WEBVTT 00:00:00.040 --> 00:00:02.460 The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 Commons license. 00:00:03.870 --> 00:00:06.910 Your support will help MIT OpenCourseWare continue to 00:00:06.910 --> 00:00:10.560 offer high quality educational resources for free. 00:00:10.560 --> 00:00:13.460 To make a donation or view additional materials from 00:00:13.460 --> 00:00:19.290 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.540 ocw.mit.edu. 00:00:22.840 --> 00:00:25.260 JOHN TSITSIKLIS: Today we're going to finish our discussion 00:00:25.260 --> 00:00:27.480 of the Poisson process. 00:00:27.480 --> 00:00:31.280 We're going to see a few of its properties, do a few 00:00:31.280 --> 00:00:35.660 interesting problems, some more interesting than others. 00:00:35.660 --> 00:00:39.000 So go through a few examples and then we're going to talk 00:00:39.000 --> 00:00:42.170 about some quite strange things that happen with the 00:00:42.170 --> 00:00:43.980 Poisson process. 00:00:43.980 --> 00:00:46.500 So the first thing is to remember what the Poisson 00:00:46.500 --> 00:00:48.200 processes is. 00:00:48.200 --> 00:00:52.300 It's a model, let's say, of arrivals of customers that 00:00:52.300 --> 00:00:55.940 are, in some sense, quote unquote, completely random, 00:00:55.940 --> 00:00:58.990 that is a customer can arrive at any point in time. 00:00:58.990 --> 00:01:01.670 All points in time are equally likely. 00:01:01.670 --> 00:01:05.330 And different points in time are sort of independent of 00:01:05.330 --> 00:01:06.450 other points in time. 00:01:06.450 --> 00:01:10.160 So the fact that I got an arrival now doesn't tell me 00:01:10.160 --> 00:01:13.050 anything about whether there's going to be an arrival at some 00:01:13.050 --> 00:01:14.950 other time. 00:01:14.950 --> 00:01:18.230 In some sense, it's a continuous time version of the 00:01:18.230 --> 00:01:19.760 Bernoulli process. 00:01:19.760 --> 00:01:23.130 So the best way to think about the Poisson process is that we 00:01:23.130 --> 00:01:26.240 divide time into extremely tiny slots. 00:01:26.240 --> 00:01:29.750 And in each time slot, there's an independent possibility of 00:01:29.750 --> 00:01:31.120 having an arrival. 00:01:31.120 --> 00:01:33.860 Different time slots are independent of each other. 00:01:33.860 --> 00:01:36.660 On the other hand, when the slot is tiny, the probability 00:01:36.660 --> 00:01:39.760 for obtaining an arrival during that tiny slot is 00:01:39.760 --> 00:01:41.910 itself going to be tiny. 00:01:41.910 --> 00:01:45.950 So we capture these properties into a formal definition what 00:01:45.950 --> 00:01:48.380 the Poisson process is. 00:01:48.380 --> 00:01:51.390 We have a probability mass function for the number of 00:01:51.390 --> 00:01:56.250 arrivals, k, during an interval of a given length. 00:01:56.250 --> 00:02:00.590 So this is the sort of basic description of the 00:02:00.590 --> 00:02:03.200 distribution of the number of arrivals. 00:02:03.200 --> 00:02:07.520 So tau is fixed. 00:02:07.520 --> 00:02:09.350 And k is the parameter. 00:02:09.350 --> 00:02:13.660 So when we add over all k's, the sum of these probabilities 00:02:13.660 --> 00:02:15.780 has to be equal to 1. 00:02:15.780 --> 00:02:19.330 There's a time homogeneity assumption, which is hidden in 00:02:19.330 --> 00:02:22.840 this, namely, the only thing that matters is the duration 00:02:22.840 --> 00:02:25.880 of the time interval, not where the time interval sits 00:02:25.880 --> 00:02:28.230 on the real axis. 00:02:28.230 --> 00:02:31.260 Then we have an independence assumption. 00:02:31.260 --> 00:02:34.570 Intervals that are disjoint are statistically independent 00:02:34.570 --> 00:02:35.650 from each other. 00:02:35.650 --> 00:02:39.110 So any information you give me about arrivals during this 00:02:39.110 --> 00:02:43.200 time interval doesn't change my beliefs about what's going 00:02:43.200 --> 00:02:45.930 to happen during another time interval. 00:02:45.930 --> 00:02:49.050 So this is a generalization of the idea that we had in 00:02:49.050 --> 00:02:51.970 Bernoulli processes that different time slots are 00:02:51.970 --> 00:02:53.930 independent of each other. 00:02:53.930 --> 00:02:56.770 And then to specify this function, the distribution of 00:02:56.770 --> 00:03:00.120 the number of arrivals, we sort of go in stages. 00:03:00.120 --> 00:03:03.750 We first specify this function for the case where the time 00:03:03.750 --> 00:03:05.920 interval is very small. 00:03:05.920 --> 00:03:09.630 And I'm telling you what those probabilities will be. 00:03:09.630 --> 00:03:13.540 And based on these then, we do some calculations and to find 00:03:13.540 --> 00:03:16.510 the formula for the distribution of the number of 00:03:16.510 --> 00:03:19.420 arrivals for intervals of a general duration. 00:03:19.420 --> 00:03:23.000 So for a small duration, delta, the probability of 00:03:23.000 --> 00:03:26.730 obtaining 1 arrival is lambda delta. 00:03:26.730 --> 00:03:30.220 The remaining probability is assigned to the event that we 00:03:30.220 --> 00:03:32.980 get to no arrivals during that interval. 00:03:32.980 --> 00:03:36.330 The probability of obtaining more than 1 arrival in a tiny 00:03:36.330 --> 00:03:40.630 interval is essentially 0. 00:03:40.630 --> 00:03:45.190 And when we say essentially, it's means modular, terms that 00:03:45.190 --> 00:03:47.620 of order delta squared. 00:03:47.620 --> 00:03:50.440 And when delta is very small, anything which is delta 00:03:50.440 --> 00:03:52.690 squared can be ignored. 00:03:52.690 --> 00:03:55.850 So up to delta squared terms, that's what happened during a 00:03:55.850 --> 00:03:57.660 little interval. 00:03:57.660 --> 00:04:01.210 Now if we know the probability distribution for the number of 00:04:01.210 --> 00:04:03.260 arrivals in a little interval. 00:04:03.260 --> 00:04:06.470 We can use this to get the distribution for the number of 00:04:06.470 --> 00:04:08.370 arrivals over several intervals. 00:04:08.370 --> 00:04:09.870 How do we do that? 00:04:09.870 --> 00:04:13.850 The big interval is composed of many little intervals. 00:04:13.850 --> 00:04:16.410 Each little interval is independent from any other 00:04:16.410 --> 00:04:20.720 little interval, so is it is as if we have a sequence of 00:04:20.720 --> 00:04:22.310 Bernoulli trials. 00:04:22.310 --> 00:04:24.910 Each Bernoulli trial is associated with a little 00:04:24.910 --> 00:04:29.580 interval and has a small probability of obtaining a 00:04:29.580 --> 00:04:32.850 success or an arrival during that mini-slot. 00:04:32.850 --> 00:04:35.800 On the other hand, when delta is small, and you take a big 00:04:35.800 --> 00:04:38.680 interval and chop it up, you get a large 00:04:38.680 --> 00:04:41.410 number of little intervals. 00:04:41.410 --> 00:04:45.240 So what we essentially have here is a Bernoulli process, 00:04:45.240 --> 00:04:48.690 in which is the number of trials is huge but the 00:04:48.690 --> 00:04:53.030 probability of success during any given trial is tiny. 00:04:53.030 --> 00:05:02.580 The average number of trials ends up being proportional to 00:05:02.580 --> 00:05:04.740 the length of the interval. 00:05:04.740 --> 00:05:07.410 If you have twice as large an interval, it's as if you're 00:05:07.410 --> 00:05:10.860 having twice as many over these mini-trials, so the 00:05:10.860 --> 00:05:14.080 expected number of arrivals will increase proportionately. 00:05:14.080 --> 00:05:18.380 There's also this parameter lambda, which we interpret as 00:05:18.380 --> 00:05:22.600 expected number of arrivals per unit time. 00:05:22.600 --> 00:05:25.940 And it comes in those probabilities here. 00:05:25.940 --> 00:05:28.810 When you double lambda, this means that a little interval 00:05:28.810 --> 00:05:31.160 is twice as likely to get an arrival. 00:05:31.160 --> 00:05:33.300 So you would expect to get twice as 00:05:33.300 --> 00:05:34.880 many arrivals as well. 00:05:34.880 --> 00:05:37.990 That's why the expected number of arrivals during an interval 00:05:37.990 --> 00:05:40.580 of length tau also scales proportional to 00:05:40.580 --> 00:05:42.850 this parameter lambda. 