub n of t.
626
00:38:55,670 --> 00:39:02,050
That's this point here, and then
it's this point there.
627
00:39:05,350 --> 00:39:15,290
Anyway, n of t over s sub n of t
is going to stay constant as
628
00:39:15,290 --> 00:39:19,140
t goes from here
over to there.
629
00:39:19,140 --> 00:39:20,390
That's the way I've
drawn the picture.
630
00:39:20,390 --> 00:39:24,910
I start out with any t in
this interval here.
631
00:39:24,910 --> 00:39:29,110
This slope keeps changing as
t goes from there to there.
632
00:39:29,110 --> 00:39:31,650
This slope does not change.
633
00:39:31,650 --> 00:39:36,090
This is determined just by which
particular integer value
634
00:39:36,090 --> 00:39:38,280
of n we're talking about.
635
00:39:38,280 --> 00:39:45,140
So n of t over s sub n of t
jumps at each value of n.
636
00:39:45,140 --> 00:39:51,760
So this now becomes just the
sequence of numbers.
637
00:39:51,760 --> 00:39:56,190
And that sequence of numbers
is the sequence n
638
00:39:56,190 --> 00:39:59,090
divided by s sub b.
639
00:39:59,090 --> 00:40:01,650
Why is that important?
640
00:40:01,650 --> 00:40:04,660
That's the thing we have
some control over.
641
00:40:04,660 --> 00:40:08,560
That's what appears up here.
642
00:40:08,560 --> 00:40:10,410
So we know how to
deal with that.
643
00:40:12,930 --> 00:40:19,040
That's what this result about
convergence added to this
644
00:40:19,040 --> 00:40:23,890
result about functions of
converging functions tells us.
645
00:40:27,350 --> 00:40:33,120
So redrawing the same figure,
we've observed that n of t
646
00:40:33,120 --> 00:40:38,070
over t is less than or equal to
n of t over s sub n of t.
647
00:40:38,070 --> 00:40:42,330
It goes through the same set of
values as n over s sub n,
648
00:40:42,330 --> 00:40:46,900
and therefore the limit as t
goes to infinity of n over t
649
00:40:46,900 --> 00:40:52,190
over s sub n of t is the same
as the limit as n goes to
650
00:40:52,190 --> 00:40:55,350
infinity of n over s sub n.
651
00:40:55,350 --> 00:40:57,990
And that limit, with
probability
652
00:40:57,990 --> 00:40:59,940
1, is 1 over x bar.
653
00:40:59,940 --> 00:41:03,650
That's the thing that this
theorem, we just
654
00:41:03,650 --> 00:41:06,810
"pfed" said to us.
655
00:41:06,810 --> 00:41:11,890
There's a bit of a pf in here
too, because you really ought
656
00:41:11,890 --> 00:41:14,740
to show that as t goes
to infinity, n
657
00:41:14,740 --> 00:41:16,670
of t goes to infinity.
658
00:41:16,670 --> 00:41:17,930
And that's not hard to do.
659
00:41:17,930 --> 00:41:19,580
It's done in the notes.
660
00:41:19,580 --> 00:41:21,350
You need to do it.
661
00:41:21,350 --> 00:41:24,990
You can almost see intuitively
that it has to happen.
662
00:41:24,990 --> 00:41:28,050
And in fact, it does
have to happen.
663
00:41:28,050 --> 00:41:28,650
OK.
664
00:41:28,650 --> 00:41:33,170
So this, in fact, is a limit.
665
00:41:33,170 --> 00:41:35,670
It does exist.
666
00:41:35,670 --> 00:41:40,440
Now, we go on the other y, and
we look at n of t over t,
667
00:41:40,440 --> 00:41:45,680
which is now greater than or
equal to n of t over s sub n
668
00:41:45,680 --> 00:41:47,070
of t plus 1.
669
00:41:47,070 --> 00:41:51,370
s sub b of t plus 1 is the
arrival epoch which is just
670
00:41:51,370 --> 00:41:54,620
larger than t.
671
00:41:54,620 --> 00:41:58,980
Now, n of t over s sub n of t
plus 1 goes through the same
672
00:41:58,980 --> 00:42:04,170
set of values as n over
s sub n plus 1.
673
00:42:04,170 --> 00:42:08,620
Namely, each time n increases,
this goes up by 1.
674
00:42:08,620 --> 00:42:13,410
So the limit as t goes to
infinity of n of t over s sub
675
00:42:13,410 --> 00:42:17,190
n of t plus 1 is the same as
the limit as n goes to
676
00:42:17,190 --> 00:42:26,570
infinity of n over the epoch
right after n of t.
677
00:42:26,570 --> 00:42:34,730
This you can rewrite as n plus
1 over s sub n plus 1 times n
678
00:42:34,730 --> 00:42:35,930
over n plus 1.
679
00:42:35,930 --> 00:42:38,320
Why do I want to rewrite
it this way?
680
00:42:38,320 --> 00:42:41,600
Because this quantity
I have a handle on.
681
00:42:41,600 --> 00:42:45,770
This is the same as the limit
of n over s sub n.
682
00:42:45,770 --> 00:42:47,760
I know what that limit is.
683
00:42:47,760 --> 00:42:52,100
This quantity, I have an even
better handle on, because this
684
00:42:52,100 --> 00:42:55,780
n over n plus 1 just moves--
685
00:42:58,380 --> 00:43:00,090
it's something that
starts out low.
686
00:43:00,090 --> 00:43:05,110
And as n gets bigger, it just
moves up towards 1.
687
00:43:05,110 --> 00:43:09,230
And therefore, when you look at
this limit, this has to be
688
00:43:09,230 --> 00:43:12,320
1 over x bar also.
689
00:43:12,320 --> 00:43:15,760
Since n of t over t is between
these two quantities, they
690
00:43:15,760 --> 00:43:17,560
both have the same limit.
691
00:43:17,560 --> 00:43:23,440
The limit of n of t over t is
equal to 1 over the expected
692
00:43:23,440 --> 00:43:26,088
value of x.
693
00:43:26,088 --> 00:43:27,080
STUDENT: Professor Gallager?
694
00:43:27,080 --> 00:43:28,072
ROBERT GALLAGER: Yeah?
695
00:43:28,072 --> 00:43:31,212
STUDENT: Excuse me if this is
a dumb question, but in the
696
00:43:31,212 --> 00:43:35,264
previous slide it said the limit
as t goes to infinity of
697
00:43:35,264 --> 00:43:39,805
the accounting process n of
t, would equal infinity.
