WEBVTT 00:00:00.000 --> 00:00:00.040 align:middle line:90% 00:00:00.040 --> 00:00:02.460 align:middle line:84% The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 align:middle line:90% Commons license. 00:00:03.870 --> 00:00:06.910 align:middle line:84% Your support will help MIT OpenCourseWare continue to 00:00:06.910 --> 00:00:10.560 align:middle line:84% offer high quality educational resources for free. 00:00:10.560 --> 00:00:13.460 align:middle line:84% To make a donation or view additional materials from 00:00:13.460 --> 00:00:19.290 align:middle line:84% hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.290 --> 00:00:20.540 align:middle line:90% ocw.mit.edu. 00:00:20.540 --> 00:00:22.840 align:middle line:90% 00:00:22.840 --> 00:00:25.260 align:middle line:84% JOHN TSITSIKLIS: Today we're going to finish our discussion 00:00:25.260 --> 00:00:27.480 align:middle line:90% of the Poisson process. 00:00:27.480 --> 00:00:31.280 align:middle line:84% We're going to see a few of its properties, do a few 00:00:31.280 --> 00:00:35.660 align:middle line:84% interesting problems, some more interesting than others. 00:00:35.660 --> 00:00:39.000 align:middle line:84% So go through a few examples and then we're going to talk 00:00:39.000 --> 00:00:42.170 align:middle line:84% about some quite strange things that happen with the 00:00:42.170 --> 00:00:43.980 align:middle line:90% Poisson process. 00:00:43.980 --> 00:00:46.500 align:middle line:84% So the first thing is to remember what the Poisson 00:00:46.500 --> 00:00:48.200 align:middle line:90% processes is. 00:00:48.200 --> 00:00:52.300 align:middle line:84% It's a model, let's say, of arrivals of customers that 00:00:52.300 --> 00:00:55.940 align:middle line:84% are, in some sense, quote unquote, completely random, 00:00:55.940 --> 00:00:58.990 align:middle line:84% that is a customer can arrive at any point in time. 00:00:58.990 --> 00:01:01.670 align:middle line:84% All points in time are equally likely. 00:01:01.670 --> 00:01:05.330 align:middle line:84% And different points in time are sort of independent of 00:01:05.330 --> 00:01:06.450 align:middle line:90% other points in time. 00:01:06.450 --> 00:01:10.160 align:middle line:84% So the fact that I got an arrival now doesn't tell me 00:01:10.160 --> 00:01:13.050 align:middle line:84% anything about whether there's going to be an arrival at some 00:01:13.050 --> 00:01:14.950 align:middle line:90% other time. 00:01:14.950 --> 00:01:18.230 align:middle line:84% In some sense, it's a continuous time version of the 00:01:18.230 --> 00:01:19.760 align:middle line:90% Bernoulli process. 00:01:19.760 --> 00:01:23.130 align:middle line:84% So the best way to think about the Poisson process is that we 00:01:23.130 --> 00:01:26.240 align:middle line:84% divide time into extremely tiny slots. 00:01:26.240 --> 00:01:29.750 align:middle line:84% And in each time slot, there's an independent possibility of 00:01:29.750 --> 00:01:31.120 align:middle line:90% having an arrival. 00:01:31.120 --> 00:01:33.860 align:middle line:84% Different time slots are independent of each other. 00:01:33.860 --> 00:01:36.660 align:middle line:84% On the other hand, when the slot is tiny, the probability 00:01:36.660 --> 00:01:39.760 align:middle line:84% for obtaining an arrival during that tiny slot is 00:01:39.760 --> 00:01:41.910 align:middle line:90% itself going to be tiny. 00:01:41.910 --> 00:01:45.950 align:middle line:84% So we capture these properties into a formal definition what 00:01:45.950 --> 00:01:48.380 align:middle line:90% the Poisson process is. 00:01:48.380 --> 00:01:51.390 align:middle line:84% We have a probability mass function for the number of 00:01:51.390 --> 00:01:56.250 align:middle line:84% arrivals, k, during an interval of a given length. 00:01:56.250 --> 00:02:00.590 align:middle line:84% So this is the sort of basic description of the 00:02:00.590 --> 00:02:03.200 align:middle line:84% distribution of the number of arrivals. 00:02:03.200 --> 00:02:07.520 align:middle line:90% So tau is fixed. 00:02:07.520 --> 00:02:09.350 align:middle line:90% And k is the parameter. 00:02:09.350 --> 00:02:13.660 align:middle line:84% So when we add over all k's, the sum of these probabilities 00:02:13.660 --> 00:02:15.780 align:middle line:90% has to be equal to 1. 00:02:15.780 --> 00:02:19.330 align:middle line:84% There's a time homogeneity assumption, which is hidden in 00:02:19.330 --> 00:02:22.840 align:middle line:84% this, namely, the only thing that matters is the duration 00:02:22.840 --> 00:02:25.880 align:middle line:84% of the time interval, not where the time interval sits 00:02:25.880 --> 00:02:28.230 align:middle line:90% on the real axis. 00:02:28.230 --> 00:02:31.260 align:middle line:84% Then we have an independence assumption. 00:02:31.260 --> 00:02:34.570 align:middle line:84% Intervals that are disjoint are statistically independent 00:02:34.570 --> 00:02:35.650 align:middle line:90% from each other. 00:02:35.650 --> 00:02:39.110 align:middle line:84% So any information you give me about arrivals during this 00:02:39.110 --> 00:02:43.200 align:middle line:84% time interval doesn't change my beliefs about what's going 00:02:43.200 --> 00:02:45.930 align:middle line:84% to happen during another time interval. 