1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,910 Your support will help MIT OpenCourseWare continue to 4 00:00:06,910 --> 00:00:10,560 offer high quality educational resources for free. 5 00:00:10,560 --> 00:00:13,460 To make a donation or view additional materials from 6 00:00:13,460 --> 00:00:17,390 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,390 --> 00:00:18,640 ocw.mit.edu. 8 00:00:22,710 --> 00:00:26,630 PROFESSOR: So by now you have seen pretty much every 9 00:00:26,630 --> 00:00:30,273 possible trick there is in basic probability theory, 10 00:00:30,273 --> 00:00:33,100 about how to calculate distributions, and so on. 11 00:00:33,100 --> 00:00:37,230 You have the basic tools to do pretty much anything. 12 00:00:37,230 --> 00:00:40,610 So what's coming after this? 13 00:00:40,610 --> 00:00:45,370 Well, probability is useful for developing the science of 14 00:00:45,370 --> 00:00:48,120 inference, and this is a subject to which we're going 15 00:00:48,120 --> 00:00:51,280 to come back at the end of the semester. 16 00:00:51,280 --> 00:00:55,240 Another chapter, which is what we will be doing over the next 17 00:00:55,240 --> 00:00:59,210 few weeks, is to deal with phenomena that evolve in time. 18 00:00:59,210 --> 00:01:03,300 So so-called random processes or stochastic processes. 19 00:01:03,300 --> 00:01:05,069 So what is this about? 20 00:01:05,069 --> 00:01:08,100 So in the real world, you don't just throw two random 21 00:01:08,100 --> 00:01:09,520 variables and go home. 22 00:01:09,520 --> 00:01:11,410 Rather the world goes on. 23 00:01:11,410 --> 00:01:14,880 So you generate the random variable, then you get more 24 00:01:14,880 --> 00:01:18,100 random variables, and things evolve in time. 25 00:01:18,100 --> 00:01:21,560 And random processes are supposed to be models that 26 00:01:21,560 --> 00:01:25,680 capture the evolution of random phenomena over time. 27 00:01:25,680 --> 00:01:27,620 So that's what we will be doing. 28 00:01:27,620 --> 00:01:31,230 Now when we have evolution in time, mathematically speaking, 29 00:01:31,230 --> 00:01:34,840 you can use discrete time or continuous time. 30 00:01:34,840 --> 00:01:36,800 Of course, discrete time is easier. 31 00:01:36,800 --> 00:01:39,280 And that's where we're going to start from. 32 00:01:39,280 --> 00:01:43,000 And we're going to start from the easiest, simplest random 33 00:01:43,000 --> 00:01:46,740 process, which is the so-called Bernoulli process, 34 00:01:46,740 --> 00:01:50,250 which is nothing but just a sequence of coin flips. 35 00:01:50,250 --> 00:01:54,650 You keep flipping a coin and keep going forever. 36 00:01:54,650 --> 00:01:56,380 That's what the Bernoulli process is. 37 00:01:56,380 --> 00:01:58,290 So in some sense it's something that you have 38 00:01:58,290 --> 00:01:59,160 already seen. 39 00:01:59,160 --> 00:02:03,180 But we're going to introduce a few additional ideas here that 40 00:02:03,180 --> 00:02:07,460 will be useful and relevant as we go along and we move on to 41 00:02:07,460 --> 00:02:09,949 continuous time processes. 42 00:02:09,949 --> 00:02:12,640 So we're going to define the Bernoulli process, talk about 43 00:02:12,640 --> 00:02:17,800 some basic properties that the process has, and derive a few 44 00:02:17,800 --> 00:02:21,440 formulas, and exploit the special structure that it has 45 00:02:21,440 --> 00:02:24,930 to do a few quite interesting things. 46 00:02:24,930 --> 00:02:29,550 By the way, where does the word Bernoulli come from? 47 00:02:29,550 --> 00:02:33,010 Well the Bernoulli's were a family of mathematicians, 48 00:02:33,010 --> 00:02:37,200 Swiss mathematicians and scientists around the 1700s. 49 00:02:37,200 --> 00:02:39,680 There were so many of them that actually-- 50 00:02:39,680 --> 00:02:42,440 and some of them had the same first name-- 51 00:02:42,440 --> 00:02:46,920 historians even have difficulty of figuring out who 52 00:02:46,920 --> 00:02:48,660 exactly did what. 53 00:02:48,660 --> 00:02:51,430 But in any case, you can imagine that at the dinner 54 00:02:51,430 --> 00:02:53,710 table they were probably flipping coins and doing 55 00:02:53,710 --> 00:02:55,570 Bernoulli trials. 56 00:02:55,570 --> 00:02:58,290 So maybe that was their pass-time. 57 00:02:58,290 --> 00:02:58,610 OK. 58 00:02:58,610 --> 00:03:02,160 So what is the Bernoulli process? 59 00:03:02,160 --> 00:03:05,750 The Bernoulli process is nothing but a sequence of 60 00:03:05,750 --> 00:03:08,700 independent Bernoulli trials that you can 61 00:03:08,700 --> 00:03:11,120 think of as coin flips. 62 00:03:11,120 --> 00:03:13,530 So you can think the result of each trial 63 00:03:13,530 --> 00:03:15,350 being heads or tails. 64 00:03:15,350 --> 00:03:19,370 It's a little more convenient maybe to talk about successes 65 00:03:19,370 --> 00:03:21,690 and failures instead of heads or tails. 66 00:03:21,690 --> 00:03:24,470 Or if you wish numerical values, to use a 1 for a 67 00:03:24,470 --> 00:03:27,330 success and 0 for a failure. 68 00:03:27,330 --> 00:03:30,820 So the model is that each one of these trials has the same 69 00:03:30,820 --> 00:03:34,240 probability of success, p. 70 00:03:34,240 --> 00:03:37,050 And the other assumption is that these trials are 71 00:03:37,050 --> 00:03:40,880 statistically independent of each other. 72 00:03:40,880 --> 00:03:43,520 So what could be some examples of Bernoulli trials? 73 00:03:43,520 --> 00:03:48,010 You buy a lottery ticket every week and you win or lose. 74 00:03:48,010 --> 00:03:50,080 Presumably, these are independent of each other. 75 00:03:50,080 --> 00:03:53,240 And if it's the same kind of lottery, the probability of 76 00:03:53,240 --> 00:03:55,930 winning should be the same during every week. 77 00:03:55,930 --> 00:03:58,840 Maybe you want to model the financial markets. 78 00:03:58,840 --> 00:04:02,810 And a crude model could be that on any given day the Dow 79 00:04:02,810 --> 00:04:05,860 Jones is going to go up or down with a certain 80 00:04:05,860 --> 00:04:07,300 probability. 81 00:04:07,300 --> 00:04:11,770 Well that probability must be somewhere around 0.5, or so. 82 00:04:11,770 --> 00:04:14,890 This is a crude model of financial markets. 83 00:04:14,890 --> 00:04:17,910 You say, probably there is more into them. 84 00:04:17,910 --> 00:04:19,670 Life is not that simple. 85 00:04:19,670 --> 00:04:23,770 But actually it's a pretty reasonable model. 86 00:04:23,770 --> 00:04:26,260 It takes quite a bit of work to come up with more 87 00:04:26,260 --> 00:04:29,590 sophisticated models that can do better predictions than 88 00:04:29,590 --> 00:04:32,150 just pure heads and tails. 89 00:04:32,150 --> 00:04:36,270 Now more interesting, perhaps to the examples we will be 90 00:04:36,270 --> 00:04:37,975 dealing with in this class-- 91 00:04:37,975 --> 00:04:43,760 a Bernoulli process is a good model for streams of arrivals 92 00:04:43,760 --> 00:04:45,940 of any kind to a facility. 93 00:04:45,940 --> 00:04:49,790 So it could be a bank, and you are sitting at 94 00:04:49,790 --> 00:04:50,710 the door of the bank. 95 00:04:50,710 --> 00:04:54,180 And at every second, you check whether a customer came in 96 00:04:54,180 --> 00:04:56,210 during that second or not. 97 00:04:56,210 --> 00:05:00,670 Or you can think about arrivals of jobs to a server. 98 00:05:00,670 --> 00:05:04,890 Or any other kind of requests to a service system. 99 00:05:04,890 --> 00:05:08,560 So requests, or jobs, arrive at random times. 100 00:05:08,560 --> 00:05:12,360 You split the time into time slots. 101 00:05:12,360 --> 00:05:15,180 And during each time slot something comes or something 102 00:05:15,180 --> 00:05:16,610 does not come. 103 00:05:16,610 --> 00:05:20,110 And for many applications, it's a reasonable assumption 104 00:05:20,110 --> 00:05:24,510 to make that arrivals on any given slot are independent of 105 00:05:24,510 --> 00:05:27,530 arrivals in any other time slot. 106 00:05:27,530 --> 00:05:30,910 So each time slot can be viewed as a trial, where 107 00:05:30,910 --> 00:05:32,810 either something comes or doesn't come. 108 00:05:32,810 --> 00:05:35,830 And different trials are independent of each other. 109 00:05:35,830 --> 00:05:38,000 Now there's two assumptions that we're making here. 110 00:05:38,000 --> 00:05:40,000 One is the independence assumption. 111 00:05:40,000 --> 00:05:42,190 The other is that this number, p, probability 112 00:05:42,190 --> 00:05:44,520 of success, is constant. 