698
00:43:39,805 --> 00:43:43,186
We've also been talking a lot
over the last week about the
699
00:43:43,186 --> 00:43:46,567
defectiveness and
non-defectiveness of these
700
00:43:46,567 --> 00:43:48,499
counting processes.
701
00:43:48,499 --> 00:43:55,300
So we can still find an n that's
sufficiently high, such
702
00:43:55,300 --> 00:43:58,510
that the probability of n of t
being greater than that n is
703
00:43:58,510 --> 00:44:00,541
0, so it's not defective.
704
00:44:00,541 --> 00:44:02,002
But I don't know.
705
00:44:02,002 --> 00:44:03,950
How do you--
706
00:44:03,950 --> 00:44:07,710
ROBERT GALLAGER: n of t is
either a random variable or a
707
00:44:07,710 --> 00:44:11,085
defective random variable
of each value of t.
708
00:44:14,724 --> 00:44:20,260
And what I'm claiming
here, which is not--
709
00:44:20,260 --> 00:44:22,110
This is something you
have to prove.
710
00:44:22,110 --> 00:44:26,400
But what I would like to show is
that for each value of t, n
711
00:44:26,400 --> 00:44:27,745
of t is not defective.
712
00:44:27,745 --> 00:44:30,120
In other words, these
arrivals have to
713
00:44:30,120 --> 00:44:32,320
come sometime or other.
714
00:44:35,680 --> 00:44:36,540
OK.
715
00:44:36,540 --> 00:44:41,160
Well, let's backtrack from
that a little bit.
716
00:44:45,230 --> 00:44:49,280
For n of t to be defective, I
would have to have an infinite
717
00:44:49,280 --> 00:44:54,010
number of arrivals come in
in some finite time.
718
00:44:54,010 --> 00:44:56,080
You did a problem in the
homework where that could
719
00:44:56,080 --> 00:45:02,250
happen, because the x sub i's
you were looking at were not
720
00:45:02,250 --> 00:45:12,600
identically distributed, so
that as t increased, the
721
00:45:12,600 --> 00:45:15,480
number of arrivals you had were
722
00:45:15,480 --> 00:45:18,830
increasing very, very rapidly.
723
00:45:18,830 --> 00:45:20,890
Here, that can't happen.
724
00:45:20,890 --> 00:45:25,120
And the reason it can't happen
is because we've started out
725
00:45:25,120 --> 00:45:29,190
with a renewal process where
by definition the
726
00:45:29,190 --> 00:45:34,080
inter-arrival intervals all have
the same distribution.
727
00:45:34,080 --> 00:45:36,710
So the rate of arrivals,
in a sense, is
728
00:45:36,710 --> 00:45:39,270
staying constant forever.
729
00:45:39,270 --> 00:45:41,860
Now, that's not a proof.
730
00:45:41,860 --> 00:45:44,435
If you look at the notes, the
notes have a proof of this.
731
00:45:47,950 --> 00:45:49,900
After you go through the proof,
you say, that's a
732
00:45:49,900 --> 00:45:51,900
little bit tedious.
733
00:45:51,900 --> 00:45:55,840
But you either have to go
through the tedious proof to
734
00:45:55,840 --> 00:46:01,430
see what is after you go through
it is obvious, or you
735
00:46:01,430 --> 00:46:04,450
have to say it's obvious,
which is a
736
00:46:04,450 --> 00:46:05,700
subject to some question.
737
00:46:09,220 --> 00:46:11,030
So yes, it was not a
stupid question.
738
00:46:11,030 --> 00:46:13,110
It was a very good question.
739
00:46:13,110 --> 00:46:15,310
And in fact, you do have
to trace that out.
740
00:46:15,310 --> 00:46:18,600
And that's involved here in
this in what we've done.
741
00:46:21,770 --> 00:46:26,020
I want to talk a little bit
about the central limit
742
00:46:26,020 --> 00:46:29,310
theorem for renewals.
743
00:46:29,310 --> 00:46:31,310
The notes don't prove
the central
744
00:46:31,310 --> 00:46:32,890
limit theorem for renewals.
745
00:46:32,890 --> 00:46:36,280
I'm not going to
prove it here.
746
00:46:36,280 --> 00:46:41,100
All I'm going to do is give you
an argument why you can
747
00:46:41,100 --> 00:46:46,280
see that it sort of has to be
true, if you don't look at any
748
00:46:46,280 --> 00:46:49,410
of the weird special cases you
might want to look at.
749
00:46:49,410 --> 00:46:53,500
So there is a reference given
in the text for where
750
00:46:53,500 --> 00:46:54,750
you can find it.
751
00:46:59,070 --> 00:47:03,950
I mean, I like to give proofs
of very important things.
752
00:47:03,950 --> 00:47:07,710
I didn't give a proof of this
because the amount of work to
753
00:47:07,710 --> 00:47:13,680
prove it was far greater than
the importance of the result,
754
00:47:13,680 --> 00:47:17,600
which means it's a very, very
tricky and very difficult
755
00:47:17,600 --> 00:47:23,070
thing to prove, even when you're
only talking about
756
00:47:23,070 --> 00:47:25,910
things like Bernoulli.
757
00:47:25,910 --> 00:47:26,380
OK.
758
00:47:26,380 --> 00:47:27,920
But here's the picture.
759
00:47:27,920 --> 00:47:30,750
And the picture, I think, will
make it sort of clear
760
00:47:30,750 --> 00:47:32,000
what's going on.
761
00:47:34,980 --> 00:47:38,740
We're talking now about an
underlying random variable x.
762
00:47:38,740 --> 00:47:42,590
We assume it has a second
moment, which we need to make
763
00:47:42,590 --> 00:47:46,000
the central limit
theorem true.
764
00:47:46,000 --> 00:47:54,480
The probability that s sub n is
less than or equal to t for
765
00:47:54,480 --> 00:48:00,490
n very large and for the
difference between t--
766
00:48:07,470 --> 00:48:10,510
Let's look at the
whole statement.
767
00:48:10,510 --> 00:48:15,270
What it's saying is if you look
at values of t which are
768
00:48:15,270 --> 00:48:21,070
equal to the mean for s sub n,
which is n x bar, plus some
769
00:48:21,070 --> 00:48:26,220
quantity alpha times sigma times
the square root of n,
770
00:48:26,220 --> 00:48:31,350
then as n gets large and t gets
correspondingly large,
771
00:48:31,350 --> 00:48:34,900
this probability is
approximately equal to the
772
00:48:34,900 --> 00:48:37,760
normal distribution function.