00:02:45.930 --> 00:02:49.050 align:middle line:84% So this is a generalization of the idea that we had in 00:02:49.050 --> 00:02:51.970 align:middle line:84% Bernoulli processes that different time slots are 00:02:51.970 --> 00:02:53.930 align:middle line:90% independent of each other. 00:02:53.930 --> 00:02:56.770 align:middle line:84% And then to specify this function, the distribution of 00:02:56.770 --> 00:03:00.120 align:middle line:84% the number of arrivals, we sort of go in stages. 00:03:00.120 --> 00:03:03.750 align:middle line:84% We first specify this function for the case where the time 00:03:03.750 --> 00:03:05.920 align:middle line:90% interval is very small. 00:03:05.920 --> 00:03:09.630 align:middle line:84% And I'm telling you what those probabilities will be. 00:03:09.630 --> 00:03:13.540 align:middle line:84% And based on these then, we do some calculations and to find 00:03:13.540 --> 00:03:16.510 align:middle line:84% the formula for the distribution of the number of 00:03:16.510 --> 00:03:19.420 align:middle line:84% arrivals for intervals of a general duration. 00:03:19.420 --> 00:03:23.000 align:middle line:84% So for a small duration, delta, the probability of 00:03:23.000 --> 00:03:26.730 align:middle line:84% obtaining 1 arrival is lambda delta. 00:03:26.730 --> 00:03:30.220 align:middle line:84% The remaining probability is assigned to the event that we 00:03:30.220 --> 00:03:32.980 align:middle line:84% get to no arrivals during that interval. 00:03:32.980 --> 00:03:36.330 align:middle line:84% The probability of obtaining more than 1 arrival in a tiny 00:03:36.330 --> 00:03:40.630 align:middle line:90% interval is essentially 0. 00:03:40.630 --> 00:03:45.190 align:middle line:84% And when we say essentially, it's means modular, terms that 00:03:45.190 --> 00:03:47.620 align:middle line:90% of order delta squared. 00:03:47.620 --> 00:03:50.440 align:middle line:84% And when delta is very small, anything which is delta 00:03:50.440 --> 00:03:52.690 align:middle line:90% squared can be ignored. 00:03:52.690 --> 00:03:55.850 align:middle line:84% So up to delta squared terms, that's what happened during a 00:03:55.850 --> 00:03:57.660 align:middle line:90% little interval. 00:03:57.660 --> 00:04:01.210 align:middle line:84% Now if we know the probability distribution for the number of 00:04:01.210 --> 00:04:03.260 align:middle line:90% arrivals in a little interval. 00:04:03.260 --> 00:04:06.470 align:middle line:84% We can use this to get the distribution for the number of 00:04:06.470 --> 00:04:08.370 align:middle line:84% arrivals over several intervals. 00:04:08.370 --> 00:04:09.870 align:middle line:90% How do we do that? 00:04:09.870 --> 00:04:13.850 align:middle line:84% The big interval is composed of many little intervals. 00:04:13.850 --> 00:04:16.410 align:middle line:84% Each little interval is independent from any other 00:04:16.410 --> 00:04:20.720 align:middle line:84% little interval, so is it is as if we have a sequence of 00:04:20.720 --> 00:04:22.310 align:middle line:90% Bernoulli trials. 00:04:22.310 --> 00:04:24.910 align:middle line:84% Each Bernoulli trial is associated with a little 00:04:24.910 --> 00:04:29.580 align:middle line:84% interval and has a small probability of obtaining a 00:04:29.580 --> 00:04:32.850 align:middle line:84% success or an arrival during that mini-slot. 00:04:32.850 --> 00:04:35.800 align:middle line:84% On the other hand, when delta is small, and you take a big 00:04:35.800 --> 00:04:38.680 align:middle line:84% interval and chop it up, you get a large 00:04:38.680 --> 00:04:41.410 align:middle line:90% number of little intervals. 00:04:41.410 --> 00:04:45.240 align:middle line:84% So what we essentially have here is a Bernoulli process, 00:04:45.240 --> 00:04:48.690 align:middle line:84% in which is the number of trials is huge but the 00:04:48.690 --> 00:04:53.030 align:middle line:84% probability of success during any given trial is tiny. 00:04:53.030 --> 00:05:02.580 align:middle line:84% The average number of trials ends up being proportional to 00:05:02.580 --> 00:05:04.740 align:middle line:90% the length of the interval. 00:05:04.740 --> 00:05:07.410 align:middle line:84% If you have twice as large an interval, it's as if you're 00:05:07.410 --> 00:05:10.860 align:middle line:84% having twice as many over these mini-trials, so the 00:05:10.860 --> 00:05:14.080 align:middle line:84% expected number of arrivals will increase proportionately. 00:05:14.080 --> 00:05:18.380 align:middle line:84% There's also this parameter lambda, which we interpret as 00:05:18.380 --> 00:05:22.600 align:middle line:84% expected number of arrivals per unit time. 00:05:22.600 --> 00:05:25.940 align:middle line:84% And it comes in those probabilities here. 00:05:25.940 --> 00:05:28.810 align:middle line:84% When you double lambda, this means that a little interval 00:05:28.810 --> 00:05:31.160 align:middle line:84% is twice as likely to get an arrival. 