00:05:42.850 --> 00:05:45.740 Somewhat unexpectedly, it turns out that the variance of 00:05:45.740 --> 00:05:48.750 the number of arrivals is also the same as the mean. 00:05:48.750 --> 00:05:50.540 This is a peculiarity that happens 00:05:50.540 --> 00:05:52.310 in the Poisson process. 00:05:52.310 --> 00:05:56.100 So this is one way of thinking about Poisson process, in 00:05:56.100 --> 00:05:59.640 terms of little intervals, each one of which has a tiny 00:05:59.640 --> 00:06:01.690 probability of success. 00:06:01.690 --> 00:06:04.370 And we think of the distribution associated with 00:06:04.370 --> 00:06:07.030 that process as being described by 00:06:07.030 --> 00:06:09.100 this particular PMF. 00:06:09.100 --> 00:06:12.800 So this is the PMF for the number of arrivals during an 00:06:12.800 --> 00:06:15.880 interval of a fixed duration, tau. 00:06:15.880 --> 00:06:20.970 It's a PMF that extends all over the entire range of 00:06:20.970 --> 00:06:22.810 non-negative integers. 00:06:22.810 --> 00:06:25.730 So the number of arrivals you can get during an interval for 00:06:25.730 --> 00:06:27.890 certain length can be anything. 00:06:27.890 --> 00:06:31.690 You can get as many arrivals as you want. 00:06:31.690 --> 00:06:34.400 Of course the probability of getting a zillion arrivals is 00:06:34.400 --> 00:06:35.620 going to be tiny. 00:06:35.620 --> 00:06:38.110 But in principle, this is possible. 00:06:38.110 --> 00:06:41.440 And that's because an interval, even if it's a fixed 00:06:41.440 --> 00:06:47.460 length, consists of an infinite number of mini-slots 00:06:47.460 --> 00:06:48.770 in some sense. 00:06:48.770 --> 00:06:50.880 You can divide, chop it up, into as many 00:06:50.880 --> 00:06:52.130 mini-slots as you want. 00:06:52.130 --> 00:06:54.400 So in principle, it's possible that every 00:06:54.400 --> 00:06:55.860 mini-slot gets an arrival. 00:06:55.860 --> 00:06:59.190 In principle, it's possible to get an arbitrarily large 00:06:59.190 --> 00:07:01.210 number of arrivals. 00:07:01.210 --> 00:07:05.250 So this particular formula here is not very intuitive 00:07:05.250 --> 00:07:06.560 when you look at it. 00:07:06.560 --> 00:07:08.630 But it's a legitimate PMF. 00:07:08.630 --> 00:07:10.360 And it's called the Poisson PMF. 00:07:10.360 --> 00:07:13.970 It's the PMF that describes the number of arrivals. 00:07:13.970 --> 00:07:17.660 So that's one way of thinking about the Poisson process, 00:07:17.660 --> 00:07:21.650 where the basic object of interest would be this PMF and 00:07:21.650 --> 00:07:23.520 you try to work with it. 00:07:23.520 --> 00:07:26.600 There's another way of thinking about what happens in 00:07:26.600 --> 00:07:28.060 the Poisson process. 00:07:28.060 --> 00:07:31.780 And this has to do with letting things evolve in time. 00:07:31.780 --> 00:07:34.160 You start at time 0. 00:07:34.160 --> 00:07:37.080 There's going to be a time at which the first arrival 00:07:37.080 --> 00:07:40.000 occurs, and call that time T1. 00:07:40.000 --> 00:07:44.130 This time turns out to have an exponential distribution with 00:07:44.130 --> 00:07:46.340 parameter lambda. 00:07:46.340 --> 00:07:49.120 Once you get an arrival, it's as if the 00:07:49.120 --> 00:07:53.300 process starts fresh. 00:07:53.300 --> 00:07:55.830 The best way to understand why this is the case is by 00:07:55.830 --> 00:07:57.740 thinking in terms of the analogy with 00:07:57.740 --> 00:07:58.840 the Bernoulli process. 00:07:58.840 --> 00:08:01.660 If you believe that statement for the Bernoulli process, 00:08:01.660 --> 00:08:05.510 since this is a limiting case, it should also be true. 00:08:05.510 --> 00:08:09.150 So starting from this time, we're going to wait a random 00:08:09.150 --> 00:08:12.710 amount of time until we get the second arrival This random 00:08:12.710 --> 00:08:15.250 amount of time, let's call it T2. 00:08:15.250 --> 00:08:18.360 This time, T2 is also going to have an exponential 00:08:18.360 --> 00:08:21.140 distribution with the same parameter, lambda. 00:08:21.140 --> 00:08:26.615 And these two are going to be independent of each other. 00:08:26.615 --> 00:08:27.750 OK? 00:08:27.750 --> 00:08:31.520 So the Poisson process has all the same memorylessness 00:08:31.520 --> 00:08:34.820 properties that the Bernoulli process has. 00:08:34.820 --> 00:08:37.630 What's another way of thinking of this property? 00:08:37.630 --> 00:08:43.360 So think of a process where you have a light bulb. 00:08:43.360 --> 00:08:47.070 The time at the light bulb burns out, you can model it by 00:08:47.070 --> 00:08:48.855 an exponential random variable. 00:08:51.680 --> 00:08:58.170 And suppose that they tell you that so far, we're are sitting 00:08:58.170 --> 00:09:01.550 at some time, T. And I tell you that the light bulb has 00:09:01.550 --> 00:09:04.510 not yet burned out. 00:09:04.510 --> 00:09:08.290 What does this tell you about the future of the light bulb? 00:09:08.290 --> 00:09:11.700 Is the fact that they didn't burn out, so far, is it good 00:09:11.700 --> 00:09:13.720 news or is it bad news? 00:09:13.720 --> 00:09:17.640 Would you rather keep this light bulb that has worked for 00:09:17.640 --> 00:09:20.950 t times steps and is still OK? 00:09:20.950 --> 00:09:25.770 Or would you rather use a new light bulb that starts new at 00:09:25.770 --> 00:09:27.740 that point in time? 00:09:27.740 --> 00:09:30.920 Because of the memorylessness property, the past of that 00:09:30.920 --> 00:09:33.220 light bulb doesn't matter. 00:09:33.220 --> 00:09:37.040 So the future of this light bulb is statistically the same 00:09:37.040 --> 00:09:40.740 as the future of a new light bulb. 00:09:40.740 --> 00:09:43.700 For both of them, the time until they burn out is going 00:09:43.700 --> 00:09:46.580 to be described an exponential distribution. 00:09:46.580 --> 00:09:50.990 So one way that people described the situation is to 00:09:50.990 --> 00:09:55.450 say that used is exactly as good as a new. 00:09:55.450 --> 00:09:59.220 So a used on is no worse than a new one. 00:09:59.220 --> 00:10:01.950 A used one is no better than a new one. 00:10:01.950 --> 00:10:06.130 So a used light bulb that hasn't yet burnt out is 00:10:06.130 --> 00:10:09.180 exactly as good as a new light bulb. 00:10:09.180 --> 00:10:11.740 So that's another way of thinking about the 00:10:11.740 --> 00:10:17.150 memorylessness that we have in the Poisson process. 00:10:17.150 --> 00:10:19.350 Back to this picture. 00:10:19.350 --> 00:10:22.410 The time until the second arrival is the sum of two 00:10:22.410 --> 00:10:24.990 independent exponential random variables. 00:10:24.990 --> 00:10:28.050 So, in principle, you can use the convolution formula to 00:10:28.050 --> 00:10:32.330 find the distribution of T1 plus T2, and that would be 00:10:32.330 --> 00:10:36.750 what we call Y2, the time until the second arrival. 00:10:36.750 --> 00:10:39.210 But there's also a direct way of obtaining to the 00:10:39.210 --> 00:10:42.580 distribution of Y2, and this is the calculation that we did 00:10:42.580 --> 00:10:44.340 last time on the blackboard. 00:10:44.340 --> 00:10:46.320 And actually, we did it more generally. 00:10:46.320 --> 00:10:49.990 We found the time until the case arrival occurs. 00:10:49.990 --> 00:10:53.860 It has a closed form formula, which is called the Erlang 00:10:53.860 --> 00:10:56.960 distribution with k degrees of freedom. 00:10:56.960 --> 00:11:00.170 So let's see what's going on here. 00:11:00.170 --> 00:11:03.230 It's a distribution Of what kind? 00:11:03.230 --> 00:11:05.210 It's a continuous distribution. 00:11:05.210 --> 00:11:07.150 It's a probability density function. 00:11:07.150 --> 00:11:10.620 This is because the time is a continuous random variable. 00:11:10.620 --> 00:11:11.580 Time is continuous. 00:11:11.580 --> 00:11:14.320 Arrivals can happen at any time. 00:11:14.320 --> 00:11:17.090 So we're talking about the PDF. 00:11:17.090 --> 00:11:20.230 This k is just the parameter of the distribution. 00:11:20.230 --> 00:11:22.450 We're talking about the k-th arrival, so 00:11:22.450 --> 00:11:24.210 k is a fixed number. 00:11:24.210 --> 00:11:27.440 Lambda is another parameter of the distribution, which is the 00:11:27.440 --> 00:11:32.660 arrival rate So it's a PDF over the Y's, whereas lambda 00:11:32.660 --> 00:11:36.060 and k are parameters of the distribution. 