113 00:05:44,520 --> 00:05:47,840 Now if you think about the bank example, if you stand 114 00:05:47,840 --> 00:05:53,330 outside the bank at 9:30 in the morning, you'll see 115 00:05:53,330 --> 00:05:55,830 arrivals happening at a certain rate. 116 00:05:55,830 --> 00:05:59,370 If you stand outside the bank at 12:00 noon, probably 117 00:05:59,370 --> 00:06:01,240 arrivals are more frequent. 118 00:06:01,240 --> 00:06:03,680 Which means that the given time slot has a higher 119 00:06:03,680 --> 00:06:08,030 probability of seeing an arrival around noon time. 120 00:06:08,030 --> 00:06:11,780 This means that the assumption of a constant p is probably 121 00:06:11,780 --> 00:06:15,080 not correct in that setting, if you're talking 122 00:06:15,080 --> 00:06:16,630 about the whole day. 123 00:06:16,630 --> 00:06:19,670 So the probability of successes or arrivals in the 124 00:06:19,670 --> 00:06:24,340 morning is going to be smaller than what it would be at noon. 125 00:06:24,340 --> 00:06:27,775 But if you're talking about a time period, let's say 10:00 126 00:06:27,775 --> 00:06:31,980 to 10:15, probably all slots have the same probability of 127 00:06:31,980 --> 00:06:34,880 seeing an arrival and it's a good approximation. 128 00:06:34,880 --> 00:06:37,450 So we're going to stick with the assumption that p is 129 00:06:37,450 --> 00:06:41,110 constant, doesn't change with time. 130 00:06:41,110 --> 00:06:44,480 Now that we have our model what do we do with it? 131 00:06:44,480 --> 00:06:47,200 Well, we start talking about the statistical properties 132 00:06:47,200 --> 00:06:48,510 that it has. 133 00:06:48,510 --> 00:06:52,400 And here there's two slightly different perspectives of 134 00:06:52,400 --> 00:06:55,680 thinking about what a random process is. 135 00:06:55,680 --> 00:06:59,060 The simplest version is to think about the random process 136 00:06:59,060 --> 00:07:01,605 as being just a sequence of random variables. 137 00:07:04,210 --> 00:07:06,900 We know what random variables are. 138 00:07:06,900 --> 00:07:09,440 We know what multiple random variables are. 139 00:07:09,440 --> 00:07:12,760 So it's just an experiment that has associated with it a 140 00:07:12,760 --> 00:07:14,730 bunch of random variables. 141 00:07:14,730 --> 00:07:17,410 So once you have random variables, what do you do 142 00:07:17,410 --> 00:07:18,480 instinctively? 143 00:07:18,480 --> 00:07:20,140 You talk about the distribution of 144 00:07:20,140 --> 00:07:21,610 these random variables. 145 00:07:21,610 --> 00:07:25,690 We already specified for the Bernoulli process that each Xi 146 00:07:25,690 --> 00:07:27,830 is a Bernoulli random variable, with probability of 147 00:07:27,830 --> 00:07:29,480 success equal to p. 148 00:07:29,480 --> 00:07:31,640 That specifies the distribution of the random 149 00:07:31,640 --> 00:07:35,500 variable X, or Xt, for general time t. 150 00:07:35,500 --> 00:07:37,780 Then you can calculate expected values and 151 00:07:37,780 --> 00:07:39,420 variances, and so on. 152 00:07:39,420 --> 00:07:44,050 So the expected value is, with probability p, you get a 1. 153 00:07:44,050 --> 00:07:46,650 And with probability 1 - p, you get a 0. 154 00:07:46,650 --> 00:07:49,510 So the expected value is equal to p. 155 00:07:49,510 --> 00:07:52,890 And then we have seen before a formula for the variance of 156 00:07:52,890 --> 00:07:57,040 the Bernoulli random variable, which is p times 1-p. 157 00:07:57,040 --> 00:08:00,350 So this way we basically now have all the statistical 158 00:08:00,350 --> 00:08:04,680 properties of the random variable Xt, and we have those 159 00:08:04,680 --> 00:08:06,670 properties for every t. 160 00:08:06,670 --> 00:08:10,140 Is this enough of a probabilistic description of a 161 00:08:10,140 --> 00:08:11,580 random process? 162 00:08:11,580 --> 00:08:12,240 Well, no. 163 00:08:12,240 --> 00:08:15,070 You need to know how the different random variables 164 00:08:15,070 --> 00:08:16,820 relate to each other. 165 00:08:16,820 --> 00:08:20,870 If you're talking about a general random process, you 166 00:08:20,870 --> 00:08:23,490 would like to know things. 167 00:08:23,490 --> 00:08:26,370 For example, the joint distribution of X2, 168 00:08:26,370 --> 00:08:29,220 with X5, and X7. 169 00:08:29,220 --> 00:08:31,680 For example, that might be something that you're 170 00:08:31,680 --> 00:08:32,809 interested in. 171 00:08:32,809 --> 00:08:38,820 And the way you specify it is by giving the joint PMF of 172 00:08:38,820 --> 00:08:40,530 these random variables. 173 00:08:40,530 --> 00:08:44,440 And you have to do that for every collection, or any 174 00:08:44,440 --> 00:08:46,150 subset, of the random variables you 175 00:08:46,150 --> 00:08:47,300 are interested in. 176 00:08:47,300 --> 00:08:49,860 So to have a complete description of a random 177 00:08:49,860 --> 00:08:54,770 processes, you need to specify for me all the possible joint 178 00:08:54,770 --> 00:08:55,960 distributions. 179 00:08:55,960 --> 00:08:58,780 And once you have all the possible joint distributions, 180 00:08:58,780 --> 00:09:01,580 then you can answer, in principle, any questions you 181 00:09:01,580 --> 00:09:03,280 might be interested in. 182 00:09:03,280 --> 00:09:05,670 How did we get around this issue for 183 00:09:05,670 --> 00:09:06,690 the Bernoulli process? 184 00:09:06,690 --> 00:09:10,030 I didn't give you the joint distributions explicitly. 185 00:09:10,030 --> 00:09:12,160 But I gave them to you implicitly. 186 00:09:12,160 --> 00:09:15,250 And this is because I told you that the different random 187 00:09:15,250 --> 00:09:18,390 variables are independent of each other. 188 00:09:18,390 --> 00:09:21,490 So at least for the Bernoulli process, where we make the 189 00:09:21,490 --> 00:09:24,320 independence assumption, we know that this is going to be 190 00:09:24,320 --> 00:09:25,970 the product of the PMFs. 191 00:09:29,000 --> 00:09:33,850 And since I have told you what the individual PMFs are, this 192 00:09:33,850 --> 00:09:37,090 means that you automatically know all the joint PMFs. 193 00:09:37,090 --> 00:09:40,940 And we can go to business based on that. 194 00:09:40,940 --> 00:09:41,310 All right. 195 00:09:41,310 --> 00:09:45,160 So this is one view of what a random process is, just a 196 00:09:45,160 --> 00:09:47,180 collection of random variables. 197 00:09:47,180 --> 00:09:50,660 There's another view that's a little more abstract, which is 198 00:09:50,660 --> 00:09:53,170 the following. 199 00:09:53,170 --> 00:09:57,100 The entire process is to be thought of as one long 200 00:09:57,100 --> 00:09:58,520 experiment. 201 00:09:58,520 --> 00:10:01,970 So we go back to the chapter one view of 202 00:10:01,970 --> 00:10:03,550 probabilistic models. 203 00:10:03,550 --> 00:10:06,240 So there must be a sample space involved. 204 00:10:06,240 --> 00:10:07,660 What is the sample space? 205 00:10:07,660 --> 00:10:11,760 If I do my infinite, long experiment of flipping an 206 00:10:11,760 --> 00:10:15,330 infinite number of coins, a typical outcome of the 207 00:10:15,330 --> 00:10:21,140 experiment would be a sequence of 0's and 1's. 208 00:10:21,140 --> 00:10:25,640 So this could be one possible outcome of the experiment, 209 00:10:25,640 --> 00:10:28,630 just an infinite sequence of 0's and 1's. 210 00:10:28,630 --> 00:10:33,020 My sample space is the set of all possible 211 00:10:33,020 --> 00:10:35,050 outcomes of this kind. 212 00:10:35,050 --> 00:10:40,660 Here's another possible outcome, and so on. 213 00:10:40,660 --> 00:10:44,060 And essentially we're dealing with a sample space, which is 214 00:10:44,060 --> 00:10:46,980 the space of all sequences of 0's and 1's. 215 00:10:46,980 --> 00:10:50,470 And we're making some sort of probabilistic assumption about 216 00:10:50,470 --> 00:10:53,330 what may happen in that experiment. 217 00:10:53,330 --> 00:10:56,350 So one particular sequence that we may be interested in 218 00:10:56,350 --> 00:10:59,660 is the sequence of obtaining all 1's. 219 00:10:59,660 --> 00:11:05,510 So this is the sequence that gives you 1's forever. 220 00:11:05,510 --> 00:11:08,330 Once you take the point of view that this is our sample 221 00:11:08,330 --> 00:11:10,760 space-- its the space of all infinite sequences-- 222 00:11:10,760 --> 00:11:13,770 you can start asking questions that have to do 223 00:11:13,770 --> 00:11:15,470 with infinite sequences. 