00:05:31.160 --> 00:05:33.300 align:middle line:84% So you would expect to get twice as 00:05:33.300 --> 00:05:34.880 align:middle line:90% many arrivals as well. 00:05:34.880 --> 00:05:37.990 align:middle line:84% That's why the expected number of arrivals during an interval 00:05:37.990 --> 00:05:40.580 align:middle line:84% of length tau also scales proportional to 00:05:40.580 --> 00:05:42.850 align:middle line:90% this parameter lambda. 00:05:42.850 --> 00:05:45.740 align:middle line:84% Somewhat unexpectedly, it turns out that the variance of 00:05:45.740 --> 00:05:48.750 align:middle line:84% the number of arrivals is also the same as the mean. 00:05:48.750 --> 00:05:50.540 align:middle line:84% This is a peculiarity that happens 00:05:50.540 --> 00:05:52.310 align:middle line:90% in the Poisson process. 00:05:52.310 --> 00:05:56.100 align:middle line:84% So this is one way of thinking about Poisson process, in 00:05:56.100 --> 00:05:59.640 align:middle line:84% terms of little intervals, each one of which has a tiny 00:05:59.640 --> 00:06:01.690 align:middle line:90% probability of success. 00:06:01.690 --> 00:06:04.370 align:middle line:84% And we think of the distribution associated with 00:06:04.370 --> 00:06:07.030 align:middle line:84% that process as being described by 00:06:07.030 --> 00:06:09.100 align:middle line:90% this particular PMF. 00:06:09.100 --> 00:06:12.800 align:middle line:84% So this is the PMF for the number of arrivals during an 00:06:12.800 --> 00:06:15.880 align:middle line:84% interval of a fixed duration, tau. 00:06:15.880 --> 00:06:20.970 align:middle line:84% It's a PMF that extends all over the entire range of 00:06:20.970 --> 00:06:22.810 align:middle line:90% non-negative integers. 00:06:22.810 --> 00:06:25.730 align:middle line:84% So the number of arrivals you can get during an interval for 00:06:25.730 --> 00:06:27.890 align:middle line:84% certain length can be anything. 00:06:27.890 --> 00:06:31.690 align:middle line:84% You can get as many arrivals as you want. 00:06:31.690 --> 00:06:34.400 align:middle line:84% Of course the probability of getting a zillion arrivals is 00:06:34.400 --> 00:06:35.620 align:middle line:90% going to be tiny. 00:06:35.620 --> 00:06:38.110 align:middle line:84% But in principle, this is possible. 00:06:38.110 --> 00:06:41.440 align:middle line:84% And that's because an interval, even if it's a fixed 00:06:41.440 --> 00:06:47.460 align:middle line:84% length, consists of an infinite number of mini-slots 00:06:47.460 --> 00:06:48.770 align:middle line:90% in some sense. 00:06:48.770 --> 00:06:50.880 align:middle line:84% You can divide, chop it up, into as many 00:06:50.880 --> 00:06:52.130 align:middle line:90% mini-slots as you want. 00:06:52.130 --> 00:06:54.400 align:middle line:84% So in principle, it's possible that every 00:06:54.400 --> 00:06:55.860 align:middle line:90% mini-slot gets an arrival. 00:06:55.860 --> 00:06:59.190 align:middle line:84% In principle, it's possible to get an arbitrarily large 00:06:59.190 --> 00:07:01.210 align:middle line:90% number of arrivals. 00:07:01.210 --> 00:07:05.250 align:middle line:84% So this particular formula here is not very intuitive 00:07:05.250 --> 00:07:06.560 align:middle line:90% when you look at it. 00:07:06.560 --> 00:07:08.630 align:middle line:90% But it's a legitimate PMF. 00:07:08.630 --> 00:07:10.360 align:middle line:84% And it's called the Poisson PMF. 00:07:10.360 --> 00:07:13.970 align:middle line:84% It's the PMF that describes the number of arrivals. 00:07:13.970 --> 00:07:17.660 align:middle line:84% So that's one way of thinking about the Poisson process, 00:07:17.660 --> 00:07:21.650 align:middle line:84% where the basic object of interest would be this PMF and 00:07:21.650 --> 00:07:23.520 align:middle line:90% you try to work with it. 00:07:23.520 --> 00:07:26.600 align:middle line:84% There's another way of thinking about what happens in 00:07:26.600 --> 00:07:28.060 align:middle line:90% the Poisson process. 00:07:28.060 --> 00:07:31.780 align:middle line:84% And this has to do with letting things evolve in time. 00:07:31.780 --> 00:07:34.160 align:middle line:90% You start at time 0. 00:07:34.160 --> 00:07:37.080 align:middle line:84% There's going to be a time at which the first arrival 00:07:37.080 --> 00:07:40.000 align:middle line:90% occurs, and call that time T1. 00:07:40.000 --> 00:07:44.130 align:middle line:84% This time turns out to have an exponential distribution with 00:07:44.130 --> 00:07:46.340 align:middle line:90% parameter lambda. 00:07:46.340 --> 00:07:49.120 align:middle line:84% Once you get an arrival, it's as if the 00:07:49.120 --> 00:07:53.300 align:middle line:90% process starts fresh. 00:07:53.300 --> 00:07:55.830 align:middle line:84% The best way to understand why this is the case is by 00:07:55.830 --> 00:07:57.740 align:middle line:84% thinking in terms of the analogy with 00:07:57.740 --> 00:07:58.840 align:middle line:90% the Bernoulli process. 00:07:58.840 --> 00:08:01.660 align:middle line:84% If you believe that statement for the Bernoulli process, 00:08:01.660 --> 00:08:05.510 align:middle line:84% since this is a limiting case, it should also be true. 00:08:05.510 --> 00:08:09.150 align:middle line:84% So starting from this time, we're going to wait a random 00:08:09.150 --> 00:08:12.710 align:middle line:84% amount of time until we get the second arrival This random 00:08:12.710 --> 00:08:15.250 align:middle line:84% amount of time, let's call it T2. 