00:11:40.530 --> 00:11:40.860 OK. 00:11:40.860 --> 00:11:45.630 So this was what we knew from last time. 00:11:45.630 --> 00:11:51.550 Just to get some practice, let us do a problem that's not too 00:11:51.550 --> 00:11:55.730 difficult, but just to see how we use the various formulas 00:11:55.730 --> 00:11:57.470 that we have. 00:11:57.470 --> 00:12:01.930 So Poisson was a mathematician, but Poisson 00:12:01.930 --> 00:12:04.730 also means fish in French. 00:12:04.730 --> 00:12:07.130 So Poisson goes fishing. 00:12:07.130 --> 00:12:11.680 And let's assume that fish are caught according 00:12:11.680 --> 00:12:13.420 to a Poisson process. 00:12:13.420 --> 00:12:15.310 That's not too bad an assumption. 00:12:15.310 --> 00:12:18.180 At any given point in time, you have a little probability 00:12:18.180 --> 00:12:19.840 that a fish would be caught. 00:12:19.840 --> 00:12:22.930 And whether you catch one now is sort of independent about 00:12:22.930 --> 00:12:28.210 whether at some later time a fish will be caught or not. 00:12:28.210 --> 00:12:30.030 So let's just make this assumption. 00:12:30.030 --> 00:12:35.270 And suppose that the rules of the game are that you-- 00:12:35.270 --> 00:12:40.350 Fish are being called it the certain rate of 0.6 per hour. 00:12:40.350 --> 00:12:44.390 You fish for 2 hours, no matter what. 00:12:44.390 --> 00:12:46.190 And then there are two possibilities. 00:12:46.190 --> 00:12:50.710 If I have caught a fish, I stop and go home. 00:12:50.710 --> 00:12:54.320 So if some fish have been caught, so there's at least 1 00:12:54.320 --> 00:12:57.250 arrival during this interval, I go home. 00:12:57.250 --> 00:13:01.760 Or if nothing has being caught, I continue fishing 00:13:01.760 --> 00:13:03.630 until I catch something. 00:13:03.630 --> 00:13:05.300 And then I go home. 00:13:05.300 --> 00:13:09.410 So that's the description of what is going to happen. 00:13:09.410 --> 00:13:12.940 And now let's starts asking questions of all sorts. 00:13:12.940 --> 00:13:16.450 What is the probability that I'm going to be fishing for 00:13:16.450 --> 00:13:19.060 more than 2 hours? 00:13:19.060 --> 00:13:23.200 I will be fishing for more than 2 hours, if and only if 00:13:23.200 --> 00:13:28.400 no fish were caught during those 2 hours, in which case, 00:13:28.400 --> 00:13:30.140 I will have to continue. 00:13:30.140 --> 00:13:33.600 Therefore, this is just this quantity. 00:13:33.600 --> 00:13:38.630 The probability of catching 2 fish in-- 00:13:38.630 --> 00:13:43.450 of catching 0 fish in the next 2 hours, and according to the 00:13:43.450 --> 00:13:47.170 formula that we have, this is going to be e to the minus 00:13:47.170 --> 00:13:50.820 lambda times how much time we have. 00:13:50.820 --> 00:13:53.040 There's another way of thinking about this. 00:13:53.040 --> 00:13:55.790 The probability that I fish for more than 2 hours is the 00:13:55.790 --> 00:14:01.230 probability that the first catch happens after time 2, 00:14:01.230 --> 00:14:04.990 which would be the integral from 2 to infinity of the 00:14:04.990 --> 00:14:09.610 density of the first arrival time. 00:14:09.610 --> 00:14:11.770 And that density is an exponential. 00:14:11.770 --> 00:14:14.910 So you do the integral of an exponential, and, of course, 00:14:14.910 --> 00:14:17.160 you would get the same answer. 00:14:17.160 --> 00:14:17.550 OK. 00:14:17.550 --> 00:14:18.730 That's easy. 00:14:18.730 --> 00:14:22.880 So what's the probability of fishing for more than 2 but 00:14:22.880 --> 00:14:25.420 less than 5 hours? 00:14:25.420 --> 00:14:28.570 What does it take for this to happen? 00:14:28.570 --> 00:14:35.540 For this to happen, we need to catch 0 fish from time 0 to 2 00:14:35.540 --> 00:14:43.020 and catch the first fish sometime between 2 and 5. 00:14:43.020 --> 00:14:44.400 So if you-- 00:14:44.400 --> 00:14:47.510 one way of thinking about what's happening here might be 00:14:47.510 --> 00:14:49.900 to say that there's a Poisson process that 00:14:49.900 --> 00:14:52.770 keeps going on forever. 00:14:52.770 --> 00:14:57.090 But as soon as I catch the first fish, instead of 00:14:57.090 --> 00:15:00.990 continuing fishing and obtaining those other fish I 00:15:00.990 --> 00:15:04.070 just go home right now. 00:15:04.070 --> 00:15:11.060 Now the fact that I go home before time 5 means that, if I 00:15:11.060 --> 00:15:13.990 were to stay until time 5, I would have 00:15:13.990 --> 00:15:15.850 caught at least 1 fish. 00:15:15.850 --> 00:15:18.350 I might have caught more than 1. 00:15:18.350 --> 00:15:22.970 So the event of interest here is that the first catch 00:15:22.970 --> 00:15:26.560 happens between times 2 and 5. 00:15:26.560 --> 00:15:32.050 So one way of calculating this quantity would be-- 00:15:32.050 --> 00:15:35.300 Its the probability that the first catch happens between 00:15:35.300 --> 00:15:37.700 times 2 and 5. 00:15:37.700 --> 00:15:40.060 Another way to deal with it is to say, this is the 00:15:40.060 --> 00:15:44.880 probability that I caught 0 fish in the first 2 hours and 00:15:44.880 --> 00:15:49.170 then the probability that I catch at least 1 fish during 00:15:49.170 --> 00:15:51.130 the next 3 hours. 00:15:53.890 --> 00:15:54.780 This. 00:15:54.780 --> 00:15:56.080 What is this? 00:15:56.080 --> 00:15:59.180 The probability of 0 fish in the next 3 hours is the 00:15:59.180 --> 00:16:01.600 probability of 0 fish during this time. 00:16:01.600 --> 00:16:04.480 1 minus this is the probability of catching at 00:16:04.480 --> 00:16:07.850 least 1 fish, of having at least 1 arrival, 00:16:07.850 --> 00:16:09.730 between times 2 and 5. 00:16:09.730 --> 00:16:13.310 If there's at least 1 arrival between times 2 and 5, then I 00:16:13.310 --> 00:16:17.140 would have gone home by time 5. 00:16:17.140 --> 00:16:20.660 So both of these, if you plug-in numbers and all that, 00:16:20.660 --> 00:16:24.170 of course, are going to give you the same answer. 00:16:24.170 --> 00:16:26.820 Now next, what's the probability that I catch at 00:16:26.820 --> 00:16:29.560 least 2 fish? 00:16:29.560 --> 00:16:32.370 In which scenario are we? 00:16:32.370 --> 00:16:36.570 Under this scenario, I go home when I catch my first fish. 00:16:36.570 --> 00:16:39.560 So in order to catch at least 2 fish, it 00:16:39.560 --> 00:16:41.340 must be in this case. 00:16:41.340 --> 00:16:44.830 So this is the same as the event that I catch at least 2 00:16:44.830 --> 00:16:49.020 fish during the first 2 time steps. 00:16:49.020 --> 00:16:52.410 So it's going to be the probability from 2 to 00:16:52.410 --> 00:16:56.780 infinity, the probability that I catch 2 fish, or that I 00:16:56.780 --> 00:17:01.860 catch 3 fish, or I catch more than that. 00:17:01.860 --> 00:17:04.109 So it's this quantity. 00:17:04.109 --> 00:17:06.730 k is the number of fish that I catch. 00:17:06.730 --> 00:17:09.599 At least 2, so k goes from 2 to infinity. 00:17:09.599 --> 00:17:13.180 These are the probabilities of catching a number k of fish 00:17:13.180 --> 00:17:14.859 during this interval. 00:17:14.859 --> 00:17:17.920 And if you want a simpler form without an infinite sum, this 00:17:17.920 --> 00:17:20.619 would be 1 minus the probability of catching 0 00:17:20.619 --> 00:17:24.880 fish, minus the probability of catching 1 fish, during a time 00:17:24.880 --> 00:17:28.050 interval of length 2. 00:17:28.050 --> 00:17:29.520 Another way to think of it. 00:17:29.520 --> 00:17:34.230 I'm going to catch 2 fish, at least 2 fish, if and only if 00:17:34.230 --> 00:17:40.630 the second fish caught in this process happens before time 2. 00:17:40.630 --> 00:17:43.950 So that's another way of thinking about the same event. 00:17:43.950 --> 00:17:46.230 So it's going to be the probability that the random 00:17:46.230 --> 00:17:51.440 variable Y2, the arrival time over the second fish, is less 00:17:51.440 --> 00:17:52.690 than or equal to 2. 00:17:55.387 --> 00:17:56.310 OK. 00:17:56.310 --> 00:18:00.000 The next one is a little trickier. 00:18:00.000 --> 00:18:03.380 Here we need to do a little bit of divide and conquer. 00:18:03.380 --> 00:18:06.490 Overall, in this expedition, what the expected number of 00:18:06.490 --> 00:18:08.840 fish to be caught? 00:18:08.840 --> 00:18:11.550 One way to think about it is to try to use the total 00:18:11.550 --> 00:18:13.100 expectations theorem. 