224 00:11:15,470 --> 00:11:19,120 Such as the question, what's the probability of obtaining 225 00:11:19,120 --> 00:11:23,180 the infinite sequence that consists of all 1's? 226 00:11:23,180 --> 00:11:24,690 So what is this probability? 227 00:11:24,690 --> 00:11:27,240 Let's see how we could calculate it. 228 00:11:27,240 --> 00:11:34,000 So the probability of obtaining all 1's is certainly 229 00:11:34,000 --> 00:11:39,890 less than or equal to the probability of obtaining 1's, 230 00:11:39,890 --> 00:11:42,335 just in the first 10 tosses. 231 00:11:45,075 --> 00:11:47,030 OK. 232 00:11:47,030 --> 00:11:50,810 This is asking for more things to happen than this. 233 00:11:50,810 --> 00:11:55,780 If this event is true, then this is also true. 234 00:11:55,780 --> 00:11:58,660 Therefore the probability of this is smaller than the 235 00:11:58,660 --> 00:11:59,540 probability of that. 236 00:11:59,540 --> 00:12:03,410 This event is contained in that event. 237 00:12:03,410 --> 00:12:04,950 This implies this. 238 00:12:04,950 --> 00:12:06,880 So we have this inequality. 239 00:12:06,880 --> 00:12:12,360 Now what's the probability of obtaining 1's in 10 trials? 240 00:12:12,360 --> 00:12:15,980 This is just p to the 10th because the trials are 241 00:12:15,980 --> 00:12:18,530 independent. 242 00:12:18,530 --> 00:12:22,780 Now of course there's no reason why I chose 10 here. 243 00:12:22,780 --> 00:12:26,160 The same argument goes through if I use an 244 00:12:26,160 --> 00:12:29,850 arbitrary number, k. 245 00:12:29,850 --> 00:12:34,250 And this has to be true for all k. 246 00:12:34,250 --> 00:12:38,690 So this probability is less than p to the k, no matter 247 00:12:38,690 --> 00:12:41,670 what k I choose. 248 00:12:41,670 --> 00:12:46,350 Therefore, this must be less than or equal to the limit of 249 00:12:46,350 --> 00:12:48,660 this, as k goes to infinity. 250 00:12:48,660 --> 00:12:51,210 This is smaller than that for all k's. 251 00:12:51,210 --> 00:12:55,860 Let k go to infinity, take k arbitrarily large, this number 252 00:12:55,860 --> 00:12:57,770 is going to become arbitrarily small. 253 00:12:57,770 --> 00:12:59,190 It goes to 0. 254 00:12:59,190 --> 00:13:02,480 And that proves that the probability of an infinite 255 00:13:02,480 --> 00:13:06,080 sequence of 1's is equal to 0. 256 00:13:06,080 --> 00:13:09,800 So take limits of both sides. 257 00:13:13,220 --> 00:13:16,217 It's going to be less than or equal to the limit-- 258 00:13:16,217 --> 00:13:18,380 I shouldn't take a limit here. 259 00:13:18,380 --> 00:13:21,745 The probability is less than or equal to the limit of p to 260 00:13:21,745 --> 00:13:26,000 the k, as k goes to infinity, which is 0. 261 00:13:26,000 --> 00:13:30,880 So this proves in a formal way that the sequence of all 1's 262 00:13:30,880 --> 00:13:32,650 has 0 probability. 263 00:13:32,650 --> 00:13:35,770 If you have an infinite number of coin flips, what's the 264 00:13:35,770 --> 00:13:40,610 probability that all of the coin flips result in heads? 265 00:13:40,610 --> 00:13:43,690 The probability of this happening is equal to zero. 266 00:13:43,690 --> 00:13:48,310 So this particular sequence has 0 probability. 267 00:13:48,310 --> 00:13:51,480 Of course, I'm assuming here that p is less than 1, 268 00:13:51,480 --> 00:13:53,420 strictly less than 1. 269 00:13:53,420 --> 00:13:56,280 Now the interesting thing is that if you look at any other 270 00:13:56,280 --> 00:13:59,690 infinite sequence, and you try to calculate the probability 271 00:13:59,690 --> 00:14:03,526 of that infinite sequence, you would get a product of (1-p) 272 00:14:03,526 --> 00:14:07,600 times 1, 1-p times 1, 1-p, times p times p, 273 00:14:07,600 --> 00:14:09,570 times 1-p and so on. 274 00:14:09,570 --> 00:14:13,560 You keep multiplying numbers that are less than 1. 275 00:14:13,560 --> 00:14:16,760 Again, I'm making the assumption that p is 276 00:14:16,760 --> 00:14:17,940 between 0 and 1. 277 00:14:17,940 --> 00:14:21,140 So 1-p is less than 1, p is less than 1. 278 00:14:21,140 --> 00:14:23,600 You keep multiplying numbers less than 1. 279 00:14:23,600 --> 00:14:26,190 If you multiply infinitely many such numbers, the 280 00:14:26,190 --> 00:14:28,470 infinite product becomes 0. 281 00:14:28,470 --> 00:14:33,310 So any individual sequence in this sample space actually has 282 00:14:33,310 --> 00:14:35,230 0 probability. 283 00:14:35,230 --> 00:14:39,560 And that is a little bit counter-intuitive perhaps. 284 00:14:39,560 --> 00:14:42,670 But the situation is more like the situation where we deal 285 00:14:42,670 --> 00:14:44,680 with continuous random variables. 286 00:14:44,680 --> 00:14:47,430 So if you could draw a continuous random variable, 287 00:14:47,430 --> 00:14:50,810 every possible outcome has 0 probability. 288 00:14:50,810 --> 00:14:52,320 And that's fine. 289 00:14:52,320 --> 00:14:54,860 But all of the outcomes collectively still have 290 00:14:54,860 --> 00:14:56,280 positive probability. 291 00:14:56,280 --> 00:14:59,580 So the situation here is very much similar. 292 00:14:59,580 --> 00:15:03,940 So the space of infinite sequences of 0's and 1's, that 293 00:15:03,940 --> 00:15:07,340 sample space is very much like a continuous space. 294 00:15:07,340 --> 00:15:10,340 If you want to push that analogy further, you could 295 00:15:10,340 --> 00:15:15,830 think of this as the expansion of a real number. 296 00:15:15,830 --> 00:15:18,950 Or the representation of a real number in binary. 297 00:15:18,950 --> 00:15:22,540 Take a real number, write it down in binary, you are going 298 00:15:22,540 --> 00:15:25,580 to get an infinite sequence of 0's and 1's. 299 00:15:25,580 --> 00:15:28,780 So you can think of each possible outcome here 300 00:15:28,780 --> 00:15:30,920 essentially as a real number. 301 00:15:30,920 --> 00:15:36,060 So the experiment of doing an infinite number of coin flips 302 00:15:36,060 --> 00:15:39,670 is sort of similar to the experiment of picking a real 303 00:15:39,670 --> 00:15:41,060 number at random. 304 00:15:41,060 --> 00:15:44,990 When you pick real numbers at random, any particular real 305 00:15:44,990 --> 00:15:46,500 number has 0 probability. 306 00:15:46,500 --> 00:15:49,780 So similarly here, any particular infinite sequence 307 00:15:49,780 --> 00:15:52,440 has 0 probability. 308 00:15:52,440 --> 00:15:55,260 So if we were to push that analogy further, there would 309 00:15:55,260 --> 00:15:57,290 be a few interesting things we could do. 310 00:15:57,290 --> 00:15:59,880 But we will not push it further. 311 00:15:59,880 --> 00:16:05,400 This is just to give you an indication that things can get 312 00:16:05,400 --> 00:16:08,710 pretty subtle and interesting once you start talking about 313 00:16:08,710 --> 00:16:12,640 random processes that involve forever, over the infinite 314 00:16:12,640 --> 00:16:13,900 time horizon. 315 00:16:13,900 --> 00:16:17,960 So things get interesting even in this context of the simple 316 00:16:17,960 --> 00:16:19,750 Bernoulli process. 317 00:16:19,750 --> 00:16:23,130 Just to give you a preview of what's coming further, today 318 00:16:23,130 --> 00:16:26,170 we're going to talk just about the Bernoulli process. 319 00:16:26,170 --> 00:16:30,810 And you should make sure before the next lecture-- 320 00:16:30,810 --> 00:16:34,590 I guess between the exam and the next lecture-- 321 00:16:34,590 --> 00:16:36,740 to understand everything we do today. 322 00:16:36,740 --> 00:16:39,930 Because next time we're going to do everything once more, 323 00:16:39,930 --> 00:16:41,640 but in continuous time. 324 00:16:41,640 --> 00:16:46,360 And in continuous time, things become more subtle and a 325 00:16:46,360 --> 00:16:47,700 little more difficult. 326 00:16:47,700 --> 00:16:50,580 But we are going to build on what we understand for the 327 00:16:50,580 --> 00:16:52,030 discrete time case. 328 00:16:52,030 --> 00:16:55,370 Now both the Bernoulli process and its continuous time analog 329 00:16:55,370 --> 00:16:58,470 has a property that we call memorylessness, whatever 330 00:16:58,470 --> 00:17:01,590 happened in the past does not affect the future. 331 00:17:01,590 --> 00:17:03,920 Later on in this class we're going to talk about more 332 00:17:03,920 --> 00:17:07,310 general random processes, so-called Markov chains, in 333 00:17:07,310 --> 00:17:10,890 which there are certain dependences across time. 334 00:17:10,890 --> 00:17:15,349 That is, what has happened in the past will have some 335 00:17:15,349 --> 00:17:18,390 bearing on what may happen in the future. 