00:08:15.250 --> 00:08:18.360 align:middle line:84% This time, T2 is also going to have an exponential 00:08:18.360 --> 00:08:21.140 align:middle line:84% distribution with the same parameter, lambda. 00:08:21.140 --> 00:08:26.615 align:middle line:84% And these two are going to be independent of each other. 00:08:26.615 --> 00:08:27.750 align:middle line:90% OK? 00:08:27.750 --> 00:08:31.520 align:middle line:84% So the Poisson process has all the same memorylessness 00:08:31.520 --> 00:08:34.820 align:middle line:84% properties that the Bernoulli process has. 00:08:34.820 --> 00:08:37.630 align:middle line:84% What's another way of thinking of this property? 00:08:37.630 --> 00:08:43.360 align:middle line:84% So think of a process where you have a light bulb. 00:08:43.360 --> 00:08:47.070 align:middle line:84% The time at the light bulb burns out, you can model it by 00:08:47.070 --> 00:08:48.855 align:middle line:84% an exponential random variable. 00:08:48.855 --> 00:08:51.680 align:middle line:90% 00:08:51.680 --> 00:08:58.170 align:middle line:84% And suppose that they tell you that so far, we're are sitting 00:08:58.170 --> 00:09:01.550 align:middle line:84% at some time, T. And I tell you that the light bulb has 00:09:01.550 --> 00:09:04.510 align:middle line:90% not yet burned out. 00:09:04.510 --> 00:09:08.290 align:middle line:84% What does this tell you about the future of the light bulb? 00:09:08.290 --> 00:09:11.700 align:middle line:84% Is the fact that they didn't burn out, so far, is it good 00:09:11.700 --> 00:09:13.720 align:middle line:90% news or is it bad news? 00:09:13.720 --> 00:09:17.640 align:middle line:84% Would you rather keep this light bulb that has worked for 00:09:17.640 --> 00:09:20.950 align:middle line:90% t times steps and is still OK? 00:09:20.950 --> 00:09:25.770 align:middle line:84% Or would you rather use a new light bulb that starts new at 00:09:25.770 --> 00:09:27.740 align:middle line:90% that point in time? 00:09:27.740 --> 00:09:30.920 align:middle line:84% Because of the memorylessness property, the past of that 00:09:30.920 --> 00:09:33.220 align:middle line:90% light bulb doesn't matter. 00:09:33.220 --> 00:09:37.040 align:middle line:84% So the future of this light bulb is statistically the same 00:09:37.040 --> 00:09:40.740 align:middle line:84% as the future of a new light bulb. 00:09:40.740 --> 00:09:43.700 align:middle line:84% For both of them, the time until they burn out is going 00:09:43.700 --> 00:09:46.580 align:middle line:84% to be described an exponential distribution. 00:09:46.580 --> 00:09:50.990 align:middle line:84% So one way that people described the situation is to 00:09:50.990 --> 00:09:55.450 align:middle line:84% say that used is exactly as good as a new. 00:09:55.450 --> 00:09:59.220 align:middle line:84% So a used on is no worse than a new one. 00:09:59.220 --> 00:10:01.950 align:middle line:84% A used one is no better than a new one. 00:10:01.950 --> 00:10:06.130 align:middle line:84% So a used light bulb that hasn't yet burnt out is 00:10:06.130 --> 00:10:09.180 align:middle line:84% exactly as good as a new light bulb. 00:10:09.180 --> 00:10:11.740 align:middle line:84% So that's another way of thinking about the 00:10:11.740 --> 00:10:17.150 align:middle line:84% memorylessness that we have in the Poisson process. 00:10:17.150 --> 00:10:19.350 align:middle line:90% Back to this picture. 00:10:19.350 --> 00:10:22.410 align:middle line:84% The time until the second arrival is the sum of two 00:10:22.410 --> 00:10:24.990 align:middle line:84% independent exponential random variables. 00:10:24.990 --> 00:10:28.050 align:middle line:84% So, in principle, you can use the convolution formula to 00:10:28.050 --> 00:10:32.330 align:middle line:84% find the distribution of T1 plus T2, and that would be 00:10:32.330 --> 00:10:36.750 align:middle line:84% what we call Y2, the time until the second arrival. 00:10:36.750 --> 00:10:39.210 align:middle line:84% But there's also a direct way of obtaining to the 00:10:39.210 --> 00:10:42.580 align:middle line:84% distribution of Y2, and this is the calculation that we did 00:10:42.580 --> 00:10:44.340 align:middle line:90% last time on the blackboard. 00:10:44.340 --> 00:10:46.320 align:middle line:84% And actually, we did it more generally. 00:10:46.320 --> 00:10:49.990 align:middle line:84% We found the time until the case arrival occurs. 00:10:49.990 --> 00:10:53.860 align:middle line:84% It has a closed form formula, which is called the Erlang 00:10:53.860 --> 00:10:56.960 align:middle line:84% distribution with k degrees of freedom. 00:10:56.960 --> 00:11:00.170 align:middle line:84% So let's see what's going on here. 00:11:00.170 --> 00:11:03.230 align:middle line:84% It's a distribution Of what kind? 00:11:03.230 --> 00:11:05.210 align:middle line:84% It's a continuous distribution. 00:11:05.210 --> 00:11:07.150 align:middle line:84% It's a probability density function. 00:11:07.150 --> 00:11:10.620 align:middle line:84% This is because the time is a continuous random variable. 