00:18:13.100 --> 00:18:17.830 And think of expected number of fish, given this scenario, 00:18:17.830 --> 00:18:21.010 or expected number of fish, given this scenario. 00:18:21.010 --> 00:18:24.190 That's a little more complicated than the way I'm 00:18:24.190 --> 00:18:25.290 going to do it. 00:18:25.290 --> 00:18:28.240 The way I'm going to do is to think as follows-- 00:18:28.240 --> 00:18:32.310 Expected number of fish is the expected number of fish caught 00:18:32.310 --> 00:18:37.520 between times 0 and 2 plus expected number of fish caught 00:18:37.520 --> 00:18:39.800 after time 2. 00:18:39.800 --> 00:18:45.580 So what's the expected number caught between time 0 and 2? 00:18:45.580 --> 00:18:47.860 This is lambda t. 00:18:47.860 --> 00:18:52.310 So lambda is 0.6 times 2. 00:18:52.310 --> 00:18:55.380 This is the expected number of fish that are caught between 00:18:55.380 --> 00:18:57.260 times 0 and 2. 00:18:57.260 --> 00:19:00.440 Now let's think about the expected number of fish caught 00:19:00.440 --> 00:19:01.630 afterwards. 00:19:01.630 --> 00:19:04.300 How many fish are being caught afterwards? 00:19:04.300 --> 00:19:06.110 Well it depends on the scenario. 00:19:06.110 --> 00:19:08.750 If we're in this scenario, we've gone home 00:19:08.750 --> 00:19:10.800 and we catch 0. 00:19:10.800 --> 00:19:14.570 If we're in this scenario, then we continue fishing until 00:19:14.570 --> 00:19:15.980 we catch one. 00:19:15.980 --> 00:19:19.970 So the expected number of fish to be caught after time 2 is 00:19:19.970 --> 00:19:24.520 going to be the probability of this scenario times 1. 00:19:24.520 --> 00:19:29.020 And the probability of that scenario is the probability 00:19:29.020 --> 00:19:33.490 that they call it's 0 fish during the first 2 time steps 00:19:33.490 --> 00:19:37.420 times 1, which is the number of fish I'm going to catch if 00:19:37.420 --> 00:19:39.790 I continue. 00:19:39.790 --> 00:19:43.960 The expected total fishing time we can calculate exactly 00:19:43.960 --> 00:19:46.150 the same way. 00:19:46.150 --> 00:19:47.890 I'm jumping to the last one. 00:19:47.890 --> 00:19:51.580 My total fishing time has a period of 2 time steps. 00:19:51.580 --> 00:19:54.910 I'm going to fish for 2 time steps no matter what. 00:19:54.910 --> 00:19:59.190 And then if I caught 0 fish, which happens with this 00:19:59.190 --> 00:20:04.540 probability, my expected time is going to be the expected 00:20:04.540 --> 00:20:08.920 time from here onwards, which is the expected value of this 00:20:08.920 --> 00:20:12.490 geometric random variable with parameter lambda. 00:20:12.490 --> 00:20:15.430 So the expected time is 1 over lambda. 00:20:15.430 --> 00:20:22.460 And in our case this, is 1/0.6. 00:20:22.460 --> 00:20:31.180 Finally, if I tell you that I have been fishing for 4 hours 00:20:31.180 --> 00:20:37.800 and nothing has been caught so far, how much do you expect 00:20:37.800 --> 00:20:41.630 this quantity to be? 00:20:41.630 --> 00:20:46.330 Here is the story that, again, that for the Poisson process 00:20:46.330 --> 00:20:48.720 used is as good as new. 00:20:48.720 --> 00:20:51.060 The process does not have any memory. 00:20:51.060 --> 00:20:54.930 Given what happens in the past doesn't matter for the future. 00:20:54.930 --> 00:20:58.430 It's as if the process starts new at this point in time. 00:20:58.430 --> 00:21:02.420 So this one is going to be, again, the same exponentially 00:21:02.420 --> 00:21:04.910 distributed random variable with the 00:21:04.910 --> 00:21:08.270 same parameter lambda. 00:21:08.270 --> 00:21:12.270 So expected time until an arrival comes is an 00:21:12.270 --> 00:21:13.740 exponential distribut -- 00:21:13.740 --> 00:21:15.910 has an exponential distribution with parameter 00:21:15.910 --> 00:21:19.660 lambda, no matter what has happened in the past. 00:21:19.660 --> 00:21:22.730 Starting from now and looking into the future, it's as if 00:21:22.730 --> 00:21:24.910 the process has just started. 00:21:24.910 --> 00:21:32.440 So it's going to be 1 over lambda, which is 1/0.6. 00:21:32.440 --> 00:21:33.690 OK. 00:21:37.540 --> 00:21:41.500 Now our next example is going to be a little more 00:21:41.500 --> 00:21:43.780 complicated or subtle. 00:21:43.780 --> 00:21:46.800 But before we get to the example, let's refresh our 00:21:46.800 --> 00:21:50.300 memory about what we discussed last time about merging 00:21:50.300 --> 00:21:53.110 Poisson independent Poisson processes. 00:21:53.110 --> 00:21:56.090 Instead of drawing the picture that way, another way we could 00:21:56.090 --> 00:21:58.260 draw it could be this. 00:21:58.260 --> 00:22:01.260 We have a Poisson process with rate lambda1, and a Poisson 00:22:01.260 --> 00:22:03.440 process with rate lambda2. 00:22:03.440 --> 00:22:07.320 They have, each one of these, have their arrivals. 00:22:07.320 --> 00:22:09.780 And then we form the merged process. 00:22:09.780 --> 00:22:13.580 And the merged process records an arrival whenever there's an 00:22:13.580 --> 00:22:16.930 arrival in either of the two processes. 00:22:19.990 --> 00:22:23.730 This process in that process are assumed to be independent 00:22:23.730 --> 00:22:26.760 of each other. 00:22:26.760 --> 00:22:32.590 Now different times in this process and that process are 00:22:32.590 --> 00:22:34.780 independent of each other. 00:22:34.780 --> 00:22:39.400 So what happens in these two time intervals is independent 00:22:39.400 --> 00:22:41.780 from what happens in these two time intervals. 00:22:41.780 --> 00:22:45.560 These two time intervals to determine what happens here. 00:22:45.560 --> 00:22:48.750 These two time intervals determine what happens there. 00:22:48.750 --> 00:22:53.740 So because these are independent from these, this 00:22:53.740 --> 00:22:56.600 means that this is also independent from that. 00:22:56.600 --> 00:22:59.020 So the independence assumption is satisfied 00:22:59.020 --> 00:23:01.150 for the merged process. 00:23:01.150 --> 00:23:05.030 And the merged process turns out to be a Poisson process. 00:23:05.030 --> 00:23:10.340 And if you want to find the arrival rate for that process, 00:23:10.340 --> 00:23:12.550 you argue as follows. 00:23:12.550 --> 00:23:15.000 During a little interval of length delta, we have 00:23:15.000 --> 00:23:17.280 probability lambda1 delta of having an 00:23:17.280 --> 00:23:18.620 arrival in this process. 00:23:18.620 --> 00:23:21.700 We have probability lambda2 delta of an arrival in this 00:23:21.700 --> 00:23:24.890 process, plus second order terms in 00:23:24.890 --> 00:23:26.860 delta, which we're ignoring. 00:23:26.860 --> 00:23:29.270 And then you do the calculation and you find that 00:23:29.270 --> 00:23:31.870 in this process, you're going to have an arrival 00:23:31.870 --> 00:23:37.830 probability, which is lambda1 plus lambda2, again ignoring 00:23:37.830 --> 00:23:40.490 second order in delta-- 00:23:40.490 --> 00:23:42.650 terms that are second order in delta. 00:23:42.650 --> 00:23:46.130 So the merged process is a Poisson process whose arrival 00:23:46.130 --> 00:23:48.760 rate is the sum of the arrival rates of 00:23:48.760 --> 00:23:52.080 the individual processes. 00:23:52.080 --> 00:23:55.290 And the calculation we did at the end of the last lecture-- 00:23:55.290 --> 00:23:59.240 If I tell you that the new arrival happened here, where 00:23:59.240 --> 00:24:00.610 did that arrival come from? 00:24:00.610 --> 00:24:02.910 Did it come from here or from there? 00:24:02.910 --> 00:24:06.720 If the lambda1 is equal to lambda2, then by symmetry you 00:24:06.720 --> 00:24:09.240 would say that it's equally likely to have come from here 00:24:09.240 --> 00:24:10.660 or to come from there. 00:24:10.660 --> 00:24:13.720 But if this lambda is much bigger than that lambda, the 00:24:13.720 --> 00:24:16.850 fact that they saw an arrival is more likely to have come 00:24:16.850 --> 00:24:17.850 from there. 00:24:17.850 --> 00:24:22.410 And the formula that captures this is the following. 00:24:22.410 --> 00:24:27.300 This is the probability that my arrival has come from this 00:24:27.300 --> 00:24:32.360 particular stream rather than that particular stream. 