336 00:17:18,390 --> 00:17:22,440 So it's like having coin flips where the outcome of the next 337 00:17:22,440 --> 00:17:25,720 coin flip has some dependence on the previous coin flip. 338 00:17:25,720 --> 00:17:28,400 And that gives us a richer class of models. 339 00:17:28,400 --> 00:17:31,670 And once we get there, essentially we will have 340 00:17:31,670 --> 00:17:34,400 covered all possible models. 341 00:17:34,400 --> 00:17:38,070 So for random processes that are practically useful and 342 00:17:38,070 --> 00:17:41,480 which you can manipulate, Markov chains are a pretty 343 00:17:41,480 --> 00:17:43,250 general class of models. 344 00:17:43,250 --> 00:17:47,260 And almost any real world phenomenon that evolves in 345 00:17:47,260 --> 00:17:52,190 time can be approximately modeled using Markov chains. 346 00:17:52,190 --> 00:17:55,580 So even though this is a first class in probability, we will 347 00:17:55,580 --> 00:17:59,560 get pretty far in that direction. 348 00:17:59,560 --> 00:17:59,950 All right. 349 00:17:59,950 --> 00:18:04,300 So now let's start doing a few calculations and answer some 350 00:18:04,300 --> 00:18:06,690 questions about the Bernoulli process. 351 00:18:06,690 --> 00:18:11,010 So again, the best way to think in terms of models that 352 00:18:11,010 --> 00:18:13,350 correspond to the Bernoulli process is in terms of 353 00:18:13,350 --> 00:18:15,870 arrivals of jobs to a facility. 354 00:18:15,870 --> 00:18:18,230 And there's two types of questions that you can ask. 355 00:18:18,230 --> 00:18:21,990 In a given amount of time, how many jobs arrived? 356 00:18:21,990 --> 00:18:26,250 Or conversely, for a given number of jobs, how much time 357 00:18:26,250 --> 00:18:28,720 did it take for them to arrive? 358 00:18:28,720 --> 00:18:31,420 So we're going to deal with these two questions, starting 359 00:18:31,420 --> 00:18:32,450 with the first. 360 00:18:32,450 --> 00:18:34,370 For a given amount of time-- 361 00:18:34,370 --> 00:18:37,870 that is, for a given number of time periods-- 362 00:18:37,870 --> 00:18:40,150 how many arrivals have we had? 363 00:18:40,150 --> 00:18:44,180 How many of those Xi's happen to be 1's? 364 00:18:44,180 --> 00:18:46,270 We fix the number of time slots-- 365 00:18:46,270 --> 00:18:47,970 let's say n time slots-- 366 00:18:47,970 --> 00:18:50,920 and you measure the number of successes. 367 00:18:50,920 --> 00:18:54,320 Well this is a very familiar random variable. 368 00:18:54,320 --> 00:18:58,990 The number of successes in n independent coin flips-- 369 00:18:58,990 --> 00:19:01,430 or in n independent trials-- 370 00:19:01,430 --> 00:19:03,660 is a binomial random variable. 371 00:19:03,660 --> 00:19:10,380 So we know its distribution is given by the binomial PMF, and 372 00:19:10,380 --> 00:19:15,820 it's just this, for k going from 0 up to n. 373 00:19:15,820 --> 00:19:18,940 And we know everything by now about this random variable. 374 00:19:18,940 --> 00:19:21,990 We know its expected value is n times p. 375 00:19:21,990 --> 00:19:27,980 And we know the variance, which is n times p, times 1-p. 376 00:19:27,980 --> 00:19:31,740 So there's nothing new here. 377 00:19:31,740 --> 00:19:34,290 That's the easy part. 378 00:19:34,290 --> 00:19:37,590 So now let's look at the opposite kind of question. 379 00:19:37,590 --> 00:19:42,210 Instead of fixing the time and asking how many arrivals, now 380 00:19:42,210 --> 00:19:46,160 let us fix the number of arrivals and ask how much time 381 00:19:46,160 --> 00:19:47,810 did it take. 382 00:19:47,810 --> 00:19:52,780 And let's start with the time until the first arrival. 383 00:19:52,780 --> 00:19:59,470 So the process starts. 384 00:19:59,470 --> 00:20:00,720 We got our slots. 385 00:20:04,070 --> 00:20:08,250 And we see, perhaps, a sequence of 0's and then at 386 00:20:08,250 --> 00:20:10,660 some point we get a 1. 387 00:20:10,660 --> 00:20:14,810 The number of trials it took until we get a 1, we're going 388 00:20:14,810 --> 00:20:16,500 to call it T1. 389 00:20:16,500 --> 00:20:19,020 And it's the time of the first arrival. 390 00:20:23,020 --> 00:20:23,680 OK. 391 00:20:23,680 --> 00:20:27,130 What is the probability distribution of T1? 392 00:20:27,130 --> 00:20:30,430 What kind of random variable is it? 393 00:20:30,430 --> 00:20:31,835 We've gone through this before. 394 00:20:34,360 --> 00:20:40,560 The event that the first arrival happens at time little 395 00:20:40,560 --> 00:20:48,400 t is the event that the first t-1 trials were failures, and 396 00:20:48,400 --> 00:20:52,850 the trial number t happens to be a success. 397 00:20:52,850 --> 00:20:57,960 So for the first success to happen at time slot number 5, 398 00:20:57,960 --> 00:21:02,390 it means that the first 4 slots had failures and the 5th 399 00:21:02,390 --> 00:21:04,800 slot had a success. 400 00:21:04,800 --> 00:21:08,440 So the probability of this happening is the probability 401 00:21:08,440 --> 00:21:13,580 of having failures in the first t -1 trials, and having 402 00:21:13,580 --> 00:21:16,300 a success at trial number 1. 403 00:21:16,300 --> 00:21:20,060 And this is the formula for t equal 1,2, and so on. 404 00:21:20,060 --> 00:21:22,460 So we know what this distribution is. 405 00:21:22,460 --> 00:21:25,200 It's the so-called geometric distribution. 406 00:21:30,900 --> 00:21:35,010 Let me jump this through this for a minute. 407 00:21:35,010 --> 00:21:38,360 In the past, we did calculate the expected value of the 408 00:21:38,360 --> 00:21:42,240 geometric distribution, and it's 1/p. 409 00:21:42,240 --> 00:21:46,010 Which means that if p is small, you expect to take a 410 00:21:46,010 --> 00:21:48,810 long time until the first success. 411 00:21:48,810 --> 00:21:52,540 And then there's a formula also for the variance of T1, 412 00:21:52,540 --> 00:21:56,410 which we never formally derived in class, but it was 413 00:21:56,410 --> 00:22:01,600 in your textbook and it just happens to be this. 414 00:22:01,600 --> 00:22:02,310 All right. 415 00:22:02,310 --> 00:22:05,920 So nothing new until this point. 416 00:22:05,920 --> 00:22:08,700 Now, let's talk about this property, the 417 00:22:08,700 --> 00:22:10,210 memorylessness property. 418 00:22:10,210 --> 00:22:13,240 We kind of touched on this property when we discussed-- 419 00:22:13,240 --> 00:22:15,760 when we did the derivation in class of the 420 00:22:15,760 --> 00:22:18,010 expected value of T1. 421 00:22:18,010 --> 00:22:20,040 Now what is the memoryless property? 422 00:22:20,040 --> 00:22:22,980 It's essentially a consequence of independence. 423 00:22:22,980 --> 00:22:26,740 If I tell you the results of my coin flips up to a certain 424 00:22:26,740 --> 00:22:30,550 time, this, because of independence, doesn't give you 425 00:22:30,550 --> 00:22:34,410 any information about the coin flips after that time. 426 00:22:37,180 --> 00:22:41,180 So knowing that we had lots of 0's here does not change what 427 00:22:41,180 --> 00:22:44,920 I believe about the future coin flips, because the future 428 00:22:44,920 --> 00:22:47,580 coin flips are going to be just independent coin flips 429 00:22:47,580 --> 00:22:53,170 with a given probability, p, for obtaining tails. 430 00:22:53,170 --> 00:22:58,240 So this is a statement that I made about a specific time. 431 00:22:58,240 --> 00:23:02,270 That is, you do coin flips until 12 o'clock. 432 00:23:02,270 --> 00:23:05,210 And then at 12 o'clock, you start watching. 433 00:23:05,210 --> 00:23:09,900 No matter what happens before 12 o'clock, after 12:00, what 434 00:23:09,900 --> 00:23:12,890 you're going to see is just a sequence of independent 435 00:23:12,890 --> 00:23:15,610 Bernoulli trials with the same probability, p. 436 00:23:15,610 --> 00:23:18,450 Whatever happened in the past is irrelevant. 437 00:23:18,450 --> 00:23:21,590 Now instead of talking about the fixed time at which you 438 00:23:21,590 --> 00:23:26,940 start watching, let's think about a situation where your 439 00:23:26,940 --> 00:23:31,240 sister sits in the next room, flips the coins until she 440 00:23:31,240 --> 00:23:35,760 observes the first success, and then calls you inside. 441 00:23:35,760 --> 00:23:38,700 And you start watching after this time. 442 00:23:38,700 --> 00:23:40,970 What are you're going to see? 443 00:23:40,970 --> 00:23:45,160 Well, you're going to see a coin flip with probability p 444 00:23:45,160 --> 00:23:46,850 of success. 