00:11:10.620 --> 00:11:11.580 align:middle line:90% Time is continuous. 00:11:11.580 --> 00:11:14.320 align:middle line:84% Arrivals can happen at any time. 00:11:14.320 --> 00:11:17.090 align:middle line:84% So we're talking about the PDF. 00:11:17.090 --> 00:11:20.230 align:middle line:84% This k is just the parameter of the distribution. 00:11:20.230 --> 00:11:22.450 align:middle line:84% We're talking about the k-th arrival, so 00:11:22.450 --> 00:11:24.210 align:middle line:90% k is a fixed number. 00:11:24.210 --> 00:11:27.440 align:middle line:84% Lambda is another parameter of the distribution, which is the 00:11:27.440 --> 00:11:32.660 align:middle line:84% arrival rate So it's a PDF over the Y's, whereas lambda 00:11:32.660 --> 00:11:36.060 align:middle line:84% and k are parameters of the distribution. 00:11:36.060 --> 00:11:40.530 align:middle line:90% 00:11:40.530 --> 00:11:40.860 align:middle line:90% OK. 00:11:40.860 --> 00:11:45.630 align:middle line:84% So this was what we knew from last time. 00:11:45.630 --> 00:11:51.550 align:middle line:84% Just to get some practice, let us do a problem that's not too 00:11:51.550 --> 00:11:55.730 align:middle line:84% difficult, but just to see how we use the various formulas 00:11:55.730 --> 00:11:57.470 align:middle line:90% that we have. 00:11:57.470 --> 00:12:01.930 align:middle line:84% So Poisson was a mathematician, but Poisson 00:12:01.930 --> 00:12:04.730 align:middle line:90% also means fish in French. 00:12:04.730 --> 00:12:07.130 align:middle line:90% So Poisson goes fishing. 00:12:07.130 --> 00:12:11.680 align:middle line:84% And let's assume that fish are caught according 00:12:11.680 --> 00:12:13.420 align:middle line:90% to a Poisson process. 00:12:13.420 --> 00:12:15.310 align:middle line:84% That's not too bad an assumption. 00:12:15.310 --> 00:12:18.180 align:middle line:84% At any given point in time, you have a little probability 00:12:18.180 --> 00:12:19.840 align:middle line:90% that a fish would be caught. 00:12:19.840 --> 00:12:22.930 align:middle line:84% And whether you catch one now is sort of independent about 00:12:22.930 --> 00:12:28.210 align:middle line:84% whether at some later time a fish will be caught or not. 00:12:28.210 --> 00:12:30.030 align:middle line:84% So let's just make this assumption. 00:12:30.030 --> 00:12:35.270 align:middle line:84% And suppose that the rules of the game are that you-- 00:12:35.270 --> 00:12:40.350 align:middle line:84% Fish are being called it the certain rate of 0.6 per hour. 00:12:40.350 --> 00:12:44.390 align:middle line:84% You fish for 2 hours, no matter what. 00:12:44.390 --> 00:12:46.190 align:middle line:84% And then there are two possibilities. 00:12:46.190 --> 00:12:50.710 align:middle line:84% If I have caught a fish, I stop and go home. 00:12:50.710 --> 00:12:54.320 align:middle line:84% So if some fish have been caught, so there's at least 1 00:12:54.320 --> 00:12:57.250 align:middle line:84% arrival during this interval, I go home. 00:12:57.250 --> 00:13:01.760 align:middle line:84% Or if nothing has being caught, I continue fishing 00:13:01.760 --> 00:13:03.630 align:middle line:90% until I catch something. 00:13:03.630 --> 00:13:05.300 align:middle line:90% And then I go home. 00:13:05.300 --> 00:13:09.410 align:middle line:84% So that's the description of what is going to happen. 00:13:09.410 --> 00:13:12.940 align:middle line:84% And now let's starts asking questions of all sorts. 00:13:12.940 --> 00:13:16.450 align:middle line:84% What is the probability that I'm going to be fishing for 00:13:16.450 --> 00:13:19.060 align:middle line:90% more than 2 hours? 00:13:19.060 --> 00:13:23.200 align:middle line:84% I will be fishing for more than 2 hours, if and only if 00:13:23.200 --> 00:13:28.400 align:middle line:84% no fish were caught during those 2 hours, in which case, 00:13:28.400 --> 00:13:30.140 align:middle line:90% I will have to continue. 00:13:30.140 --> 00:13:33.600 align:middle line:84% Therefore, this is just this quantity. 00:13:33.600 --> 00:13:38.630 align:middle line:84% The probability of catching 2 fish in-- 00:13:38.630 --> 00:13:43.450 align:middle line:84% of catching 0 fish in the next 2 hours, and according to the 00:13:43.450 --> 00:13:47.170 align:middle line:84% formula that we have, this is going to be e to the minus 00:13:47.170 --> 00:13:50.820 align:middle line:84% lambda times how much time we have. 00:13:50.820 --> 00:13:53.040 align:middle line:84% There's another way of thinking about this. 00:13:53.040 --> 00:13:55.790 align:middle line:84% The probability that I fish for more than 2 hours is the 00:13:55.790 --> 00:14:01.230 align:middle line:84% probability that the first catch happens after time 2, 00:14:01.230 --> 00:14:04.990 align:middle line:84% which would be the integral from 2 to infinity of the 00:14:04.990 --> 00:14:09.610 align:middle line:84% density of the first arrival time. 00:14:09.610 --> 00:14:11.770 align:middle line:84% And that density is an exponential. 00:14:11.770 --> 00:14:14.910 align:middle line:84% So you do the integral of an exponential, and, of course, 00:14:14.910 --> 00:14:17.160 align:middle line:90% you would get the same answer. 