773
00:48:37,760 --> 00:48:43,400
In other words, what that's
saying is as I'm looking at
774
00:48:43,400 --> 00:48:48,820
the random variable, s sub n,
and taking n very, very large.
775
00:48:48,820 --> 00:48:53,710
The expected value of s sub n is
equal to n times x bar, so
776
00:48:53,710 --> 00:48:58,175
I'm moving way out where this
number is very, very large.
777
00:49:02,590 --> 00:49:11,370
As n gets larger and larger, n
increases and x bar increases,
778
00:49:11,370 --> 00:49:16,370
and they increase on a
slope 1 over x bar.
779
00:49:16,370 --> 00:49:19,170
So this is n, this
is n over x bar.
780
00:49:19,170 --> 00:49:24,610
The slope is n over n over x
bar, which is this slope here.
781
00:49:24,610 --> 00:49:32,510
Now, when you look at this
picture, what it sort of
782
00:49:32,510 --> 00:49:37,250
involves is you can choose
any n you want.
783
00:49:37,250 --> 00:49:40,780
We will assume the
x bar is fixed.
784
00:49:40,780 --> 00:49:43,580
You can choose any t
that you want to.
785
00:49:43,580 --> 00:49:50,230
Let's first hold b fixed and
look at a third dimension now,
786
00:49:50,230 --> 00:49:53,680
where for this particular
value of n, I
787
00:49:53,680 --> 00:49:54,930
want to look at the--
788
00:49:57,480 --> 00:50:01,370
And instead of looking at the
distribution function, let me
789
00:50:01,370 --> 00:50:04,050
look at a probability density,
which makes the argument
790
00:50:04,050 --> 00:50:05,660
easier to see.
791
00:50:05,660 --> 00:50:22,320
As I look at this for a
particular value of n, what
792
00:50:22,320 --> 00:50:25,820
I'm going to get is b x bar.
793
00:50:28,900 --> 00:50:38,840
This will be the probability
density of s sub n of x.
794
00:50:38,840 --> 00:50:44,820
And that's going to look like,
when n is large enough, it's
795
00:50:44,820 --> 00:50:50,560
going to look like a Gaussian
probability density.
796
00:50:50,560 --> 00:50:55,250
And the mean of that Gaussian
probability density will be
797
00:50:55,250 --> 00:50:59,690
mean n x bar.
798
00:50:59,690 --> 00:51:04,880
And the variance of this
probability density, now, is
799
00:51:04,880 --> 00:51:12,500
going to be the square root
of n times sigma.
800
00:51:18,230 --> 00:51:19,480
What else do I need?
801
00:51:23,090 --> 00:51:24,690
I guess that's it.
802
00:51:24,690 --> 00:51:26,050
This is the standard
deviation.
803
00:51:33,370 --> 00:51:33,740
OK.
804
00:51:33,740 --> 00:51:36,990
So you can visualize
what happens, now.
805
00:51:36,990 --> 00:51:42,030
As you start letting n get
bigger and bigger, you have
806
00:51:42,030 --> 00:51:45,930
this Gaussian density
for each value of n.
807
00:51:45,930 --> 00:51:50,030
Think of drawing this again for
some larger value of n.
808
00:51:50,030 --> 00:51:53,880
The mean will shift out
corresponding to a linear
809
00:51:53,880 --> 00:51:55,340
increase in n.
810
00:51:55,340 --> 00:51:59,570
The standard deviation will
shift out, but only according
811
00:51:59,570 --> 00:52:01,200
to the square root of n.
812
00:52:01,200 --> 00:52:04,490
So what's happening is the same
thing that always happens
813
00:52:04,490 --> 00:52:09,100
in the central limit theorem, is
that as n gets large, this
814
00:52:09,100 --> 00:52:14,810
density here is moving out with
n, and it's getting wider
815
00:52:14,810 --> 00:52:16,080
with the square root of n.
816
00:52:16,080 --> 00:52:18,840
So it's getting wider much
more slowly than
817
00:52:18,840 --> 00:52:21,280
it's getting bigger.
818
00:52:21,280 --> 00:52:24,330
Than It's getting wider
much more slowly
819
00:52:24,330 --> 00:52:26,580
than it's moving out.
820
00:52:26,580 --> 00:52:30,080
So if I try to look at what
happens, what's the
821
00:52:30,080 --> 00:52:34,600
probability that n of t is
greater than or equal to n?
822
00:52:34,600 --> 00:52:39,960
I now want to look at the same
curve here, but instead of
823
00:52:39,960 --> 00:52:44,160
looking at it here for a fixed
value of n, I want to look at
824
00:52:44,160 --> 00:52:50,660
the probability density out here
at some fixed value of t.
825
00:52:50,660 --> 00:52:52,460
So what's going to happen?
826
00:52:52,460 --> 00:52:58,450
The probability density here is
going to be the probability
827
00:52:58,450 --> 00:53:01,960
density that we we're talking
about here, but for
828
00:53:01,960 --> 00:53:04,490
this value up here.
829
00:53:04,490 --> 00:53:08,200
The probability density here is
going to correspond to the
830
00:53:08,200 --> 00:53:12,700
probability density here, and
so forth as we move down.
831
00:53:12,700 --> 00:53:16,700
So what's happening to this
probability density is that as
832
00:53:16,700 --> 00:53:20,720
we move up, the standard
deviation is getting
833
00:53:20,720 --> 00:53:22,070
a little bit wider.
834
00:53:22,070 --> 00:53:25,120
As we move down, standard
deviation is getting a little
835
00:53:25,120 --> 00:53:26,320
bit smaller.
836
00:53:26,320 --> 00:53:29,750
And as n gets bigger and bigger,
this shouldn't make
837
00:53:29,750 --> 00:53:32,510
any difference.
838
00:53:32,510 --> 00:53:37,440
So therefore, if you buy for the
moment the fact that this
839
00:53:37,440 --> 00:53:41,040
doesn't make any difference,
you have a Gaussian density
840
00:53:41,040 --> 00:53:43,790
going this way, you
have a Gaussian
841
00:53:43,790 --> 00:53:45,570
density centered here.
842
00:53:45,570 --> 00:53:51,270
Up here you have a Gaussian
density centered
843
00:53:51,270 --> 00:53:53,380
here at this point.
844
00:53:53,380 --> 00:53:56,030
And all those Gaussian densities
are the same, which
845
00:53:56,030 --> 00:53:59,300
means you have a Gaussian
density going this way, which
846
00:53:59,300 --> 00:54:02,640
is centered here.