445 00:23:46,850 --> 00:23:49,870 You're going to see another trial that has probability p 446 00:23:49,870 --> 00:23:53,250 as a success, and these are all independent of each other. 447 00:23:53,250 --> 00:23:56,800 So what you're going to see starting at that time is going 448 00:23:56,800 --> 00:24:02,610 to be just a sequence of independent Bernoulli trials, 449 00:24:02,610 --> 00:24:06,190 as if the process was starting at this time. 450 00:24:06,190 --> 00:24:10,880 How long it took for the first success to occur doesn't have 451 00:24:10,880 --> 00:24:15,850 any bearing on what is going to happen afterwards. 452 00:24:15,850 --> 00:24:19,170 What happens afterwards is still a sequence of 453 00:24:19,170 --> 00:24:21,230 independent coin flips. 454 00:24:21,230 --> 00:24:24,680 And this story is actually even more general. 455 00:24:24,680 --> 00:24:28,980 So your sister watches the coin flips and at some point 456 00:24:28,980 --> 00:24:31,690 tells you, oh, something really interesting is 457 00:24:31,690 --> 00:24:32,440 happening here. 458 00:24:32,440 --> 00:24:35,250 I got this string of a hundred 1's in a row. 459 00:24:35,250 --> 00:24:37,250 Come and watch. 460 00:24:37,250 --> 00:24:40,260 Now when you go in there and you start watching, do you 461 00:24:40,260 --> 00:24:43,890 expect to see something unusual? 462 00:24:43,890 --> 00:24:46,830 There were unusual things that happened before 463 00:24:46,830 --> 00:24:48,180 you were called in. 464 00:24:48,180 --> 00:24:50,620 Does this means that you're going to see unusual things 465 00:24:50,620 --> 00:24:51,780 afterwards? 466 00:24:51,780 --> 00:24:52,180 No. 467 00:24:52,180 --> 00:24:55,060 Afterwards, what you're going to see is, again, just a 468 00:24:55,060 --> 00:24:57,700 sequence of independent coin flips. 469 00:24:57,700 --> 00:25:00,640 The fact that some strange things happened before doesn't 470 00:25:00,640 --> 00:25:03,940 have any bearing as to what is going to happen in the future. 471 00:25:03,940 --> 00:25:08,560 So if the roulettes in the casino are properly made, the 472 00:25:08,560 --> 00:25:12,610 fact that there were 3 reds in a row doesn't affect the odds 473 00:25:12,610 --> 00:25:16,430 of whether in the next roll it's going to 474 00:25:16,430 --> 00:25:19,570 be a red or a black. 475 00:25:19,570 --> 00:25:22,850 So whatever happens in the past-- no matter 476 00:25:22,850 --> 00:25:25,010 how unusual it is-- 477 00:25:25,010 --> 00:25:28,910 at the time when you're called in, what's going to happen in 478 00:25:28,910 --> 00:25:32,510 the future is going to be just independent Bernoulli trials, 479 00:25:32,510 --> 00:25:34,170 with the same probability, p. 480 00:25:36,730 --> 00:25:41,900 The only case where this story changes is if your sister has 481 00:25:41,900 --> 00:25:43,850 a little bit of foresight. 482 00:25:43,850 --> 00:25:48,500 So your sister can look ahead into the future and knows that 483 00:25:48,500 --> 00:25:54,230 the next 10 coin flips will be heads, and calls you before 484 00:25:54,230 --> 00:25:56,430 those 10 flips will happen. 485 00:25:56,430 --> 00:25:59,660 If she calls you in, then what are you going to see? 486 00:25:59,660 --> 00:26:02,310 You're not going to see independent Bernoulli trials, 487 00:26:02,310 --> 00:26:05,510 since she has psychic powers and she knows that the next 488 00:26:05,510 --> 00:26:06,910 ones would be 1's. 489 00:26:06,910 --> 00:26:12,770 She called you in and you will see a sequence of 1's. 490 00:26:12,770 --> 00:26:15,470 So it's no more independent Bernoulli trials. 491 00:26:15,470 --> 00:26:19,420 So what's the subtle difference here? 492 00:26:19,420 --> 00:26:24,010 The future is independent from the past, provided that the 493 00:26:24,010 --> 00:26:28,310 time that you are called and asked to start watching is 494 00:26:28,310 --> 00:26:31,460 determined by someone who doesn't have any foresight, 495 00:26:31,460 --> 00:26:33,470 who cannot see the future. 496 00:26:33,470 --> 00:26:36,960 If you are called in, just on the basis of what has happened 497 00:26:36,960 --> 00:26:39,630 so far, then you don't have any 498 00:26:39,630 --> 00:26:41,430 information about the future. 499 00:26:41,430 --> 00:26:44,870 And one special case is the picture here. 500 00:26:44,870 --> 00:26:47,240 You have your coin flips. 501 00:26:47,240 --> 00:26:51,060 Once you see a one that happens, once you see a 502 00:26:51,060 --> 00:26:53,310 success, you are called in. 503 00:26:53,310 --> 00:26:57,690 You are called in on the basis of what happened in the past, 504 00:26:57,690 --> 00:26:59,070 but without any foresight. 505 00:27:02,208 --> 00:27:03,140 OK. 506 00:27:03,140 --> 00:27:07,020 And this subtle distinction is what's going to make our next 507 00:27:07,020 --> 00:27:10,420 example interesting and subtle. 508 00:27:10,420 --> 00:27:13,380 So here's the question. 509 00:27:13,380 --> 00:27:17,790 You buy a lottery ticket every day, so we have a Bernoulli 510 00:27:17,790 --> 00:27:21,880 process that's running in time. 511 00:27:21,880 --> 00:27:26,050 And you're interested in the length of the first string of 512 00:27:26,050 --> 00:27:26,860 losing days. 513 00:27:26,860 --> 00:27:28,410 What does that mean? 514 00:27:28,410 --> 00:27:33,700 So suppose that a typical sequence of events 515 00:27:33,700 --> 00:27:35,040 could be this one. 516 00:27:40,970 --> 00:27:43,470 So what are we discussing here? 517 00:27:43,470 --> 00:27:47,180 We're looking at the first string of losing days, where 518 00:27:47,180 --> 00:27:49,140 losing days means 0's. 519 00:27:51,940 --> 00:27:56,910 So the string of losing days is this string here. 520 00:27:56,910 --> 00:28:02,030 Let's call the length of that string, L. We're interested in 521 00:28:02,030 --> 00:28:06,520 the random variable, which is the length of this interval. 522 00:28:06,520 --> 00:28:08,670 What kind of random variable is it? 523 00:28:11,230 --> 00:28:11,600 OK. 524 00:28:11,600 --> 00:28:16,190 Here's one possible way you might think about the problem. 525 00:28:16,190 --> 00:28:16,550 OK. 526 00:28:16,550 --> 00:28:24,140 Starting from this time, and looking until this time here, 527 00:28:24,140 --> 00:28:27,420 what are we looking at? 528 00:28:27,420 --> 00:28:31,640 We're looking at the time, starting from here, until the 529 00:28:31,640 --> 00:28:35,040 first success. 530 00:28:35,040 --> 00:28:40,530 So the past doesn't matter. 531 00:28:40,530 --> 00:28:43,550 Starting from here we have coin flips 532 00:28:43,550 --> 00:28:45,870 until the first success. 533 00:28:45,870 --> 00:28:48,160 The time until the first success 534 00:28:48,160 --> 00:28:50,280 in a Bernoulli process-- 535 00:28:50,280 --> 00:28:54,700 we just discussed that it's a geometric random variable. 536 00:28:54,700 --> 00:28:58,090 So your first conjecture would be that this random variable 537 00:28:58,090 --> 00:29:02,960 here, which is 1 longer than the one we are interested in, 538 00:29:02,960 --> 00:29:06,310 that perhaps is a geometric random variable. 539 00:29:14,040 --> 00:29:18,460 And if this were so, then you could say that the random 540 00:29:18,460 --> 00:29:23,300 variable, L, is a geometric, minus 1. 541 00:29:23,300 --> 00:29:26,160 Can that be the correct answer? 542 00:29:26,160 --> 00:29:29,590 A geometric random variable, what values does it take? 543 00:29:29,590 --> 00:29:33,160 It takes values 1, 2, 3, and so on. 544 00:29:33,160 --> 00:29:37,576 1 minus a geometric would take values from 0, 545 00:29:37,576 --> 00:29:40,940 1, 2, and so on. 546 00:29:40,940 --> 00:29:45,200 Can the random variable L be 0? 547 00:29:45,200 --> 00:29:45,920 No. 548 00:29:45,920 --> 00:29:48,500 The random variable L is the length of a 549 00:29:48,500 --> 00:29:50,390 string of losing days. 550 00:29:50,390 --> 00:29:56,190 So the shortest that L could be, would be just 1. 551 00:29:56,190 --> 00:29:59,770 If you get just one losing day and then you start winning, L 552 00:29:59,770 --> 00:30:01,520 would be equal to 1. 553 00:30:01,520 --> 00:30:05,338 So L cannot be 0 by definition, which means that L 554 00:30:05,338 --> 00:30:09,820 + 1 cannot be 1, by definition. 555 00:30:09,820 --> 00:30:14,310 But if L +1 were geometric, it could be equal to 1. 556 00:30:14,310 --> 00:30:16,226 Therefore this random variable, L 557 00:30:16,226 --> 00:30:18,830 + 1, is not a geometric. 558 00:30:23,550 --> 00:30:23,920 OK. 559 00:30:23,920 --> 00:30:26,720 Why is it not geometric? 