00:24:32.360 --> 00:24:38.900 So when an arrival comes and you ask, what is the origin of 00:24:38.900 --> 00:24:39.690 that arrival? 00:24:39.690 --> 00:24:43.760 It's as if I'm flipping a coin with these odds. 00:24:43.760 --> 00:24:46.910 And depending on outcome of that coin, I'm going to tell 00:24:46.910 --> 00:24:49.790 you came from there or it came from there. 00:24:49.790 --> 00:24:53.850 So the origin of an arrival is either this 00:24:53.850 --> 00:24:55.610 stream or that stream. 00:24:55.610 --> 00:24:58.190 And this is the probability that the origin of the arrival 00:24:58.190 --> 00:24:59.510 is that one. 00:24:59.510 --> 00:25:04.160 Now if we look at 2 different arrivals, and we ask about 00:25:04.160 --> 00:25:05.570 their origins-- 00:25:05.570 --> 00:25:08.130 So let's think about the origin of this arrival and 00:25:08.130 --> 00:25:12.060 compare it with the origin that arrival. 00:25:12.060 --> 00:25:14.010 The origin of this arrival is random. 00:25:14.010 --> 00:25:16.720 It could be right be either this or that. 00:25:16.720 --> 00:25:18.840 And this is the relevant probability. 00:25:18.840 --> 00:25:20.750 The origin of that arrival is random. 00:25:20.750 --> 00:25:24.360 It could be either here or is there, and again, with the 00:25:24.360 --> 00:25:26.880 same relevant probability. 00:25:26.880 --> 00:25:27.730 Question. 00:25:27.730 --> 00:25:31.780 The origin of this arrival, is it dependent or independent 00:25:31.780 --> 00:25:34.710 from the origin that arrival? 00:25:34.710 --> 00:25:37.500 And here's how the argument goes. 00:25:37.500 --> 00:25:40.740 Separate times are independent. 00:25:40.740 --> 00:25:45.050 Whatever has happened in the process during this set of 00:25:45.050 --> 00:25:48.040 times is independent from whatever happened in the 00:25:48.040 --> 00:25:50.980 process during that set of times. 00:25:50.980 --> 00:25:55.040 Because different times have nothing to do with each other, 00:25:55.040 --> 00:25:59.650 the origin of this, of an arrival here, has nothing to 00:25:59.650 --> 00:26:02.480 do with the origin of an arrival there. 00:26:02.480 --> 00:26:06.890 So the origins of different arrivals are also independent 00:26:06.890 --> 00:26:08.850 random variables. 00:26:08.850 --> 00:26:12.710 So if I tell you that-- 00:26:12.710 --> 00:26:14.150 yeah. 00:26:14.150 --> 00:26:15.310 OK. 00:26:15.310 --> 00:26:19.600 So it as if that each time that you have an arrival in 00:26:19.600 --> 00:26:22.820 the merge process, it's as if you're flipping a coin to 00:26:22.820 --> 00:26:26.410 determine where did that arrival came from and these 00:26:26.410 --> 00:26:31.516 coins are independent of each other. 00:26:31.516 --> 00:26:32.766 OK. 00:26:35.550 --> 00:26:35.920 OK. 00:26:35.920 --> 00:26:37.770 Now we're going to use this-- 00:26:37.770 --> 00:26:42.970 what we know about merged processes to solve the problem 00:26:42.970 --> 00:26:48.240 that would be harder to do, if you were not using ideas from 00:26:48.240 --> 00:26:49.720 Poisson processes. 00:26:49.720 --> 00:26:52.250 So the formulation of the problem has nothing to do with 00:26:52.250 --> 00:26:54.370 the Poisson process. 00:26:54.370 --> 00:26:57.450 The formulation is the following. 00:26:57.450 --> 00:26:59.870 We have 3 light-bulbs. 00:26:59.870 --> 00:27:03.490 And each light bulb is independent and is going to 00:27:03.490 --> 00:27:07.920 die out at the time that's exponentially distributed. 00:27:07.920 --> 00:27:11.170 So 3 light bulbs. 00:27:11.170 --> 00:27:16.630 They start their lives and then at some point 00:27:16.630 --> 00:27:21.260 they die or burn out. 00:27:21.260 --> 00:27:26.150 So let's think of this as X, this as Y, and this as Z. 00:27:26.150 --> 00:27:31.220 And we're interested in the time until the last 00:27:31.220 --> 00:27:33.200 light-bulb burns out. 00:27:33.200 --> 00:27:36.930 So we're interested in the maximum of the 3 random 00:27:36.930 --> 00:27:41.480 variables, X, Y, and Z. And in particular, we want to find 00:27:41.480 --> 00:27:43.170 the expected value of this maximum. 00:27:45.770 --> 00:27:47.490 OK. 00:27:47.490 --> 00:27:50.760 So you can do derived distribution, use the expected 00:27:50.760 --> 00:27:52.880 value rule, anything you want. 00:27:52.880 --> 00:27:56.230 You can get this answer using the tools that you already 00:27:56.230 --> 00:27:58.180 have in your hands. 00:27:58.180 --> 00:28:02.070 But now let us see how we can connect to this picture with a 00:28:02.070 --> 00:28:05.550 Poisson picture and come up with the answer in a very 00:28:05.550 --> 00:28:07.240 simple way. 00:28:07.240 --> 00:28:09.630 What is an exponential random variable? 00:28:09.630 --> 00:28:14.450 An exponential random variable is the first act in the long 00:28:14.450 --> 00:28:19.570 play that involves a whole Poisson process. 00:28:19.570 --> 00:28:23.020 So an exponential random variable is the first act of a 00:28:23.020 --> 00:28:24.650 Poisson movie. 00:28:24.650 --> 00:28:25.660 Same thing here. 00:28:25.660 --> 00:28:29.700 You can think of this random variable as being part of some 00:28:29.700 --> 00:28:31.850 Poisson process that has been running. 00:28:35.360 --> 00:28:38.040 So it's part of this bigger picture. 00:28:38.040 --> 00:28:42.370 We're still interested in the maximum of the 3. 00:28:42.370 --> 00:28:45.780 The other arrivals are not going to affect our answers. 00:28:45.780 --> 00:28:49.640 It's just, conceptually speaking, we can think of the 00:28:49.640 --> 00:28:52.840 exponential random variable as being embedded in a bigger 00:28:52.840 --> 00:28:55.110 Poisson picture. 00:28:55.110 --> 00:29:00.980 So we have 3 Poisson process that are running in parallel. 00:29:00.980 --> 00:29:06.150 Let us split the expected time until the last burnout into 00:29:06.150 --> 00:29:09.800 pieces, which is time until the first burnout, time from 00:29:09.800 --> 00:29:11.810 the first until the second, and time from the 00:29:11.810 --> 00:29:13.690 second until the third. 00:29:16.780 --> 00:29:20.570 And find the expected values of each one of these pieces. 00:29:20.570 --> 00:29:24.620 What can we say about the expected value of this? 00:29:24.620 --> 00:29:29.310 This is the first arrival out of all of 00:29:29.310 --> 00:29:31.540 these 3 Poisson processes. 00:29:31.540 --> 00:29:34.070 It's the first event that happens when you look at all 00:29:34.070 --> 00:29:36.080 of these processes simultaneously. 00:29:36.080 --> 00:29:39.660 So 3 Poisson processes running in parallel. 00:29:39.660 --> 00:29:43.750 We're interested in the time until one of them, any one of 00:29:43.750 --> 00:29:46.380 them, gets in arrival. 00:29:46.380 --> 00:29:47.690 Rephrase. 00:29:47.690 --> 00:29:51.330 We merged the 3 Poisson processes, and we ask for the 00:29:51.330 --> 00:29:56.820 time until we observe an arrival in the merged process. 00:29:56.820 --> 00:30:01.250 When 1 of the 3 gets an arrival for the first time, 00:30:01.250 --> 00:30:03.880 the merged process gets its first arrival. 00:30:03.880 --> 00:30:06.300 So what's the expected value of this time 00:30:06.300 --> 00:30:08.820 until the first burnout? 00:30:08.820 --> 00:30:11.940 It's going to be the expected value of a 00:30:11.940 --> 00:30:13.720 Poisson random variable. 00:30:13.720 --> 00:30:17.050 So the first burnout is going to have an expected 00:30:17.050 --> 00:30:20.430 value, which is-- 00:30:20.430 --> 00:30:21.540 OK. 00:30:21.540 --> 00:30:23.690 It's a Poisson process. 00:30:23.690 --> 00:30:28.530 The merged process of the 3 has a collective arrival rate, 00:30:28.530 --> 00:30:32.750 which is 3 times lambda. 00:30:32.750 --> 00:30:36.250 So this is the parameter over the exponential distribution 00:30:36.250 --> 00:30:39.870 that describes the time until the first arrival in the 00:30:39.870 --> 00:30:41.220 merged process. 00:30:41.220 --> 00:30:42.990 And the expected value of this random 00:30:42.990 --> 00:30:45.670 variable is 1 over that. 00:30:45.670 --> 00:30:48.190 When you have an exponential random variable with parameter 00:30:48.190 --> 00:30:50.150 lambda, the expected value of that random 00:30:50.150 --> 00:30:52.330 variable is 1 over lambda. 00:30:52.330 --> 00:30:56.660 Here we're talking about the first arrival time in a 00:30:56.660 --> 00:30:58.720 process with rate 3 lambda. 