847
00:54:02,640 --> 00:54:05,150
Here's the upper tail of
that Gaussian density.
848
00:54:05,150 --> 00:54:07,300
Here's the lower tail of
that Gaussian density.
849
00:54:11,600 --> 00:54:15,640
Now, to put that analytically,
it's saying that the
850
00:54:15,640 --> 00:54:21,040
probability that n of t is
greater than or equal to n,
851
00:54:21,040 --> 00:54:25,580
that's the same as the
probability that s sub n is
852
00:54:25,580 --> 00:54:27,560
less than or equal to t.
853
00:54:27,560 --> 00:54:33,460
So that is the distribution
function of s sub n less than
854
00:54:33,460 --> 00:54:34,350
or equal to t.
855
00:54:34,350 --> 00:54:39,640
When we go from n to t, what we
find is that n is equal to
856
00:54:39,640 --> 00:54:41,360
t over x bar--
857
00:54:41,360 --> 00:54:43,450
that's the mean we have here--
858
00:54:43,450 --> 00:54:46,860
minus alpha times sigma
times the square
859
00:54:46,860 --> 00:54:49,160
root of n over x bar.
860
00:54:49,160 --> 00:54:51,110
In other words, what
is happening is
861
00:54:51,110 --> 00:54:52,270
the following thing.
862
00:54:52,270 --> 00:54:59,900
We have a density going this
way, which has variance, which
863
00:54:59,900 --> 00:55:02,560
has standard deviation
proportional to the
864
00:55:02,560 --> 00:55:04,130
square root of n.
865
00:55:04,130 --> 00:55:07,460
When we look at that same
density going this way,
866
00:55:07,460 --> 00:55:10,550
ignoring the fact that this
distance here that we're
867
00:55:10,550 --> 00:55:15,030
looking at is very small, this
density here is going to be
868
00:55:15,030 --> 00:55:18,600
compressed by this slope here.
869
00:55:18,600 --> 00:55:22,380
In other words, what we have is
the probability that n of t
870
00:55:22,380 --> 00:55:24,590
greater than or equal to
n is approximately
871
00:55:24,590 --> 00:55:27,160
equal to phi of alpha.
872
00:55:27,160 --> 00:55:32,220
n is equal to t over x bar minus
this alpha here times
873
00:55:32,220 --> 00:55:36,480
sigma times the square
root of n over x bar.
874
00:55:36,480 --> 00:55:39,720
Nasty equation, because
we have an n on both
875
00:55:39,720 --> 00:55:41,520
sides of the equation.
876
00:55:41,520 --> 00:55:44,350
So we will try to solve
this equation.
877
00:55:44,350 --> 00:55:48,080
And this is approximately equal
to t over x bar minus
878
00:55:48,080 --> 00:55:52,670
alpha times sigma times the
square root of n over x bar
879
00:55:52,670 --> 00:55:54,935
times the square root of x.
880
00:55:54,935 --> 00:55:56,630
Why is that true?
881
00:55:56,630 --> 00:56:14,500
Because it's approximately equal
to the square root--
882
00:56:14,500 --> 00:56:22,290
Well, it is equal to the square
root of t over x bar,
883
00:56:22,290 --> 00:56:25,820
which is this quantity here.
884
00:56:25,820 --> 00:56:30,630
And since this quantity here
is small relative to this
885
00:56:30,630 --> 00:56:34,770
quantity here, when you solve
this equation for t, you're
886
00:56:34,770 --> 00:56:38,760
going to ignore this term
and just get this small
887
00:56:38,760 --> 00:56:40,310
correction term here.
888
00:56:40,310 --> 00:56:43,620
That's exactly the same thing
that I said when I was looking
889
00:56:43,620 --> 00:56:47,620
at this graphically, when I was
saying that if you look at
890
00:56:47,620 --> 00:56:52,370
the density at larger values
than n, you get a standard
891
00:56:52,370 --> 00:56:54,440
deviation which is larger.
892
00:56:54,440 --> 00:56:58,360
When you look at a smaller
value of n, you get is a
893
00:56:58,360 --> 00:57:01,370
standard deviation
which is smaller.
894
00:57:01,370 --> 00:57:11,290
Which means that when you look
at it along here, you're going
895
00:57:11,290 --> 00:57:16,570
to get what looks like a
Gaussian density, except the
896
00:57:16,570 --> 00:57:20,370
standard deviation is a little
expanded up there and little
897
00:57:20,370 --> 00:57:22,110
shrunk down here.
898
00:57:22,110 --> 00:57:24,610
But that doesn't make any
difference as n gets very
899
00:57:24,610 --> 00:57:28,840
large, because that shrinking
factor is proportional to the
900
00:57:28,840 --> 00:57:30,985
square root of n
rather than n.
901
00:57:35,160 --> 00:57:37,270
Now beyond that, you just
have to look at this
902
00:57:37,270 --> 00:57:39,780
and live with it.
903
00:57:39,780 --> 00:57:42,650
Or else you have to look up a
proof of it, which I don't
904
00:57:42,650 --> 00:57:45,580
particularly recommend.
905
00:57:45,580 --> 00:57:52,950
So this is the central limit
theorem for renewal processes.
906
00:57:52,950 --> 00:57:58,230
n of t tends to Gaussian with
a mean t over x bar and a
907
00:57:58,230 --> 00:58:03,970
standard deviation sigma times
square root of t over x bar
908
00:58:03,970 --> 00:58:07,280
times 1 over square root of x.
909
00:58:09,860 --> 00:58:14,540
And now you sort of understand
why that is, I hope.
910
00:58:14,540 --> 00:58:15,230
OK.
911
00:58:15,230 --> 00:58:17,910
Next thing I want to
go to is the time
912
00:58:17,910 --> 00:58:20,940
average residual life.
913
00:58:20,940 --> 00:58:24,990
You were probably somewhat
bothered when you saw with
914
00:58:24,990 --> 00:58:31,060
Poisson processes that if you
arrived to wait for a bus, the
915
00:58:31,060 --> 00:58:36,980
expected time between buses
turned out to be twice the
916
00:58:36,980 --> 00:58:39,060
expected time from one
bus to the next.
917
00:58:39,060 --> 00:58:43,160
Namely, whenever you arrive to
look for a bus, the time until
918
00:58:43,160 --> 00:58:48,280
the next bus was an exponential
random variable.
919
00:58:48,280 --> 00:58:53,020
The time back to the last bus,
if you're far enough in, was
920
00:58:53,020 --> 00:58:55,160
an exponential random
variable.