560 00:30:26,720 --> 00:30:29,100 I started watching at this time. 561 00:30:29,100 --> 00:30:33,810 From this time until the first success, that should be a 562 00:30:33,810 --> 00:30:35,690 geometric random variable. 563 00:30:35,690 --> 00:30:38,180 Where's the catch? 564 00:30:38,180 --> 00:30:42,670 If I'm asked to start watching at this time, it's because my 565 00:30:42,670 --> 00:30:48,260 sister knows that the next one was a failure. 566 00:30:48,260 --> 00:30:52,360 This is the time where the string of failures starts. 567 00:30:52,360 --> 00:30:56,050 In order to know that they should start watching here, 568 00:30:56,050 --> 00:30:58,770 it's the same as if I'm told that the 569 00:30:58,770 --> 00:31:01,240 next one is a failure. 570 00:31:01,240 --> 00:31:05,180 So to be asked to start watching at this time requires 571 00:31:05,180 --> 00:31:08,050 that someone looked in the future. 572 00:31:08,050 --> 00:31:13,210 And in that case, it's no longer true that these will be 573 00:31:13,210 --> 00:31:14,850 independent Bernoulli trials. 574 00:31:14,850 --> 00:31:16,000 In fact, they're not. 575 00:31:16,000 --> 00:31:18,990 If you start watching here, you're certain that the next 576 00:31:18,990 --> 00:31:20,210 one is a failure. 577 00:31:20,210 --> 00:31:23,510 The next one is not an independent Bernoulli trial. 578 00:31:23,510 --> 00:31:26,860 That's why the argument that would claim that this L + 1 is 579 00:31:26,860 --> 00:31:30,400 geometric would be incorrect. 580 00:31:30,400 --> 00:31:33,680 So if this is not the correct answer, which 581 00:31:33,680 --> 00:31:35,150 is the correct answer? 582 00:31:35,150 --> 00:31:37,700 The correct answer goes as follows. 583 00:31:37,700 --> 00:31:39,400 Your sister is watching. 584 00:31:39,400 --> 00:31:44,080 Your sister sees the first failure, and then tells you, 585 00:31:44,080 --> 00:31:45,530 OK, the failures-- 586 00:31:45,530 --> 00:31:46,670 or losing days-- 587 00:31:46,670 --> 00:31:47,790 have started. 588 00:31:47,790 --> 00:31:49,260 Come in and watch. 589 00:31:49,260 --> 00:31:51,550 So you start to watching at this time. 590 00:31:51,550 --> 00:31:56,060 And you start watching until the first success comes. 591 00:31:56,060 --> 00:31:59,290 This will be a geometric random variable. 592 00:31:59,290 --> 00:32:05,005 So from here to here, this will be geometric. 593 00:32:09,430 --> 00:32:11,560 So things happen. 594 00:32:11,560 --> 00:32:14,270 You are asked to start watching. 595 00:32:14,270 --> 00:32:18,660 After you start watching, the future is just a sequence of 596 00:32:18,660 --> 00:32:20,540 independent Bernoulli trials. 597 00:32:20,540 --> 00:32:23,930 And the time until the first failure occurs, this is going 598 00:32:23,930 --> 00:32:27,470 to be a geometric random variable with parameter p. 599 00:32:27,470 --> 00:32:31,830 And then you notice that the interval of interest is 600 00:32:31,830 --> 00:32:35,030 exactly the same as the length of this interval. 601 00:32:35,030 --> 00:32:37,550 This starts one time step later, and ends 602 00:32:37,550 --> 00:32:39,260 one time step later. 603 00:32:39,260 --> 00:32:43,370 So conclusion is that L is actually geometric, with 604 00:32:43,370 --> 00:32:44,620 parameter p. 605 00:33:33,090 --> 00:33:36,160 OK, it looks like I'm missing one slide. 606 00:33:36,160 --> 00:33:38,540 Can I cheat a little from here? 607 00:33:46,830 --> 00:33:48,080 OK. 608 00:33:48,080 --> 00:33:52,550 So now that we dealt with the time until the first arrival, 609 00:33:52,550 --> 00:33:56,110 we can start talking about the time until the second 610 00:33:56,110 --> 00:33:57,190 arrival, and so on. 611 00:33:57,190 --> 00:33:59,550 How do we define these? 612 00:33:59,550 --> 00:34:02,920 After the first arrival happens, we're going to have a 613 00:34:02,920 --> 00:34:06,100 sequence of time slots with no arrivals, and then the next 614 00:34:06,100 --> 00:34:08,199 arrival is going to happen. 615 00:34:08,199 --> 00:34:11,080 So we call this time that elapses-- 616 00:34:11,080 --> 00:34:14,760 or number of time slots after the first arrival 617 00:34:14,760 --> 00:34:16,730 until the next one-- 618 00:34:16,730 --> 00:34:18,500 we call it T2. 619 00:34:18,500 --> 00:34:22,070 This is the second inter-arrival time, that is, 620 00:34:22,070 --> 00:34:23,780 time between arrivals. 621 00:34:23,780 --> 00:34:28,260 Once this arrival has happened, then we wait and see 622 00:34:28,260 --> 00:34:31,600 how many more it takes until the third arrival. 623 00:34:31,600 --> 00:34:37,040 And we call this time here, T3. 624 00:34:37,040 --> 00:34:42,429 We're interested in the time of the k-th arrival, which is 625 00:34:42,429 --> 00:34:45,230 going to be just the sum of the first k 626 00:34:45,230 --> 00:34:46,889 inter-arrival times. 627 00:34:46,889 --> 00:34:51,909 So for example, let's say Y3 is the time that the third 628 00:34:51,909 --> 00:34:53,510 arrival comes. 629 00:34:53,510 --> 00:34:58,940 Y3 is just the sum of T1, plus T2, plus T3. 630 00:35:01,840 --> 00:35:06,310 So we're interested in this random variable, Y3, and it's 631 00:35:06,310 --> 00:35:08,765 the sum of inter-arrival times. 632 00:35:08,765 --> 00:35:12,190 To understand what kind of random variable it is, I guess 633 00:35:12,190 --> 00:35:16,970 we should understand what kind of random variables these are 634 00:35:16,970 --> 00:35:18,230 going to be. 635 00:35:18,230 --> 00:35:22,040 So what kind of random variable is T2? 636 00:35:22,040 --> 00:35:27,440 Your sister is doing her coin flips until a success is 637 00:35:27,440 --> 00:35:29,750 observed for the first time. 638 00:35:29,750 --> 00:35:32,540 Based on that information about what has happened so 639 00:35:32,540 --> 00:35:34,730 far, you are called into the room. 640 00:35:34,730 --> 00:35:39,320 And you start watching until a success is observed again. 641 00:35:39,320 --> 00:35:42,160 So after you start watching, what you have is just a 642 00:35:42,160 --> 00:35:45,190 sequence of independent Bernoulli trials. 643 00:35:45,190 --> 00:35:47,110 So each one of these has probability 644 00:35:47,110 --> 00:35:49,400 p of being a success. 645 00:35:49,400 --> 00:35:52,180 The time it's going to take until the first success, this 646 00:35:52,180 --> 00:35:57,600 number, T2, is going to be again just another geometric 647 00:35:57,600 --> 00:35:58,840 random variable. 648 00:35:58,840 --> 00:36:01,860 It's as if the process just started. 649 00:36:01,860 --> 00:36:06,280 After you are called into the room, you have no foresight, 650 00:36:06,280 --> 00:36:09,540 you don't have any information about the future, other than 651 00:36:09,540 --> 00:36:11,020 the fact that these are going to be 652 00:36:11,020 --> 00:36:13,280 independent Bernoulli trials. 653 00:36:13,280 --> 00:36:18,480 So T2 itself is going to be geometric with the same 654 00:36:18,480 --> 00:36:20,590 parameter p. 655 00:36:20,590 --> 00:36:24,700 And then you can continue the arguments and argue that T3 is 656 00:36:24,700 --> 00:36:27,880 also geometric with the same parameter p. 657 00:36:27,880 --> 00:36:30,850 Furthermore, whatever happened, how long it took 658 00:36:30,850 --> 00:36:34,330 until you were called in, it doesn't change the statistics 659 00:36:34,330 --> 00:36:36,810 about what's going to happen in the future. 660 00:36:36,810 --> 00:36:39,130 So whatever happens in the future is 661 00:36:39,130 --> 00:36:41,460 independent from the past. 662 00:36:41,460 --> 00:36:47,340 So T1, T2, and T3 are independent random variables. 663 00:36:47,340 --> 00:36:54,220 So conclusion is that the time until the third arrival is the 664 00:36:54,220 --> 00:37:00,510 sum of 3 independent geometric random variables, with the 665 00:37:00,510 --> 00:37:02,150 same parameter. 666 00:37:02,150 --> 00:37:04,550 And this is true more generally. 667 00:37:04,550 --> 00:37:08,820 The time until the k-th arrival is going to be the sum 668 00:37:08,820 --> 00:37:14,900 of k independent random variables. 669 00:37:14,900 --> 00:37:21,540 So in general, Yk is going to be T1 plus Tk, where the Ti's 670 00:37:21,540 --> 00:37:26,850 are geometric, with the same parameter p, and independent. 671 00:37:30,440 --> 00:37:33,520 So now what's more natural than trying to find the 672 00:37:33,520 --> 00:37:37,100 distribution of the random variable Yk? 673 00:37:37,100 --> 00:37:38,350 How can we find it? 674 00:37:38,350 --> 00:37:40,260 So I fixed k for you. 675 00:37:40,260 --> 00:37:41,680 Let's say k is 100. 