00:14:17.160 --> 00:14:17.550 align:middle line:90% OK. 00:14:17.550 --> 00:14:18.730 align:middle line:90% That's easy. 00:14:18.730 --> 00:14:22.880 align:middle line:84% So what's the probability of fishing for more than 2 but 00:14:22.880 --> 00:14:25.420 align:middle line:90% less than 5 hours? 00:14:25.420 --> 00:14:28.570 align:middle line:84% What does it take for this to happen? 00:14:28.570 --> 00:14:35.540 align:middle line:84% For this to happen, we need to catch 0 fish from time 0 to 2 00:14:35.540 --> 00:14:43.020 align:middle line:84% and catch the first fish sometime between 2 and 5. 00:14:43.020 --> 00:14:44.400 align:middle line:90% So if you-- 00:14:44.400 --> 00:14:47.510 align:middle line:84% one way of thinking about what's happening here might be 00:14:47.510 --> 00:14:49.900 align:middle line:84% to say that there's a Poisson process that 00:14:49.900 --> 00:14:52.770 align:middle line:90% keeps going on forever. 00:14:52.770 --> 00:14:57.090 align:middle line:84% But as soon as I catch the first fish, instead of 00:14:57.090 --> 00:15:00.990 align:middle line:84% continuing fishing and obtaining those other fish I 00:15:00.990 --> 00:15:04.070 align:middle line:90% just go home right now. 00:15:04.070 --> 00:15:11.060 align:middle line:84% Now the fact that I go home before time 5 means that, if I 00:15:11.060 --> 00:15:13.990 align:middle line:84% were to stay until time 5, I would have 00:15:13.990 --> 00:15:15.850 align:middle line:90% caught at least 1 fish. 00:15:15.850 --> 00:15:18.350 align:middle line:84% I might have caught more than 1. 00:15:18.350 --> 00:15:22.970 align:middle line:84% So the event of interest here is that the first catch 00:15:22.970 --> 00:15:26.560 align:middle line:90% happens between times 2 and 5. 00:15:26.560 --> 00:15:32.050 align:middle line:84% So one way of calculating this quantity would be-- 00:15:32.050 --> 00:15:35.300 align:middle line:84% Its the probability that the first catch happens between 00:15:35.300 --> 00:15:37.700 align:middle line:90% times 2 and 5. 00:15:37.700 --> 00:15:40.060 align:middle line:84% Another way to deal with it is to say, this is the 00:15:40.060 --> 00:15:44.880 align:middle line:84% probability that I caught 0 fish in the first 2 hours and 00:15:44.880 --> 00:15:49.170 align:middle line:84% then the probability that I catch at least 1 fish during 00:15:49.170 --> 00:15:51.130 align:middle line:90% the next 3 hours. 00:15:51.130 --> 00:15:53.890 align:middle line:90% 00:15:53.890 --> 00:15:54.780 align:middle line:90% This. 00:15:54.780 --> 00:15:56.080 align:middle line:90% What is this? 00:15:56.080 --> 00:15:59.180 align:middle line:84% The probability of 0 fish in the next 3 hours is the 00:15:59.180 --> 00:16:01.600 align:middle line:84% probability of 0 fish during this time. 00:16:01.600 --> 00:16:04.480 align:middle line:84% 1 minus this is the probability of catching at 00:16:04.480 --> 00:16:07.850 align:middle line:84% least 1 fish, of having at least 1 arrival, 00:16:07.850 --> 00:16:09.730 align:middle line:90% between times 2 and 5. 00:16:09.730 --> 00:16:13.310 align:middle line:84% If there's at least 1 arrival between times 2 and 5, then I 00:16:13.310 --> 00:16:17.140 align:middle line:84% would have gone home by time 5. 00:16:17.140 --> 00:16:20.660 align:middle line:84% So both of these, if you plug-in numbers and all that, 00:16:20.660 --> 00:16:24.170 align:middle line:84% of course, are going to give you the same answer. 00:16:24.170 --> 00:16:26.820 align:middle line:84% Now next, what's the probability that I catch at 00:16:26.820 --> 00:16:29.560 align:middle line:90% least 2 fish? 00:16:29.560 --> 00:16:32.370 align:middle line:90% In which scenario are we? 00:16:32.370 --> 00:16:36.570 align:middle line:84% Under this scenario, I go home when I catch my first fish. 00:16:36.570 --> 00:16:39.560 align:middle line:84% So in order to catch at least 2 fish, it 00:16:39.560 --> 00:16:41.340 align:middle line:90% must be in this case. 00:16:41.340 --> 00:16:44.830 align:middle line:84% So this is the same as the event that I catch at least 2 00:16:44.830 --> 00:16:49.020 align:middle line:84% fish during the first 2 time steps. 00:16:49.020 --> 00:16:52.410 align:middle line:84% So it's going to be the probability from 2 to 00:16:52.410 --> 00:16:56.780 align:middle line:84% infinity, the probability that I catch 2 fish, or that I 00:16:56.780 --> 00:17:01.860 align:middle line:84% catch 3 fish, or I catch more than that. 00:17:01.860 --> 00:17:04.109 align:middle line:90% So it's this quantity. 00:17:04.109 --> 00:17:06.730 align:middle line:84% k is the number of fish that I catch. 00:17:06.730 --> 00:17:09.599 align:middle line:84% At least 2, so k goes from 2 to infinity. 00:17:09.599 --> 00:17:13.180 align:middle line:84% These are the probabilities of catching a number k of fish 00:17:13.180 --> 00:17:14.859 align:middle line:90% during this interval. 00:17:14.859 --> 00:17:17.920 align:middle line:84% And if you want a simpler form without an infinite sum, this 00:17:17.920 --> 00:17:20.619 align:middle line:84% would be 1 minus the probability of catching 0 00:17:20.619 --> 00:17:24.880 align:middle line:84% fish, minus the probability of catching 1 fish, during a time 00:17:24.880 --> 00:17:28.050 align:middle line:90% interval of length 2. 