921
00:58:55,160 --> 00:59:00,010
The sum of two, the expected
value from the time before
922
00:59:00,010 --> 00:59:04,940
until the time later, was
twice what it should be.
923
00:59:04,940 --> 00:59:08,590
And we went through some kind
of song and dance saying
924
00:59:08,590 --> 00:59:12,760
that's because you come in at a
given point and you're more
925
00:59:12,760 --> 00:59:19,540
likely to come in during one of
these longer inter-arrival
926
00:59:19,540 --> 00:59:22,300
periods than you are to
come in during a short
927
00:59:22,300 --> 00:59:24,010
inter-arrival.
928
00:59:24,010 --> 00:59:27,330
And it has to be a song and a
dance, and it didn't really
929
00:59:27,330 --> 00:59:30,630
explain anything very well,
because we were locked into
930
00:59:30,630 --> 00:59:32,850
the exponential density.
931
00:59:32,850 --> 00:59:34,110
Now we have an advantage.
932
00:59:34,110 --> 00:59:38,310
We can explain things like that,
because we can look at
933
00:59:38,310 --> 00:59:41,700
any old distribution we want to
look at, and that will let
934
00:59:41,700 --> 00:59:45,620
us see what this thing which
is called the paradox of
935
00:59:45,620 --> 00:59:49,430
residual life really
amounts to.
936
00:59:49,430 --> 00:59:53,070
It's what tells us why we
sometimes have to wait a very
937
00:59:53,070 --> 00:59:57,190
much longer time than we think
we should if we understand
938
00:59:57,190 --> 00:59:59,075
some particular kind
of process.
939
01:00:05,310 --> 01:00:08,430
So here's where we're
going to start.
940
01:00:08,430 --> 01:00:10,400
What happened?
941
01:00:10,400 --> 01:00:12,183
I lost a slide.
942
01:00:12,183 --> 01:00:12,890
Ah.
943
01:00:12,890 --> 01:00:14,990
There we are.
944
01:00:14,990 --> 01:00:21,120
Residual life, y of t, of a
renewal process at times t, is
945
01:00:21,120 --> 01:00:24,420
the remaining time until
the next renewal.
946
01:00:24,420 --> 01:00:30,290
Namely, we have this counting
process for any
947
01:00:30,290 --> 01:00:32,990
given renewal process.
948
01:00:32,990 --> 01:00:38,090
We have this random variable,
which is the time of the first
949
01:00:38,090 --> 01:00:45,920
arrival after t, which is
s sub n of t plus 1.
950
01:00:45,920 --> 01:00:50,140
And that difference
is the duration
951
01:00:50,140 --> 01:00:52,080
until the next arrival.
952
01:00:52,080 --> 01:00:56,870
Starting at time t, there's a
random variable, which is the
953
01:00:56,870 --> 01:01:00,790
time from t until the next
arrival after t.
954
01:01:00,790 --> 01:01:07,670
That is specifically the arrival
epoch of the arrival
955
01:01:07,670 --> 01:01:12,410
after time t, which is s sub n
of t plus 1 minus the number
956
01:01:12,410 --> 01:01:15,120
of arrivals that have occurred
up until time t.
957
01:01:15,120 --> 01:01:22,490
You take any sample path of this
renewal process, and y of
958
01:01:22,490 --> 01:01:27,100
t will have some value
in that sample path.
959
01:01:27,100 --> 01:01:29,480
As I say here, this is how long
you have to wait for a
960
01:01:29,480 --> 01:01:33,740
bus if the bus arrivals were
renewal processes.
961
01:01:33,740 --> 01:01:37,626
STUDENT: Should it also be
s n t, where there is
962
01:01:37,626 --> 01:01:40,500
minus sign on that?
963
01:01:40,500 --> 01:01:43,980
ROBERT GALLAGER: No, because
just by definition, a residual
964
01:01:43,980 --> 01:01:48,500
life, the residual life starting
at time t is the time
965
01:01:48,500 --> 01:01:50,330
for the next arrival.
966
01:01:50,330 --> 01:01:52,850
There's also something called
age that we'll talk about
967
01:01:52,850 --> 01:02:02,550
later, which is how long is it
back to the last arrival.
968
01:02:02,550 --> 01:02:07,710
In other words, that age is
the age of the particular
969
01:02:07,710 --> 01:02:10,125
inter-arrival interval that
you happen to be in.
970
01:02:10,125 --> 01:02:10,490
Yes?
971
01:02:10,490 --> 01:02:16,873
STUDENT: It should be s sub n of
t plus 1 minus t instead of
972
01:02:16,873 --> 01:02:20,801
minus N, because it's
the time from t to--
973
01:02:30,130 --> 01:02:31,640
ROBERT GALLAGER: Yes,
I agree with you.
974
01:02:31,640 --> 01:02:33,022
There's something wrong there.
975
01:02:52,010 --> 01:02:52,600
I'm sorry.
976
01:02:52,600 --> 01:02:57,090
That I should be s sub n
of t plus 1 minus t.
977
01:02:57,090 --> 01:02:58,028
Good.
978
01:02:58,028 --> 01:03:01,490
That's what happens
when you make up
979
01:03:01,490 --> 01:03:02,990
slides too late at night.
980
01:03:18,230 --> 01:03:20,950
And as I said, we'll talk about
something called age,
981
01:03:20,950 --> 01:03:28,210
which is a of t is equal to
t minus s sub n of t.
982
01:03:31,900 --> 01:03:36,090
So this is a random variable
defined at every value of t.
983
01:03:36,090 --> 01:03:39,360
What we'd like to look at now is
what does that look like as
984
01:03:39,360 --> 01:03:43,280
a sample function as
a sample path.
985
01:03:45,870 --> 01:03:49,350
The residual life is
a function of t--
986
01:03:55,110 --> 01:03:59,030
Nicest way to view residual
life is that it's a reward
987
01:03:59,030 --> 01:04:02,150
function on a renewal process.
988
01:04:02,150 --> 01:04:07,370
A renewal process just
consists of these--
989
01:04:07,370 --> 01:04:09,790
Well, you can look
at in three ways.
990
01:04:09,790 --> 01:04:12,740
It's a sequence of inter-arrival
times, all
991
01:04:12,740 --> 01:04:14,390
identically distributed.
992
01:04:14,390 --> 01:04:17,470
It's the sequence of
arrival epochs.
993
01:04:17,470 --> 01:04:24,060
Or it's this unaccountably
infinite number of random
994
01:04:24,060 --> 01:04:25,360
variables, n of t.