00:30:58.720 --> 00:31:00.680 The expected time until the first arrival 00:31:00.680 --> 00:31:03.000 is 1 over (3 lambda). 00:31:03.000 --> 00:31:03.870 Alright. 00:31:03.870 --> 00:31:08.710 So at this time, this bulb, this arrival happened, this 00:31:08.710 --> 00:31:11.490 bulb has been burned. 00:31:11.490 --> 00:31:15.760 So we don't care about that bulb anymore. 00:31:15.760 --> 00:31:21.610 We start at this time, and we look forward. 00:31:21.610 --> 00:31:23.640 This bulb has been burned. 00:31:23.640 --> 00:31:27.810 So let's just look forward from now on. 00:31:27.810 --> 00:31:28.900 What have we got? 00:31:28.900 --> 00:31:34.030 We have two bulbs that are burning. 00:31:34.030 --> 00:31:37.320 We have a Poisson process that's the bigger picture of 00:31:37.320 --> 00:31:40.270 what could happen to that light bulb, if we were to keep 00:31:40.270 --> 00:31:41.190 replacing it. 00:31:41.190 --> 00:31:42.880 Another Poisson process. 00:31:42.880 --> 00:31:45.610 These two processes are, again, independent. 00:31:45.610 --> 00:31:50.850 From this time until that time, how long does it take? 00:31:50.850 --> 00:31:53.930 It's the time until either this process records an 00:31:53.930 --> 00:31:57.090 arrival or that process records and arrival. 00:31:57.090 --> 00:32:01.210 That's the same as the time that the merged process of 00:32:01.210 --> 00:32:03.810 these two records an arrival. 00:32:03.810 --> 00:32:06.430 So we're talking about the expected time until the first 00:32:06.430 --> 00:32:08.710 arrival in a merged process. 00:32:08.710 --> 00:32:11.030 The merged process is Poisson. 00:32:11.030 --> 00:32:14.240 It's Poisson with rate 2 lambda. 00:32:14.240 --> 00:32:17.690 So that extra time is going to take-- 00:32:17.690 --> 00:32:21.390 the expected value is going to be 1 over the (rate of that 00:32:21.390 --> 00:32:22.580 Poisson process). 00:32:22.580 --> 00:32:25.170 So 1 over (2 lambda) is the expected value 00:32:25.170 --> 00:32:26.980 of this random variable. 00:32:26.980 --> 00:32:30.870 So at this point, this bulb now is also burned. 00:32:30.870 --> 00:32:33.620 So we start looking from this time on. 00:32:33.620 --> 00:32:37.110 That part of the picture disappears. 00:32:37.110 --> 00:32:40.150 Starting from this time, what's the expected value 00:32:40.150 --> 00:32:43.650 until that remaining light-bulb burns out? 00:32:43.650 --> 00:32:47.130 Well, as we said before, in a Poisson process or with 00:32:47.130 --> 00:32:50.090 exponential random variables, we have memorylessness. 00:32:50.090 --> 00:32:53.120 A used bulb is as good as a new one. 00:32:53.120 --> 00:32:55.990 So it's as if we're starting from scratch here. 00:32:55.990 --> 00:32:58.700 So this is going to be an exponential random variable 00:32:58.700 --> 00:33:00.690 with parameter lambda. 00:33:00.690 --> 00:33:05.540 And the expected value of it is going to be 1 over lambda. 00:33:05.540 --> 00:33:07.990 So the beauty of approaching this problem in this 00:33:07.990 --> 00:33:10.930 particular way is, of course, that we manage to do 00:33:10.930 --> 00:33:14.100 everything without any calculus at all, without 00:33:14.100 --> 00:33:16.990 striking an integral, without trying to calculate 00:33:16.990 --> 00:33:19.220 expectations in any form. 00:33:19.220 --> 00:33:23.150 Most of the non-trivial problems that you encounter in 00:33:23.150 --> 00:33:28.540 the Poisson world basically involve tricks of these kind. 00:33:28.540 --> 00:33:31.830 You have a question and you try to rephrase it, trying to 00:33:31.830 --> 00:33:35.240 think in terms of what might happen in the Poisson setting, 00:33:35.240 --> 00:33:39.200 use memorylessness, use merging, et cetera, et cetera. 00:33:43.360 --> 00:33:46.080 Now we talked about merging. 00:33:46.080 --> 00:33:49.480 It turns out that the splitting of Poisson processes 00:33:49.480 --> 00:33:53.400 also works in a nice way. 00:33:53.400 --> 00:33:57.160 The story here is exactly the same as for 00:33:57.160 --> 00:33:58.820 the Bernoulli process. 00:33:58.820 --> 00:34:01.870 So I'm having a Poisson process. 00:34:01.870 --> 00:34:06.060 And each time, with some rate lambda, and each time that an 00:34:06.060 --> 00:34:09.790 arrival comes, I'm going to send it to that stream and the 00:34:09.790 --> 00:34:13.179 record an arrival here with some probability P. And I'm 00:34:13.179 --> 00:34:16.120 going to send it to the other stream with some probability 1 00:34:16.120 --> 00:34:19.469 minus P. So either of this will happen or that will 00:34:19.469 --> 00:34:21.940 happen, depending on the outcome of the 00:34:21.940 --> 00:34:23.550 coin flip that I do. 00:34:23.550 --> 00:34:27.449 Each time that then arrival occurs, I flip a coin and I 00:34:27.449 --> 00:34:30.929 decide whether to record it here or there. 00:34:30.929 --> 00:34:32.620 This is called splitting a Poisson 00:34:32.620 --> 00:34:34.719 process into two pieces. 00:34:34.719 --> 00:34:37.120 What kind of process do we get here? 00:34:37.120 --> 00:34:40.250 If you look at the little interval for length delta, 00:34:40.250 --> 00:34:41.810 what's the probability that this little 00:34:41.810 --> 00:34:44.090 interval gets an arrival? 00:34:44.090 --> 00:34:47.739 It's the probability that this one gets an arrival, which is 00:34:47.739 --> 00:34:51.260 lambda delta times the probability that after I get 00:34:51.260 --> 00:34:55.210 an arrival my coin flip came out to be that way, so that it 00:34:55.210 --> 00:34:56.270 sends me there. 00:34:56.270 --> 00:34:58.740 So this means that this little interval is going to have 00:34:58.740 --> 00:35:03.620 probability lambda delta P. Or maybe more suggestively, I 00:35:03.620 --> 00:35:09.480 should write it as lambda P times delta. 00:35:09.480 --> 00:35:12.350 So every little interval has a probability of an arrival 00:35:12.350 --> 00:35:13.470 proportional to delta. 00:35:13.470 --> 00:35:16.780 The proportionality factor is lambda P. So lambda P is the 00:35:16.780 --> 00:35:18.590 rate of that process. 00:35:18.590 --> 00:35:22.500 And then you go through the mental exercise that you went 00:35:22.500 --> 00:35:25.170 through for the Bernoulli process to argue that a 00:35:25.170 --> 00:35:28.520 different intervals here are independent and so on. 00:35:28.520 --> 00:35:31.710 And that completes checking that this process is going to 00:35:31.710 --> 00:35:33.360 be a Poisson process. 00:35:33.360 --> 00:35:38.060 So when you split a Poisson process by doing independent 00:35:38.060 --> 00:35:41.040 coin flips each time that something happens, the 00:35:41.040 --> 00:35:44.330 processes that you get is again a Poisson process, but 00:35:44.330 --> 00:35:46.490 of course with a reduced rate. 00:35:46.490 --> 00:35:50.040 So instead of the word splitting, sometimes people 00:35:50.040 --> 00:35:54.330 also use the words thinning-out. 00:35:54.330 --> 00:35:57.650 That is, out of the arrivals that came, you keep a few but 00:35:57.650 --> 00:35:59.000 throw away a few. 00:36:01.820 --> 00:36:02.730 OK. 00:36:02.730 --> 00:36:08.570 So now the last topic over this lecture is a quite 00:36:08.570 --> 00:36:11.270 curious phenomenon that goes under the 00:36:11.270 --> 00:36:12.595 name of random incidents. 00:36:15.550 --> 00:36:18.950 So here's the story. 00:36:18.950 --> 00:36:22.550 Buses have been running on Mass Ave. from time 00:36:22.550 --> 00:36:24.070 immemorial. 00:36:24.070 --> 00:36:29.060 And the bus company that runs the buses claims that they 00:36:29.060 --> 00:36:33.150 come as a Poisson process with some rate, let's say, of 4 00:36:33.150 --> 00:36:34.970 buses per hour. 00:36:34.970 --> 00:36:39.250 So that the expected time between bus arrivals is going 00:36:39.250 --> 00:36:42.500 to be 15 minutes. 00:36:42.500 --> 00:36:45.180 OK. 00:36:45.180 --> 00:36:45.840 Alright. 00:36:45.840 --> 00:36:48.130 So people have been complaining that they have 00:36:48.130 --> 00:36:49.150 been showing up there. 00:36:49.150 --> 00:36:51.500 They think the buses are taking too long. 00:36:51.500 --> 00:36:54.270 So you are asked to investigate. 00:36:54.270 --> 00:36:56.840 Is the company-- 00:36:56.840 --> 00:37:00.730 Does it operate according to its promises or not. 00:37:00.730 --> 00:37:05.880 So you send an undercover agent to go and check the 00:37:05.880 --> 00:37:07.940 interarrival times of the buses. 