1 00:00:00,530 --> 00:00:02,960 The following content is provided under a Creative 2 00:00:02,960 --> 00:00:04,370 Commons license. 3 00:00:04,370 --> 00:00:07,410 Your support will help MIT OpenCourseWare continue to 4 00:00:07,410 --> 00:00:11,060 offer high quality educational resources for free. 5 00:00:11,060 --> 00:00:13,960 To make a donation or view additional materials from 6 00:00:13,960 --> 00:00:17,890 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,890 --> 00:00:19,140 ocw.mit.edu. 8 00:00:21,936 --> 00:00:22,840 ROBERT GALLAGER: OK. 9 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 68 00:04:22,540 --> 00:04:26,580 in probability, rather than with probability 1. 69 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. 114 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 141 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.