995
01:04:27,980 --> 01:04:32,280
Given that process, you can
define whatever kind of reward
996
01:04:32,280 --> 01:04:36,270
you want to, which is the same
kind of reward we were talking
997
01:04:36,270 --> 01:04:40,210
about with Markov chains, where
you just define some
998
01:04:40,210 --> 01:04:46,180
kind of reward that you achieve
at each value of t.
999
01:04:46,180 --> 01:04:48,150
But that reward--
1000
01:04:48,150 --> 01:04:51,740
we'll talk about reward
on renewal processes--
1001
01:04:51,740 --> 01:04:56,190
is restricted to be a reward
which is a function only of
1002
01:04:56,190 --> 01:04:59,090
the particular inter-arrival
interval that you
1003
01:04:59,090 --> 01:05:00,540
happen to be in.
1004
01:05:00,540 --> 01:05:03,760
Now, I don't want to talk about
that too much right now,
1005
01:05:03,760 --> 01:05:08,370
because it is easier to
understand residual life than
1006
01:05:08,370 --> 01:05:12,150
it is to understand the general
idea of these renewal
1007
01:05:12,150 --> 01:05:13,640
reward functions.
1008
01:05:13,640 --> 01:05:17,460
So we'll just talk about
residual life to start with,
1009
01:05:17,460 --> 01:05:20,840
and then get back to the
more general thing.
1010
01:05:20,840 --> 01:05:25,470
We would like, sometimes, to
look at the time-average value
1011
01:05:25,470 --> 01:05:30,560
of residual life, which is you
take the residual life at time
1012
01:05:30,560 --> 01:05:34,080
tau, you integrate it at the
time t, and then you
1013
01:05:34,080 --> 01:05:36,700
divide by time t.
1014
01:05:36,700 --> 01:05:39,450
This is the time average
residual life from
1015
01:05:39,450 --> 01:05:42,360
0 up to time t.
1016
01:05:42,360 --> 01:05:45,880
We will now ask the question,
does this have a limit as t
1017
01:05:45,880 --> 01:05:47,480
goes to infinity?
1018
01:05:47,480 --> 01:05:51,480
And we will see that,
in fact, it does.
1019
01:05:51,480 --> 01:05:52,873
So let's draw a picture.
1020
01:05:55,510 --> 01:06:00,815
Here is a picture of some
arbitrary renewal process.
1021
01:06:00,815 --> 01:06:06,100
I've given the inter-arrival
times, x1, x2, so forth, the
1022
01:06:06,100 --> 01:06:11,500
arrival epochs, s1, s2,
so forth, and n of t.
1023
01:06:11,500 --> 01:06:19,330
Now, let's ask, for this
particular sample function
1024
01:06:19,330 --> 01:06:20,930
what is the residual life?
1025
01:06:20,930 --> 01:06:25,040
Namely, at each value of t,
what's the time until the next
1026
01:06:25,040 --> 01:06:26,760
arrival occurs?
1027
01:06:26,760 --> 01:06:31,700
Well, this is a perfectly
specific function of this
1028
01:06:31,700 --> 01:06:34,670
individual sample
function here.
1029
01:06:34,670 --> 01:06:40,470
This is a sample function, now
in the interval from 0 to s1,
1030
01:06:40,470 --> 01:06:43,330
the time until the
next arrival.
1031
01:06:43,330 --> 01:06:47,440
It starts out as x1,
drops down to 0.
1032
01:06:47,440 --> 01:06:53,480
Now, don't ask the question,
what is my residual life if I
1033
01:06:53,480 --> 01:06:56,330
don't know what the rest of
the sample function is?
1034
01:06:56,330 --> 01:06:59,130
That's not the question
we're asking here.
1035
01:06:59,130 --> 01:07:03,400
The question we're asking is
somebody gives you a picture
1036
01:07:03,400 --> 01:07:07,730
of this entire sample path, and
you want to find out, for
1037
01:07:07,730 --> 01:07:12,070
that particular picture, what is
the residual life at every
1038
01:07:12,070 --> 01:07:13,500
value of t.
1039
01:07:13,500 --> 01:07:19,620
And for a value of t very close
to 0, the residual life
1040
01:07:19,620 --> 01:07:21,320
is the time up to s1.
1041
01:07:21,320 --> 01:07:27,580
So it's decaying linearly
down to 0 at s1.
1042
01:07:27,580 --> 01:07:34,740
At s1, it jumps up immediately
to x2, which is the time from
1043
01:07:34,740 --> 01:07:38,340
any time after s1 to s2.
1044
01:07:38,340 --> 01:07:41,150
I have a little circle
down there.
1045
01:07:41,150 --> 01:07:45,330
And from x2, it decays
down to 0.
1046
01:07:45,330 --> 01:07:50,970
So we have a whole bunch
here of triangles.
1047
01:07:50,970 --> 01:07:55,920
So for any sample function, we
have this sample function of
1048
01:07:55,920 --> 01:08:01,400
residual life, which is, in
fact, just decaying triangles.
1049
01:08:01,400 --> 01:08:04,080
It's nothing more than that.
1050
01:08:04,080 --> 01:08:08,600
For every t in here, the amount
of time until the next
1051
01:08:08,600 --> 01:08:15,580
arrival is simply s2 minus t,
which is that value there.
1052
01:08:15,580 --> 01:08:19,870
This decay is with slope minus
1, so there's nothing to
1053
01:08:19,870 --> 01:08:23,729
finding out what this
is if you know this.
1054
01:08:23,729 --> 01:08:26,990
This is a very simple
function of that.
1055
01:08:26,990 --> 01:08:30,490
So a residual-life sample
function is a sequence of
1056
01:08:30,490 --> 01:08:33,359
isosceles triangles,
one starting at
1057
01:08:33,359 --> 01:08:35,380
each arrival epoch.
1058
01:08:35,380 --> 01:08:41,010
The time average for a given
sample function is, how do I
1059
01:08:41,010 --> 01:08:46,069
find the time average starting
from 0 going up to
1060
01:08:46,069 --> 01:08:48,189
some large value t?
1061
01:08:48,189 --> 01:08:50,876
Well, I simply integrate these
isosceles triangles.
1062
01:08:53,490 --> 01:08:58,930
And I can integrate these, and
you can integrate these, and
1063
01:08:58,930 --> 01:09:02,729
anybody who's had a high school
education can integrate
1064
01:09:02,729 --> 01:09:05,890
these, because it's just the
sum of the areas of all of
1065
01:09:05,890 --> 01:09:07,290
these triangles.