00:17:28.050 --> 00:17:29.520 align:middle line:90% Another way to think of it. 00:17:29.520 --> 00:17:34.230 align:middle line:84% I'm going to catch 2 fish, at least 2 fish, if and only if 00:17:34.230 --> 00:17:40.630 align:middle line:84% the second fish caught in this process happens before time 2. 00:17:40.630 --> 00:17:43.950 align:middle line:84% So that's another way of thinking about the same event. 00:17:43.950 --> 00:17:46.230 align:middle line:84% So it's going to be the probability that the random 00:17:46.230 --> 00:17:51.440 align:middle line:84% variable Y2, the arrival time over the second fish, is less 00:17:51.440 --> 00:17:52.690 align:middle line:90% than or equal to 2. 00:17:52.690 --> 00:17:55.387 align:middle line:90% 00:17:55.387 --> 00:17:56.310 align:middle line:90% OK. 00:17:56.310 --> 00:18:00.000 align:middle line:84% The next one is a little trickier. 00:18:00.000 --> 00:18:03.380 align:middle line:84% Here we need to do a little bit of divide and conquer. 00:18:03.380 --> 00:18:06.490 align:middle line:84% Overall, in this expedition, what the expected number of 00:18:06.490 --> 00:18:08.840 align:middle line:90% fish to be caught? 00:18:08.840 --> 00:18:11.550 align:middle line:84% One way to think about it is to try to use the total 00:18:11.550 --> 00:18:13.100 align:middle line:90% expectations theorem. 00:18:13.100 --> 00:18:17.830 align:middle line:84% And think of expected number of fish, given this scenario, 00:18:17.830 --> 00:18:21.010 align:middle line:84% or expected number of fish, given this scenario. 00:18:21.010 --> 00:18:24.190 align:middle line:84% That's a little more complicated than the way I'm 00:18:24.190 --> 00:18:25.290 align:middle line:90% going to do it. 00:18:25.290 --> 00:18:28.240 align:middle line:84% The way I'm going to do is to think as follows-- 00:18:28.240 --> 00:18:32.310 align:middle line:84% Expected number of fish is the expected number of fish caught 00:18:32.310 --> 00:18:37.520 align:middle line:84% between times 0 and 2 plus expected number of fish caught 00:18:37.520 --> 00:18:39.800 align:middle line:90% after time 2. 00:18:39.800 --> 00:18:45.580 align:middle line:84% So what's the expected number caught between time 0 and 2? 00:18:45.580 --> 00:18:47.860 align:middle line:90% This is lambda t. 00:18:47.860 --> 00:18:52.310 align:middle line:90% So lambda is 0.6 times 2. 00:18:52.310 --> 00:18:55.380 align:middle line:84% This is the expected number of fish that are caught between 00:18:55.380 --> 00:18:57.260 align:middle line:90% times 0 and 2. 00:18:57.260 --> 00:19:00.440 align:middle line:84% Now let's think about the expected number of fish caught 00:19:00.440 --> 00:19:01.630 align:middle line:90% afterwards. 00:19:01.630 --> 00:19:04.300 align:middle line:84% How many fish are being caught afterwards? 00:19:04.300 --> 00:19:06.110 align:middle line:84% Well it depends on the scenario. 00:19:06.110 --> 00:19:08.750 align:middle line:84% If we're in this scenario, we've gone home 00:19:08.750 --> 00:19:10.800 align:middle line:90% and we catch 0. 00:19:10.800 --> 00:19:14.570 align:middle line:84% If we're in this scenario, then we continue fishing until 00:19:14.570 --> 00:19:15.980 align:middle line:90% we catch one. 00:19:15.980 --> 00:19:19.970 align:middle line:84% So the expected number of fish to be caught after time 2 is 00:19:19.970 --> 00:19:24.520 align:middle line:84% going to be the probability of this scenario times 1. 00:19:24.520 --> 00:19:29.020 align:middle line:84% And the probability of that scenario is the probability 00:19:29.020 --> 00:19:33.490 align:middle line:84% that they call it's 0 fish during the first 2 time steps 00:19:33.490 --> 00:19:37.420 align:middle line:84% times 1, which is the number of fish I'm going to catch if 00:19:37.420 --> 00:19:39.790 align:middle line:90% I continue. 00:19:39.790 --> 00:19:43.960 align:middle line:84% The expected total fishing time we can calculate exactly 00:19:43.960 --> 00:19:46.150 align:middle line:90% the same way. 00:19:46.150 --> 00:19:47.890 align:middle line:90% I'm jumping to the last one. 00:19:47.890 --> 00:19:51.580 align:middle line:84% My total fishing time has a period of 2 time steps. 00:19:51.580 --> 00:19:54.910 align:middle line:84% I'm going to fish for 2 time steps no matter what. 00:19:54.910 --> 00:19:59.190 align:middle line:84% And then if I caught 0 fish, which happens with this 00:19:59.190 --> 00:20:04.540 align:middle line:84% probability, my expected time is going to be the expected 00:20:04.540 --> 00:20:08.920 align:middle line:84% time from here onwards, which is the expected value of this 00:20:08.920 --> 00:20:12.490 align:middle line:84% geometric random variable with parameter lambda. 00:20:12.490 --> 00:20:15.430 align:middle line:84% So the expected time is 1 over lambda. 00:20:15.430 --> 00:20:22.460 align:middle line:84% And in our case this, is 1/0.6. 