676 00:37:41,680 --> 00:37:43,580 I'm interested in how long it takes until 677 00:37:43,580 --> 00:37:46,180 100 customers arrive. 678 00:37:46,180 --> 00:37:48,850 How can we find the distribution of Yk? 679 00:37:48,850 --> 00:37:51,980 Well one way of doing it is to use this 680 00:37:51,980 --> 00:37:54,200 lovely convolution formula. 681 00:37:54,200 --> 00:37:57,950 Take a geometric, convolve it with another geometric, you 682 00:37:57,950 --> 00:37:59,430 get something. 683 00:37:59,430 --> 00:38:02,040 Take that something that you got, convolve it with a 684 00:38:02,040 --> 00:38:07,090 geometric once more, do this 99 times, and this gives you 685 00:38:07,090 --> 00:38:09,450 the distribution of Yk. 686 00:38:09,450 --> 00:38:14,300 So that's definitely doable, and it's extremely tedious. 687 00:38:14,300 --> 00:38:16,900 Let's try to find the distribution 688 00:38:16,900 --> 00:38:22,520 of Yk using a shortcut. 689 00:38:22,520 --> 00:38:28,660 So the probability that Yk is equal to t. 690 00:38:28,660 --> 00:38:31,620 So we're trying to find the PMF of Yk. 691 00:38:31,620 --> 00:38:34,030 k has been fixed for us. 692 00:38:34,030 --> 00:38:36,810 And we want to calculate this probability for the various 693 00:38:36,810 --> 00:38:39,580 values of t, because this is going to give 694 00:38:39,580 --> 00:38:43,642 us the PMF of Yk. 695 00:38:43,642 --> 00:38:45,850 OK. 696 00:38:45,850 --> 00:38:47,540 What is this event? 697 00:38:47,540 --> 00:38:53,210 What does it take for the k-th arrival to be at time t? 698 00:38:53,210 --> 00:38:56,580 For that to happen, we need two things. 699 00:38:56,580 --> 00:39:00,960 In the first t -1 slots, how many arrivals 700 00:39:00,960 --> 00:39:03,130 should we have gotten? 701 00:39:03,130 --> 00:39:04,830 k - 1. 702 00:39:04,830 --> 00:39:09,400 And then in the last slot, we get one more arrival, and 703 00:39:09,400 --> 00:39:11,430 that's the k-th one. 704 00:39:11,430 --> 00:39:20,340 So this is the probability that we have k - 1 arrivals in 705 00:39:20,340 --> 00:39:24,250 the time interval from 1 up to t. 706 00:39:28,670 --> 00:39:34,120 And then, an arrival at time t. 707 00:39:39,860 --> 00:39:43,210 That's the only way that it can happen, that the k-th 708 00:39:43,210 --> 00:39:45,450 arrival happens at time t. 709 00:39:45,450 --> 00:39:48,470 We need to have an arrival at time t. 710 00:39:48,470 --> 00:39:50,150 And before that time, we need to have 711 00:39:50,150 --> 00:39:53,380 exactly k - 1 arrivals. 712 00:39:53,380 --> 00:39:55,590 Now this is an event that refers-- 713 00:39:58,710 --> 00:39:59,960 t-1. 714 00:40:02,830 --> 00:40:07,460 In the previous time slots we had exactly k -1 arrivals. 715 00:40:07,460 --> 00:40:10,680 And then at the last time slot we get one more arrival. 716 00:40:10,680 --> 00:40:14,560 Now the interesting thing is that this event here has to do 717 00:40:14,560 --> 00:40:18,620 with what happened from time 1 up to time t -1. 718 00:40:18,620 --> 00:40:22,350 This event has to do with what happened at time t. 719 00:40:22,350 --> 00:40:25,510 Different time slots are independent of each other. 720 00:40:25,510 --> 00:40:31,130 So this event and that event are independent. 721 00:40:31,130 --> 00:40:34,930 So this means that we can multiply their probabilities. 722 00:40:34,930 --> 00:40:37,280 So take the probability of this. 723 00:40:37,280 --> 00:40:38,590 What is that? 724 00:40:38,590 --> 00:40:41,600 Well probability of having a certain number of arrivals in 725 00:40:41,600 --> 00:40:44,650 a certain number of time slots, these are just the 726 00:40:44,650 --> 00:40:46,530 binomial probabilities. 727 00:40:46,530 --> 00:40:51,250 So this is, out of t - 1 slots, to get exactly k - 1 728 00:40:51,250 --> 00:41:02,670 arrivals, p to the k-1, (1-p) to the t-1 - (k-1), 729 00:41:02,670 --> 00:41:05,480 this gives us t-k. 730 00:41:05,480 --> 00:41:07,960 And then we multiply with this probability, the probability 731 00:41:07,960 --> 00:41:14,300 of an arrival, at time t is equal to p. 732 00:41:14,300 --> 00:41:21,310 And so this is the formula for the PMF of the number-- 733 00:41:21,310 --> 00:41:27,410 of the time it takes until the k-th arrival happens. 734 00:41:32,760 --> 00:41:35,730 Does it agree with the formula in your handout? 735 00:41:35,730 --> 00:41:37,850 Or its not there? 736 00:41:37,850 --> 00:41:38,710 It's not there. 737 00:41:38,710 --> 00:41:39,960 OK. 738 00:41:48,018 --> 00:41:49,010 Yeah. 739 00:41:49,010 --> 00:41:50,020 OK. 740 00:41:50,020 --> 00:41:57,338 So that's the formula and it is true for what values of t? 741 00:41:57,338 --> 00:41:58,588 [INAUDIBLE]. 742 00:42:03,182 --> 00:42:08,350 It takes at least k time slots in order to get k arrivals, so 743 00:42:08,350 --> 00:42:12,370 this formula should be true for k larger 744 00:42:12,370 --> 00:42:13,990 than or equal to t. 745 00:42:20,330 --> 00:42:23,391 For t larger than or equal to k. 746 00:42:30,130 --> 00:42:31,040 All right. 747 00:42:31,040 --> 00:42:34,950 So this gives us the PMF of the random variable Yk. 748 00:42:34,950 --> 00:42:37,430 Of course, we may also be interested in the mean and 749 00:42:37,430 --> 00:42:39,260 variance of Yk. 750 00:42:39,260 --> 00:42:42,150 But this is a lot easier. 751 00:42:42,150 --> 00:42:46,350 Since Yk is the sum of independent random variables, 752 00:42:46,350 --> 00:42:50,310 the expected value of Yk is going to be just k times the 753 00:42:50,310 --> 00:42:52,980 expected value of your typical t. 754 00:42:52,980 --> 00:43:03,080 So the expected value of Yk is going to be just k times 1/p, 755 00:43:03,080 --> 00:43:06,600 which is the mean of the geometric. 756 00:43:06,600 --> 00:43:09,470 And similarly for the variance, it's going to be k 757 00:43:09,470 --> 00:43:12,960 times the variance of a geometric. 758 00:43:12,960 --> 00:43:16,950 So we have everything there is to know about the distribution 759 00:43:16,950 --> 00:43:19,965 of how long it takes until the first arrival comes. 760 00:43:23,760 --> 00:43:25,420 OK. 761 00:43:25,420 --> 00:43:27,810 Finally, let's do a few more things about 762 00:43:27,810 --> 00:43:30,680 the Bernoulli process. 763 00:43:30,680 --> 00:43:34,970 It's interesting to talk about several processes at the time. 764 00:43:34,970 --> 00:43:39,600 So in the situation here of splitting a Bernoulli process 765 00:43:39,600 --> 00:43:43,730 is where you have arrivals that come to a server. 766 00:43:43,730 --> 00:43:46,370 And that's a picture of which slots get arrivals. 767 00:43:46,370 --> 00:43:48,900 But actually maybe you have two servers. 768 00:43:48,900 --> 00:43:53,110 And whenever an arrival comes to the system, you flip a coin 769 00:43:53,110 --> 00:43:56,620 and with some probability, q, you send it to one server. 770 00:43:56,620 --> 00:44:00,390 And with probability 1-q, you send it to another server. 771 00:44:00,390 --> 00:44:03,520 So there is a single arrival stream, but 772 00:44:03,520 --> 00:44:05,040 two possible servers. 773 00:44:05,040 --> 00:44:07,280 And whenever there's an arrival, you either send it 774 00:44:07,280 --> 00:44:09,270 here or you send it there. 775 00:44:09,270 --> 00:44:13,780 And each time you decide where you send it by flipping an 776 00:44:13,780 --> 00:44:17,950 independent coin that has its own bias q. 777 00:44:17,950 --> 00:44:22,450 The coin flips that decide where do you send it are 778 00:44:22,450 --> 00:44:27,460 assumed to be independent from the arrival process itself. 779 00:44:27,460 --> 00:44:30,480 So there's two coin flips that are happening. 780 00:44:30,480 --> 00:44:33,850 At each time slot, there's a coin flip that decides whether 781 00:44:33,850 --> 00:44:37,150 you have an arrival in this process here, and that coin 782 00:44:37,150 --> 00:44:39,630 flip is with parameter p. 783 00:44:39,630 --> 00:44:43,000 And if you have something that arrives, you flip another coin 784 00:44:43,000 --> 00:44:47,050 with probabilities q, and 1-q, that decides whether you send 785 00:44:47,050 --> 00:44:49,770 it up there or you send it down there. 786 00:44:49,770 --> 00:44:55,460 So what kind of arrival process does this server see? 787 00:44:55,460 --> 00:44:59,510 At any given time slot, there's probability p that 788 00:44:59,510 --> 00:45:01,480 there's an arrival here. 789 00:45:01,480 --> 00:45:04,300 And there's a further probability q that this 790 00:45:04,300 --> 00:45:07,320 arrival gets sent up there. 