1066
01:09:07,290 --> 01:09:15,020
So this area here is 1 over 2
times x sub i squared, then we
1067
01:09:15,020 --> 01:09:16,109
divide by t.
1068
01:09:16,109 --> 01:09:19,069
So it's 1 over t times
this integral.
1069
01:09:19,069 --> 01:09:23,022
This integral here is the area
of the first triangle plus the
1070
01:09:23,022 --> 01:09:27,830
area of the second triangle plus
1/3 plus 1/4, plus this
1071
01:09:27,830 --> 01:09:33,270
little runt thing at the end,
which is, if I pick t in here,
1072
01:09:33,270 --> 01:09:39,160
this little runt thing is
going to be that little
1073
01:09:39,160 --> 01:09:42,020
trapezoid, which we could figure
out if we wanted to,
1074
01:09:42,020 --> 01:09:43,939
but we don't want to.
1075
01:09:43,939 --> 01:09:50,620
The main thing is we get this
sum of squares here, there
1076
01:09:50,620 --> 01:09:53,109
that's easy enough
to deal with.
1077
01:09:53,109 --> 01:09:55,800
So this is what we found here.
1078
01:09:55,800 --> 01:10:00,410
It is easier to bound this
quantity, instead of having
1079
01:10:00,410 --> 01:10:04,710
that little runt at the end, to
drop the runt to this side
1080
01:10:04,710 --> 01:10:06,860
and to extend the runt
on this side to the
1081
01:10:06,860 --> 01:10:10,060
entire isosceles triangles.
1082
01:10:10,060 --> 01:10:16,220
So this time average residual
life at the time t is between
1083
01:10:16,220 --> 01:10:18,640
this and this.
1084
01:10:18,640 --> 01:10:24,260
The limit of this as t goes
to infinity is what?
1085
01:10:24,260 --> 01:10:26,610
Well, it's just a limit
of a sequence
1086
01:10:26,610 --> 01:10:28,505
of IID random variables.
1087
01:10:39,490 --> 01:10:41,080
No, excuse me.
1088
01:10:41,080 --> 01:10:43,260
We are dealing here with
sample function.
1089
01:10:43,260 --> 01:10:48,300
So what we have is a limit
as t goes to infinity.
1090
01:10:48,300 --> 01:10:52,740
And I want to rewrite this
here as x sub n squared
1091
01:10:52,740 --> 01:10:57,030
divided by n of t times
n has to t over 2t.
1092
01:10:57,030 --> 01:10:59,860
I want to separate it, and
just divide it and
1093
01:10:59,860 --> 01:11:02,360
multiply by n of t.
1094
01:11:02,360 --> 01:11:04,080
I want to look at this term.
1095
01:11:04,080 --> 01:11:09,020
What happens to this term
as t gets large?
1096
01:11:09,020 --> 01:11:13,170
Well, as t gets large,
n of t gets large.
1097
01:11:13,170 --> 01:11:18,230
This quantity here just goes
through the same set of values
1098
01:11:18,230 --> 01:11:23,760
as the sum up to some finite
limit divided by that limit
1099
01:11:23,760 --> 01:11:24,200
goes through.
1100
01:11:24,200 --> 01:11:32,820
So the limit of this quantity
here is just the expected
1101
01:11:32,820 --> 01:11:35,360
value of x squared.
1102
01:11:35,360 --> 01:11:37,830
What is this quantity here?
1103
01:11:37,830 --> 01:11:41,090
Well, this is what the renewal
theorem deals with.
1104
01:11:41,090 --> 01:11:46,550
This limit here is 1 over 2
times the expected value of x.
1105
01:11:46,550 --> 01:11:50,080
That's what we showed before.
1106
01:11:50,080 --> 01:11:53,190
This goes to a limit, this goes
to a limit, the whole
1107
01:11:53,190 --> 01:11:55,060
thing goes to a limit.
1108
01:11:55,060 --> 01:11:57,630
And it goes to limit
with probability 1
1109
01:11:57,630 --> 01:12:00,140
for all sample functions.
1110
01:12:00,140 --> 01:12:04,760
So this time average residual
life has the expected value of
1111
01:12:04,760 --> 01:12:11,070
x squared divided by 2 times
the expected value of x.
1112
01:12:11,070 --> 01:12:14,350
Now if you look at this, you'll
see that what we've
1113
01:12:14,350 --> 01:12:17,580
done is something which is very
simple, because of the
1114
01:12:17,580 --> 01:12:20,310
fact we have renewal theory
at this point.
1115
01:12:20,310 --> 01:12:24,080
If we had to look at the
probabilities of where all of
1116
01:12:24,080 --> 01:12:29,290
these arrival epochs occur,
and then deal with all of
1117
01:12:29,290 --> 01:12:36,210
those random variables, and
go through some enormously
1118
01:12:36,210 --> 01:12:39,660
complex calculation to find
the expected value of this
1119
01:12:39,660 --> 01:12:43,050
residual life at the time
t, it would be an
1120
01:12:43,050 --> 01:12:45,340
incredibly hard problem.
1121
01:12:45,340 --> 01:12:48,590
But looking at it in terms of
sample paths for random
1122
01:12:48,590 --> 01:12:51,340
variables, it's an incredibly
simple problem.
1123
01:12:55,900 --> 01:13:00,000
Want to look at one example
here, because when we look at
1124
01:13:00,000 --> 01:13:05,390
this, well, first thing
is just a couple of
1125
01:13:05,390 --> 01:13:07,040
examples to work out.
1126
01:13:07,040 --> 01:13:10,960
The time average residual life
has expected value of x
1127
01:13:10,960 --> 01:13:14,040
squared over 2 times the
expected value of x.
1128
01:13:14,040 --> 01:13:19,150
If x is almost deterministic,
then the expected value of x
1129
01:13:19,150 --> 01:13:23,900
squared is just a square of
the expected value of x.
1130
01:13:23,900 --> 01:13:28,390
So we wind up with the expected
value of x over 2,
1131
01:13:28,390 --> 01:13:32,250
which is sort of what you would
expect if you look from
1132
01:13:32,250 --> 01:13:35,970
time 0 to time infinity, and
these arrivals come along
1133
01:13:35,970 --> 01:13:40,230
regularly, then the expected
time you have to wait for the
1134
01:13:40,230 --> 01:13:48,890
next arrival varies
from 0 up to x.
1135
01:13:48,890 --> 01:13:51,370
And the average of it is x/2.
1136
01:13:51,370 --> 01:13:54,190
So no problem there.