00:37:07.940 --> 00:37:09.660 Are they 15 minutes? 00:37:09.660 --> 00:37:11.690 Or are they longer? 00:37:11.690 --> 00:37:17.660 So you put your dark glasses and you show up at the bus 00:37:17.660 --> 00:37:21.110 stop at some random time. 00:37:21.110 --> 00:37:25.530 And you go and ask the guy in the falafel truck, how long 00:37:25.530 --> 00:37:28.370 has it been since the last arrival? 00:37:28.370 --> 00:37:31.310 So of course that guy works for the FBI, right? 00:37:31.310 --> 00:37:36.900 So they tell you, well, it's been, let's say, 12 minutes 00:37:36.900 --> 00:37:39.360 since the last bus arrival. 00:37:39.360 --> 00:37:40.960 And then you say, "Oh, 12 minutes. 00:37:40.960 --> 00:37:42.780 Average time is 15. 00:37:42.780 --> 00:37:47.000 So a bus should be coming any time now." 00:37:47.000 --> 00:37:48.230 Is that correct? 00:37:48.230 --> 00:37:49.660 No, you wouldn't think that way. 00:37:49.660 --> 00:37:51.010 It's a Poisson process. 00:37:51.010 --> 00:37:53.810 It doesn't matter how long it has been since 00:37:53.810 --> 00:37:55.270 the last bus arrival. 00:37:55.270 --> 00:37:56.920 So you don't go through that fallacy. 00:37:56.920 --> 00:37:59.970 Instead of predicting how long it's going to be, you just sit 00:37:59.970 --> 00:38:03.300 down there and wait and measure the time. 00:38:03.300 --> 00:38:08.820 And you find that this is, let's say, 11 minutes. 00:38:08.820 --> 00:38:13.260 And you go to your boss and report, "Well, it took-- 00:38:13.260 --> 00:38:16.410 I went there and the time from the previous bus to the next 00:38:16.410 --> 00:38:18.310 one was 23 minutes. 00:38:18.310 --> 00:38:20.360 It's more than the 15 that they said." 00:38:20.360 --> 00:38:21.830 So go and do that again. 00:38:21.830 --> 00:38:23.590 You go day after day. 00:38:23.590 --> 00:38:28.350 You keep these statistics of the length of this interval. 00:38:28.350 --> 00:38:32.160 And you tell your boss it's a lot more than 15. 00:38:32.160 --> 00:38:36.720 It tends to be more like 30 or so. 00:38:36.720 --> 00:38:39.170 So the bus company is cheating us. 00:38:39.170 --> 00:38:43.490 Does the bus company really run Poisson buses at the rate 00:38:43.490 --> 00:38:46.490 that they have promised? 00:38:46.490 --> 00:38:51.270 Well let's analyze the situation here and figure out 00:38:51.270 --> 00:38:55.010 what the length of this interval 00:38:55.010 --> 00:38:57.900 should be, on the average. 00:38:57.900 --> 00:39:01.120 The naive argument is that this interval is an 00:39:01.120 --> 00:39:02.590 interarrival time. 00:39:02.590 --> 00:39:06.410 And interarrival times, on the average, are 15 minutes, if 00:39:06.410 --> 00:39:10.610 the company runs indeed Poisson processes with these 00:39:10.610 --> 00:39:11.850 interarrival times. 00:39:11.850 --> 00:39:14.970 But actually the situation is a little more subtle because 00:39:14.970 --> 00:39:19.940 this is not a typical interarrival interval. 00:39:19.940 --> 00:39:23.440 This interarrival interval consists of two pieces. 00:39:23.440 --> 00:39:28.810 Let's call them T1 and T1 prime. 00:39:28.810 --> 00:39:32.250 What can you tell me about those two random variables? 00:39:32.250 --> 00:39:35.940 What kind of random variable is T1? 00:39:35.940 --> 00:39:39.950 Starting from this time, with the Poisson process, the past 00:39:39.950 --> 00:39:41.290 doesn't matter. 00:39:41.290 --> 00:39:43.870 It's the time until an arrival happens. 00:39:43.870 --> 00:39:49.110 So T1 is going to be an exponential random variable 00:39:49.110 --> 00:39:50.425 with parameter lambda. 00:39:53.300 --> 00:39:56.620 So in particular, the expected value of T1 is 00:39:56.620 --> 00:40:00.260 going to be 15 by itself. 00:40:00.260 --> 00:40:02.720 How about the random variable T1 prime. 00:40:02.720 --> 00:40:07.130 What kind of random variable is it? 00:40:07.130 --> 00:40:14.180 This is like the first arrival in a Poisson process that runs 00:40:14.180 --> 00:40:17.650 backwards in time. 00:40:17.650 --> 00:40:20.330 What kind of process is a Poisson process running 00:40:20.330 --> 00:40:21.200 backwards in time? 00:40:21.200 --> 00:40:23.030 Let's think of coin flips. 00:40:23.030 --> 00:40:26.130 Suppose you have a movie of coin flips. 00:40:26.130 --> 00:40:29.480 And for some accident, that fascinating movie, you happen 00:40:29.480 --> 00:40:31.100 to watch it backwards. 00:40:31.100 --> 00:40:33.610 Will it look any different statistically? 00:40:33.610 --> 00:40:33.780 No. 00:40:33.780 --> 00:40:36.940 It's going to be just the sequence of random coin flips. 00:40:36.940 --> 00:40:40.770 So a Bernoulli process that's runs in reverse time is 00:40:40.770 --> 00:40:42.410 statistically identical to a Bernoulli 00:40:42.410 --> 00:40:44.290 process in forward time. 00:40:44.290 --> 00:40:46.600 The Poisson process is a limit of the Bernoulli. 00:40:46.600 --> 00:40:48.950 So, same story with the Poisson process. 00:40:48.950 --> 00:40:51.410 If you run it backwards in time it looks the same. 00:40:51.410 --> 00:40:55.190 So looking backwards in time, this is a Poisson process. 00:40:55.190 --> 00:40:58.930 And T1 prime is the time until the first arrival in this 00:40:58.930 --> 00:41:00.260 backward process. 00:41:00.260 --> 00:41:04.910 So T1 prime is also going to be an exponential random 00:41:04.910 --> 00:41:07.340 variable with the same parameter, lambda. 00:41:07.340 --> 00:41:11.000 And the expected value of T1 prime is 15. 00:41:11.000 --> 00:41:15.860 Conclusion is that the expected length of this 00:41:15.860 --> 00:41:22.860 interval is going to be 30 minutes. 00:41:22.860 --> 00:41:26.690 And the fact that this agent found the average to be 00:41:26.690 --> 00:41:31.230 something like 30 does not contradict the claims of the 00:41:31.230 --> 00:41:35.010 bus company that they're running Poisson buses with a 00:41:35.010 --> 00:41:38.370 rate of lambda equal to 4. 00:41:38.370 --> 00:41:38.780 OK. 00:41:38.780 --> 00:41:43.390 So maybe the company can this way-- they can defend 00:41:43.390 --> 00:41:44.970 themselves in court. 00:41:44.970 --> 00:41:47.490 But there's something puzzling here. 00:41:47.490 --> 00:41:50.360 How long is the interarrival time? 00:41:50.360 --> 00:41:51.910 Is it 15? 00:41:51.910 --> 00:41:53.216 Or is it 30? 00:41:53.216 --> 00:41:55.750 On the average. 00:41:55.750 --> 00:41:59.960 The issue is what do we mean by a typical 00:41:59.960 --> 00:42:01.360 interarrival time. 00:42:01.360 --> 00:42:04.940 When we say typical, we mean some kind of average. 00:42:04.940 --> 00:42:08.690 But average over what? 00:42:08.690 --> 00:42:13.280 And here's two different ways of thinking about averages. 00:42:13.280 --> 00:42:15.080 You number the buses. 00:42:15.080 --> 00:42:17.120 And you have bus number 100. 00:42:17.120 --> 00:42:21.120 You have bus number 101, bus number 102, bus 00:42:21.120 --> 00:42:24.660 number 110, and so on. 00:42:24.660 --> 00:42:29.370 One way of thinking about averages is that you pick a 00:42:29.370 --> 00:42:32.150 bus number at random. 00:42:32.150 --> 00:42:36.070 I pick, let's say, that bus, all buses being sort of 00:42:36.070 --> 00:42:37.760 equally likely to be picked. 00:42:37.760 --> 00:42:41.610 And I measure this interarrival time. 00:42:41.610 --> 00:42:45.380 So for a typical bus. 00:42:45.380 --> 00:42:50.390 Then, starting from here until there, the expected time has 00:42:50.390 --> 00:42:56.600 to be 1 over lambda, for the Poisson process. 00:42:56.600 --> 00:42:58.370 But what we did in this experiment 00:42:58.370 --> 00:42:59.720 was something different. 00:42:59.720 --> 00:43:02.040 We didn't pick a bus at random. 00:43:02.040 --> 00:43:05.090 We picked a time at random. 00:43:05.090 --> 00:43:08.870 And if the picture is, let's say, this way, I'm much more 00:43:08.870 --> 00:43:12.770 likely to pick this interval and therefore this 00:43:12.770 --> 00:43:16.290 interarrival time, rather than that interval. 00:43:16.290 --> 00:43:20.480 Because, this interval corresponds to very few times. 