791 00:45:07,320 --> 00:45:10,860 So the probability that this server sees an arrival at any 792 00:45:10,860 --> 00:45:14,090 given time is p times q. 793 00:45:14,090 --> 00:45:18,900 So this process here is going to be a Bernoulli process, but 794 00:45:18,900 --> 00:45:21,810 with a different parameter, p times q. 795 00:45:21,810 --> 00:45:24,820 And this one down here, with the same argument, is going to 796 00:45:24,820 --> 00:45:29,860 be Bernoulli with parameter p times (1-q). 797 00:45:29,860 --> 00:45:33,500 So by taking a Bernoulli stream of arrivals and 798 00:45:33,500 --> 00:45:36,460 splitting it into two, you get two 799 00:45:36,460 --> 00:45:39,050 separate Bernoulli processes. 800 00:45:39,050 --> 00:45:40,890 This is going to be a Bernoulli process, that's 801 00:45:40,890 --> 00:45:42,980 going to be a Bernoulli process. 802 00:45:42,980 --> 00:45:45,630 Well actually, I'm running a little too fast. 803 00:45:45,630 --> 00:45:49,330 What does it take to verify that it's a Bernoulli process? 804 00:45:49,330 --> 00:45:52,650 At each time slot, it's a 0 or 1. 805 00:45:52,650 --> 00:45:55,330 And it's going to be a 1, you're going to see an arrival 806 00:45:55,330 --> 00:45:57,450 with probability p times q. 807 00:45:57,450 --> 00:46:00,510 What else do we need to verify, to be able to tell-- 808 00:46:00,510 --> 00:46:02,820 to say that it's a Bernoulli process? 809 00:46:02,820 --> 00:46:05,620 We need to make sure that whatever happens in this 810 00:46:05,620 --> 00:46:09,340 process, in different time slots, are statistically 811 00:46:09,340 --> 00:46:11,240 independent from each other. 812 00:46:11,240 --> 00:46:13,030 Is that property true? 813 00:46:13,030 --> 00:46:16,900 For example, what happens in this time slot whether you got 814 00:46:16,900 --> 00:46:20,100 an arrival or not, is it independent from what happened 815 00:46:20,100 --> 00:46:22,660 at that time slot? 816 00:46:22,660 --> 00:46:26,850 The answer is yes for the following reason. 817 00:46:26,850 --> 00:46:30,760 What happens in this time slot has to do with the coin flip 818 00:46:30,760 --> 00:46:34,840 associated with the original process at this time, and the 819 00:46:34,840 --> 00:46:38,340 coin flip that decides where to send things. 820 00:46:38,340 --> 00:46:41,370 What happens at that time slot has to do with the coin flip 821 00:46:41,370 --> 00:46:45,010 here, and the additional coin flip that decides where to 822 00:46:45,010 --> 00:46:47,130 send it if something came. 823 00:46:47,130 --> 00:46:50,570 Now all these coin flips are independent of each other. 824 00:46:50,570 --> 00:46:53,460 The coin flips that determine whether we have an arrival 825 00:46:53,460 --> 00:46:56,860 here is independent from the coin flips that determined 826 00:46:56,860 --> 00:46:59,280 whether we had an arrival there. 827 00:46:59,280 --> 00:47:02,770 And you can generalize this argument and conclude that, 828 00:47:02,770 --> 00:47:07,390 indeed, every time slot here is independent from 829 00:47:07,390 --> 00:47:09,030 any other time slot. 830 00:47:09,030 --> 00:47:12,020 And this does make it a Bernoulli process. 831 00:47:12,020 --> 00:47:15,590 And the reason is that, in the original process, every time 832 00:47:15,590 --> 00:47:18,390 slot is independent from every other time slot. 833 00:47:18,390 --> 00:47:21,000 And the additional assumption that the coin flips that we're 834 00:47:21,000 --> 00:47:24,370 using to decide where to send things, these are also 835 00:47:24,370 --> 00:47:26,020 independent of each other. 836 00:47:26,020 --> 00:47:30,220 So we're using here the basic property that functions of 837 00:47:30,220 --> 00:47:33,283 independent things remain independent. 838 00:47:36,390 --> 00:47:38,710 There's a converse picture of this. 839 00:47:38,710 --> 00:47:41,970 Instead of taking one stream and splitting it into two 840 00:47:41,970 --> 00:47:44,720 streams, you can do the opposite. 841 00:47:44,720 --> 00:47:48,040 You could start from two streams of arrivals. 842 00:47:48,040 --> 00:47:51,300 Let's say you have arrivals of men and you have arrivals of 843 00:47:51,300 --> 00:47:54,000 women, but you don't care about gender. 844 00:47:54,000 --> 00:47:57,430 And the only thing you record is whether, in a given time 845 00:47:57,430 --> 00:48:00,450 slot, you had an arrival or not. 846 00:48:00,450 --> 00:48:04,320 Notice that here we may have an arrival of a man and the 847 00:48:04,320 --> 00:48:05,790 arrival of a woman. 848 00:48:05,790 --> 00:48:11,180 We just record it with a 1, by saying there was an arrival. 849 00:48:11,180 --> 00:48:14,260 So in the merged process, we're not keeping track of how 850 00:48:14,260 --> 00:48:16,880 many arrivals we had total. 851 00:48:16,880 --> 00:48:18,840 We just record whether there was an 852 00:48:18,840 --> 00:48:21,400 arrival or not an arrival. 853 00:48:21,400 --> 00:48:25,830 So an arrival gets recorded here if, and only if, one or 854 00:48:25,830 --> 00:48:28,800 both of these streams had an arrival. 855 00:48:28,800 --> 00:48:31,780 So that we call a merging of two Bernoull-- of two 856 00:48:31,780 --> 00:48:34,470 processes, of two arrival processes. 857 00:48:34,470 --> 00:48:37,840 So let's make the assumption that this arrival process is 858 00:48:37,840 --> 00:48:41,440 independent from that arrival process. 859 00:48:41,440 --> 00:48:44,330 So what happens at the typical slot here? 860 00:48:44,330 --> 00:48:49,680 I'm going to see an arrival, unless none of 861 00:48:49,680 --> 00:48:51,950 these had an arrival. 862 00:48:51,950 --> 00:48:56,380 So the probability of an arrival in a typical time slot 863 00:48:56,380 --> 00:49:02,650 is going to be 1 minus the probability of no arrival. 864 00:49:02,650 --> 00:49:07,370 And the event of no arrival corresponds to the first 865 00:49:07,370 --> 00:49:10,330 process having no arrival, and the second 866 00:49:10,330 --> 00:49:14,350 process having no arrival. 867 00:49:14,350 --> 00:49:18,110 So there's no arrival in the merged process if, and only 868 00:49:18,110 --> 00:49:21,270 if, there's no arrival in the first process and no arrival 869 00:49:21,270 --> 00:49:22,710 in the second process. 870 00:49:22,710 --> 00:49:26,080 We're assuming that the two processes are independent and 871 00:49:26,080 --> 00:49:29,160 that's why we can multiply probabilities here. 872 00:49:29,160 --> 00:49:34,270 And then you can take this formula and it simplifies to p 873 00:49:34,270 --> 00:49:38,120 + q, minus p times q. 874 00:49:38,120 --> 00:49:41,360 So each time slot of the merged process has a certain 875 00:49:41,360 --> 00:49:44,620 probability of seeing an arrival. 876 00:49:44,620 --> 00:49:47,360 Is the merged process a Bernoulli process? 877 00:49:47,360 --> 00:49:51,260 Yes, it is after you verify the additional property that 878 00:49:51,260 --> 00:49:54,650 different slots are independent of each other. 879 00:49:54,650 --> 00:49:56,560 Why are they independent? 880 00:49:56,560 --> 00:50:01,070 What happens in this slot has to do with that slot, and that 881 00:50:01,070 --> 00:50:03,160 slot down here. 882 00:50:03,160 --> 00:50:05,790 These two slots-- 883 00:50:05,790 --> 00:50:08,570 so what happens here, has to do with what 884 00:50:08,570 --> 00:50:11,430 happens here and there. 885 00:50:11,430 --> 00:50:16,680 What happens in this slot has to do with whatever happened 886 00:50:16,680 --> 00:50:19,200 here and there. 887 00:50:19,200 --> 00:50:23,190 Now, whatever happens here and there is independent from 888 00:50:23,190 --> 00:50:25,330 whatever happens here and there. 889 00:50:25,330 --> 00:50:29,180 Therefore, what happens here is independent from what 890 00:50:29,180 --> 00:50:30,220 happens there. 891 00:50:30,220 --> 00:50:33,310 So the independence property is preserved. 892 00:50:33,310 --> 00:50:36,640 The different slots of this merged process are independent 893 00:50:36,640 --> 00:50:37,590 of each other. 894 00:50:37,590 --> 00:50:41,970 So the merged process is itself a Bernoulli process. 895 00:50:41,970 --> 00:50:45,900 So please digest these two pictures of merging and 896 00:50:45,900 --> 00:50:48,960 splitting, because we're going to revisit them in continuous 897 00:50:48,960 --> 00:50:52,510 time where things are little subtler than that. 898 00:50:52,510 --> 00:50:53,160 OK. 899 00:50:53,160 --> 00:50:56,240 Good luck on the exam and see you in a week.