1137
01:13:54,190 --> 01:13:58,860
If x is exponential, we've
already found out that the
1138
01:13:58,860 --> 01:14:02,900
expected time we have to wait
until the next arrival is the
1139
01:14:02,900 --> 01:14:08,060
expected value of x, because
these arrivals are memoryless.
1140
01:14:08,060 --> 01:14:11,500
So I start looking at this
Poisson process at a given
1141
01:14:11,500 --> 01:14:16,210
value of t, and the time until
the next arrival is
1142
01:14:16,210 --> 01:14:21,910
exponential, and it's the same
as the expected time from one
1143
01:14:21,910 --> 01:14:24,030
arrival to the next arrival.
1144
01:14:24,030 --> 01:14:26,360
So we have that quantity
there, which
1145
01:14:26,360 --> 01:14:27,830
looks a little strange.
1146
01:14:27,830 --> 01:14:32,720
This one, this is a very
peculiar random variable.
1147
01:14:32,720 --> 01:14:37,000
But this really explains what's
going on with this kind
1148
01:14:37,000 --> 01:14:42,080
of paradoxical thing, which we
found with a Poisson process,
1149
01:14:42,080 --> 01:14:47,240
where if you arrive to wait for
a bus, you're waiting time
1150
01:14:47,240 --> 01:14:51,880
is not any less because of the
fact that you've just arrived
1151
01:14:51,880 --> 01:14:54,940
between two arrivals, and it
ought to be the same distance
1152
01:14:54,940 --> 01:14:58,060
back to the last one at a
distance of first one.
1153
01:14:58,060 --> 01:15:00,210
That was always a little
surprising.
1154
01:15:00,210 --> 01:15:04,010
This, I think, explains
what's going on
1155
01:15:04,010 --> 01:15:05,530
better than most things.
1156
01:15:05,530 --> 01:15:11,850
Look at a binary random
variable, x, where the
1157
01:15:11,850 --> 01:15:17,680
probability that x is equal to
epsilon is 1 minus epsilon,
1158
01:15:17,680 --> 01:15:19,830
and the probability that
x is equal to 1
1159
01:15:19,830 --> 01:15:21,960
over epsilon is epsilon.
1160
01:15:21,960 --> 01:15:24,520
And think of epsilon as
being very large.
1161
01:15:24,520 --> 01:15:25,910
So what happens?
1162
01:15:25,910 --> 01:15:29,190
You got a whole bunch of little,
tiny inter-renewal
1163
01:15:29,190 --> 01:15:32,960
intervals, which are
epsilon apart.
1164
01:15:32,960 --> 01:15:35,990
And then with very small
probability, you get an
1165
01:15:35,990 --> 01:15:38,570
enormous one.
1166
01:15:38,570 --> 01:15:41,580
And you wait for 1
over epsilon for
1167
01:15:41,580 --> 01:15:43,100
that one to be finished.
1168
01:15:43,100 --> 01:15:45,530
Then you've got a bunch of
little ones which are all
1169
01:15:45,530 --> 01:15:46,760
epsilon apart.
1170
01:15:46,760 --> 01:15:51,300
Then you get an enormous one,
which is 1 over epsilon long.
1171
01:15:51,300 --> 01:15:56,680
And now you can see perfectly
well that if you arrive to
1172
01:15:56,680 --> 01:16:00,690
wait for a bus and the buses are
distributed this way, this
1173
01:16:00,690 --> 01:16:05,020
is sort of what happens when you
have a bus system which is
1174
01:16:05,020 --> 01:16:07,910
perfectly regular but
subject to failures.
1175
01:16:07,910 --> 01:16:09,580
Whenever you have a failure,
you have an
1176
01:16:09,580 --> 01:16:11,300
incredibly long wait.
1177
01:16:11,300 --> 01:16:13,570
Otherwise, you have
very small waits.
1178
01:16:13,570 --> 01:16:17,320
So what happens here?
1179
01:16:17,320 --> 01:16:21,380
The duration of this whole
interval here of these little,
1180
01:16:21,380 --> 01:16:25,130
tiny inter-arrival times, the
distance between failure in a
1181
01:16:25,130 --> 01:16:30,240
sense, is 1 minus epsilon,
as it turns out.
1182
01:16:30,240 --> 01:16:33,240
It's very close to 1.
1183
01:16:33,240 --> 01:16:37,370
This distance here is
1 over epsilon.
1184
01:16:37,370 --> 01:16:41,633
And this quantity here,
if you work it out--
1185
01:16:46,060 --> 01:16:46,500
Let's see.
1186
01:16:46,500 --> 01:16:48,780
What is it?
1187
01:16:48,780 --> 01:16:52,100
We take this distribution here,
we look for the expected
1188
01:16:52,100 --> 01:16:53,350
value of x squared.
1189
01:16:57,800 --> 01:16:58,450
Let's see.
1190
01:16:58,450 --> 01:17:03,970
1 over epsilon squared times
epsilon, which is 1 over
1191
01:17:03,970 --> 01:17:07,690
epsilon plus 1 minus epsilon
times something.
1192
01:17:07,690 --> 01:17:13,480
So the expected time that you
have to wait if you arrive
1193
01:17:13,480 --> 01:17:17,050
somewhere along here is
1 over 2 epsilon.
1194
01:17:17,050 --> 01:17:20,520
If epsilon is very small, you
have a very, very long waiting
1195
01:17:20,520 --> 01:17:25,900
time because of these very long
distributions in here.
1196
01:17:25,900 --> 01:17:30,180
You normally don't tend to
arrive in any of these periods
1197
01:17:30,180 --> 01:17:31,740
or any of these periods.
1198
01:17:31,740 --> 01:17:35,740
But however you want to
interpret it, this theorem
1199
01:17:35,740 --> 01:17:40,280
about renewals tells you
precisely that the time
1200
01:17:40,280 --> 01:17:45,870
average residual life
is, in fact, this
1201
01:17:45,870 --> 01:17:49,170
quantity 1 over 2 epsilon.
1202
01:17:49,170 --> 01:17:52,220
That's this paradox
of residual life.
1203
01:17:52,220 --> 01:17:55,420
Your residual life is much
larger than it looks like it
1204
01:17:55,420 --> 01:18:00,780
ought to be, because it's not
by any means the same as the
1205
01:18:00,780 --> 01:18:05,940
expected interval between
successive arrivals, which in
1206
01:18:05,940 --> 01:18:07,860
this case is very small.
1207
01:18:07,860 --> 01:18:09,730
OK I think I'll stop there.
1208
01:18:09,730 --> 01:18:13,280
And we will talk more about
this next time.