00:43:20.480 --> 00:43:23.680 So if I'm picking a time at random and, in some sense, 00:43:23.680 --> 00:43:27.430 let's say, uniform, so that all times are equally likely, 00:43:27.430 --> 00:43:31.190 I'm much more likely to fall inside a big interval rather 00:43:31.190 --> 00:43:32.710 than a small interval. 00:43:32.710 --> 00:43:37.140 So a person who shows up at the bus stop at a random time. 00:43:37.140 --> 00:43:42.040 They're selecting an interval in a biased way, with the bias 00:43:42.040 --> 00:43:44.350 favor of longer intervals. 00:43:44.350 --> 00:43:47.850 And that's why what they observe is a random variable 00:43:47.850 --> 00:43:51.830 that has a larger expected value then the ordinary 00:43:51.830 --> 00:43:53.080 expected value. 00:43:53.080 --> 00:43:56.780 So the subtlety here is to realize that we're talking 00:43:56.780 --> 00:43:59.590 between two different kinds of experiments. 00:43:59.590 --> 00:44:05.250 Picking a bus number at random verses picking an interval at 00:44:05.250 --> 00:44:11.500 random with a bias in favor of longer intervals. 00:44:11.500 --> 00:44:14.840 Lots of paradoxes that one can cook up using Poisson 00:44:14.840 --> 00:44:19.190 processes and random processes in general often have to do 00:44:19.190 --> 00:44:21.340 with the story of this kind. 00:44:21.340 --> 00:44:24.780 The phenomenon that we had in this particular example also 00:44:24.780 --> 00:44:28.970 shows up in general, whenever you have other kinds of 00:44:28.970 --> 00:44:30.470 arrival processes. 00:44:30.470 --> 00:44:34.210 So the Poisson process is the simplest arrival process there 00:44:34.210 --> 00:44:36.830 is, where the interarrival times are 00:44:36.830 --> 00:44:38.820 exponential random variables. 00:44:38.820 --> 00:44:40.280 There's a larger class of models. 00:44:40.280 --> 00:44:43.580 They're called renewal processes, in which, again, we 00:44:43.580 --> 00:44:46.900 have a sequence of successive arrivals, interarrival times 00:44:46.900 --> 00:44:50.100 are identically distributed and independent, but they may 00:44:50.100 --> 00:44:52.320 come from a general distribution. 00:44:52.320 --> 00:44:55.100 So to make the same point of the previous example but in a 00:44:55.100 --> 00:44:59.250 much simpler setting, suppose that bus interarrival times 00:44:59.250 --> 00:45:02.830 are either 5 or 10 minutes apart. 00:45:02.830 --> 00:45:05.930 So you get some intervals that are of length 5. 00:45:05.930 --> 00:45:08.790 You get some that are of length 10. 00:45:08.790 --> 00:45:12.810 And suppose that these are equally likely. 00:45:12.810 --> 00:45:16.990 So we have -- not exactly -- 00:45:16.990 --> 00:45:20.380 In the long run, we have as many 5 minute intervals as we 00:45:20.380 --> 00:45:22.490 have 10 minute intervals. 00:45:22.490 --> 00:45:30.590 So the average interarrival time is 7 and 1/2. 00:45:30.590 --> 00:45:35.850 But if a person shows up at a random time, what are they 00:45:35.850 --> 00:45:37.100 going to see? 00:45:40.520 --> 00:45:43.150 Do we have as many 5s as 10s? 00:45:43.150 --> 00:45:47.490 But every 10 covers twice as much space. 00:45:47.490 --> 00:45:52.640 So if I show up at a random time, I have probability 2/3 00:45:52.640 --> 00:45:57.180 falling inside an interval of duration 10. 00:45:57.180 --> 00:46:00.990 And I have one 1/3 probability of falling inside an interval 00:46:00.990 --> 00:46:02.460 of duration 5. 00:46:02.460 --> 00:46:06.710 That's because, out of the whole real line, 2/3 of it is 00:46:06.710 --> 00:46:08.810 covered by intervals of length 10, just 00:46:08.810 --> 00:46:09.590 because they're longer. 00:46:09.590 --> 00:46:12.280 1/3 is covered by the smaller intervals. 00:46:12.280 --> 00:46:19.530 Now if I fall inside an interval of length 10 and I 00:46:19.530 --> 00:46:23.260 measure the length of the interval that I fell into, 00:46:23.260 --> 00:46:25.030 that's going to be 10. 00:46:25.030 --> 00:46:27.780 But if I fall inside an interval of length 5 and I 00:46:27.780 --> 00:46:30.320 measure how long it is, I'm going to get a 5. 00:46:30.320 --> 00:46:37.270 And that, of course, is going to be different than 7.5. 00:46:37.270 --> 00:46:38.010 OK. 00:46:38.010 --> 00:46:42.310 And which number should be bigger? 00:46:42.310 --> 00:46:45.110 It's the second number that's bigger because this one is 00:46:45.110 --> 00:46:48.930 biased in favor of the longer intervals. 00:46:48.930 --> 00:46:51.380 So that's, again, another illustration of the different 00:46:51.380 --> 00:46:54.640 results that you get when you have this random incidence 00:46:54.640 --> 00:46:55.990 phenomenon. 00:46:55.990 --> 00:46:59.320 So the bottom line, again, is that if you talk about a 00:46:59.320 --> 00:47:03.380 typical interarrival time, one must be very precise in 00:47:03.380 --> 00:47:05.370 specifying what we mean typical. 00:47:05.370 --> 00:47:08.120 So typical means sort of random. 00:47:08.120 --> 00:47:11.250 But to use the word random, you must specify very 00:47:11.250 --> 00:47:15.070 precisely what is the random experiment that you are using. 00:47:15.070 --> 00:47:18.920 And if you're not careful, you can get into apparent puzzles, 00:47:18.920 --> 00:47:20.770 such as the following. 00:47:20.770 --> 00:47:25.170 Suppose somebody tells you the average family size is 4, but 00:47:25.170 --> 00:47:30.340 the average person lives in a family of size 6. 00:47:30.340 --> 00:47:33.330 Is that compatible? 00:47:33.330 --> 00:47:36.610 Family size is 4 on the average, but typical people 00:47:36.610 --> 00:47:40.110 live, on the average, in families of size 6. 00:47:40.110 --> 00:47:41.590 Well yes. 00:47:41.590 --> 00:47:43.080 There's no contradiction here. 00:47:43.080 --> 00:47:45.450 We're talking about two different experiments. 00:47:45.450 --> 00:47:50.000 In one experiment, I pick a family at random, and I tell 00:47:50.000 --> 00:47:51.960 you the average family is 4. 00:47:51.960 --> 00:47:55.910 In another experiment, I pick a person at random and I tell 00:47:55.910 --> 00:47:58.310 you that this person, on the average, will be in their 00:47:58.310 --> 00:48:00.080 family of size 6. 00:48:00.080 --> 00:48:01.140 And what is the catch here? 00:48:01.140 --> 00:48:05.440 That if I pick a person at random, large families are 00:48:05.440 --> 00:48:08.160 more likely to be picked. 00:48:08.160 --> 00:48:11.710 So there's a bias in favor of large families. 00:48:11.710 --> 00:48:15.270 Or if you want to survey, let's say, are trains crowded 00:48:15.270 --> 00:48:16.495 in your city? 00:48:16.495 --> 00:48:19.170 Or are buses crowded? 00:48:19.170 --> 00:48:22.040 One choice is to pick a bus at random and inspect 00:48:22.040 --> 00:48:23.220 how crowded it is. 00:48:23.220 --> 00:48:27.260 Another choice is to pick a typical person and ask them, 00:48:27.260 --> 00:48:29.080 "Did you ride the bus today? 00:48:29.080 --> 00:48:33.500 Was it's crowded?" Well suppose that in this city 00:48:33.500 --> 00:48:36.265 there's one bus that's extremely crowded and all the 00:48:36.265 --> 00:48:38.520 other buses are completely empty. 00:48:38.520 --> 00:48:42.300 If you ask a person. "Was your bus crowded?" They will tell 00:48:42.300 --> 00:48:46.040 you, "Yes, my bus was crowded." There's no witness 00:48:46.040 --> 00:48:49.460 from the empty buses to testify in their favor. 00:48:49.460 --> 00:48:52.780 So by sampling people instead of sampling buses, you're 00:48:52.780 --> 00:48:54.940 going to get different result. 00:48:54.940 --> 00:48:58.320 And in the process industry, if your job is to inspect and 00:48:58.320 --> 00:49:01.450 check cookies, you will be faced with a big dilemma. 00:49:01.450 --> 00:49:05.190 Do you want to find out how many chocolate chips there are 00:49:05.190 --> 00:49:06.940 on a typical cookie? 00:49:06.940 --> 00:49:09.990 Are you going to interview cookies or are you going to 00:49:09.990 --> 00:49:13.880 interview chocolate chips and ask them how many other chips 00:49:13.880 --> 00:49:16.520 where there on your cookie? 00:49:16.520 --> 00:49:18.020 And you're going to get different 00:49:18.020 --> 00:49:19.210 answers in these cases. 00:49:19.210 --> 00:49:22.670 So moral is, one has to be very precise on how you 00:49:22.670 --> 00:49:26.160 formulate the sampling procedure that you have. 00:49:26.160 --> 00:49:28.330 And you'll get different answers.