00:20:22.460 --> 00:20:31.180 align:middle line:84% Finally, if I tell you that I have been fishing for 4 hours 00:20:31.180 --> 00:20:37.800 align:middle line:84% and nothing has been caught so far, how much do you expect 00:20:37.800 --> 00:20:41.630 align:middle line:90% this quantity to be? 00:20:41.630 --> 00:20:46.330 align:middle line:84% Here is the story that, again, that for the Poisson process 00:20:46.330 --> 00:20:48.720 align:middle line:90% used is as good as new. 00:20:48.720 --> 00:20:51.060 align:middle line:84% The process does not have any memory. 00:20:51.060 --> 00:20:54.930 align:middle line:84% Given what happens in the past doesn't matter for the future. 00:20:54.930 --> 00:20:58.430 align:middle line:84% It's as if the process starts new at this point in time. 00:20:58.430 --> 00:21:02.420 align:middle line:84% So this one is going to be, again, the same exponentially 00:21:02.420 --> 00:21:04.910 align:middle line:84% distributed random variable with the 00:21:04.910 --> 00:21:08.270 align:middle line:90% same parameter lambda. 00:21:08.270 --> 00:21:12.270 align:middle line:84% So expected time until an arrival comes is an 00:21:12.270 --> 00:21:13.740 align:middle line:90% exponential distribut -- 00:21:13.740 --> 00:21:15.910 align:middle line:84% has an exponential distribution with parameter 00:21:15.910 --> 00:21:19.660 align:middle line:84% lambda, no matter what has happened in the past. 00:21:19.660 --> 00:21:22.730 align:middle line:84% Starting from now and looking into the future, it's as if 00:21:22.730 --> 00:21:24.910 align:middle line:90% the process has just started. 00:21:24.910 --> 00:21:32.440 align:middle line:84% So it's going to be 1 over lambda, which is 1/0.6. 00:21:32.440 --> 00:21:33.690 align:middle line:90% OK. 00:21:33.690 --> 00:21:37.540 align:middle line:90% 00:21:37.540 --> 00:21:41.500 align:middle line:84% Now our next example is going to be a little more 00:21:41.500 --> 00:21:43.780 align:middle line:90% complicated or subtle. 00:21:43.780 --> 00:21:46.800 align:middle line:84% But before we get to the example, let's refresh our 00:21:46.800 --> 00:21:50.300 align:middle line:84% memory about what we discussed last time about merging 00:21:50.300 --> 00:21:53.110 align:middle line:84% Poisson independent Poisson processes. 00:21:53.110 --> 00:21:56.090 align:middle line:84% Instead of drawing the picture that way, another way we could 00:21:56.090 --> 00:21:58.260 align:middle line:90% draw it could be this. 00:21:58.260 --> 00:22:01.260 align:middle line:84% We have a Poisson process with rate lambda1, and a Poisson 00:22:01.260 --> 00:22:03.440 align:middle line:90% process with rate lambda2. 00:22:03.440 --> 00:22:07.320 align:middle line:84% They have, each one of these, have their arrivals. 00:22:07.320 --> 00:22:09.780 align:middle line:84% And then we form the merged process. 00:22:09.780 --> 00:22:13.580 align:middle line:84% And the merged process records an arrival whenever there's an 00:22:13.580 --> 00:22:16.930 align:middle line:84% arrival in either of the two processes. 00:22:16.930 --> 00:22:19.990 align:middle line:90% 00:22:19.990 --> 00:22:23.730 align:middle line:84% This process in that process are assumed to be independent 00:22:23.730 --> 00:22:26.760 align:middle line:90% of each other. 00:22:26.760 --> 00:22:32.590 align:middle line:84% Now different times in this process and that process are 00:22:32.590 --> 00:22:34.780 align:middle line:90% independent of each other. 00:22:34.780 --> 00:22:39.400 align:middle line:84% So what happens in these two time intervals is independent 00:22:39.400 --> 00:22:41.780 align:middle line:84% from what happens in these two time intervals. 00:22:41.780 --> 00:22:45.560 align:middle line:84% These two time intervals to determine what happens here. 00:22:45.560 --> 00:22:48.750 align:middle line:84% These two time intervals determine what happens there. 00:22:48.750 --> 00:22:53.740 align:middle line:84% So because these are independent from these, this 00:22:53.740 --> 00:22:56.600 align:middle line:84% means that this is also independent from that. 00:22:56.600 --> 00:22:59.020 align:middle line:84% So the independence assumption is satisfied 00:22:59.020 --> 00:23:01.150 align:middle line:90% for the merged process. 00:23:01.150 --> 00:23:05.030 align:middle line:84% And the merged process turns out to be a Poisson process. 00:23:05.030 --> 00:23:10.340 align:middle line:84% And if you want to find the arrival rate for that process, 00:23:10.340 --> 00:23:12.550 align:middle line:90% you argue as follows. 00:23:12.550 --> 00:23:15.000 align:middle line:84% During a little interval of length delta, we have 00:23:15.000 --> 00:23:17.280 align:middle line:84% probability lambda1 delta of having an 00:23:17.280 --> 00:23:18.620 align:middle line:90% arrival in this process. 00:23:18.620 --> 00:23:21.700 align:middle line:84% We have probability lambda2 delta of an arrival in this 00:23:21.700 --> 00:23:24.890 align:middle line:84% process, plus second order terms in 00:23:24.890 --> 00:23:26.860 align:middle line:90% delta, which we're ignoring. 00:23:26.860 --> 00:23:29.270 align:middle line:84% And then you do the calculation and you find that 00:23:29.270 --> 00:23:31.870 align:middle line:84% in this process, you're going to have an arrival 00:23:31.870 --> 00:23:37.830 align:middle line:84% probability, which is lambda1 plus lambda2, again ignoring 00:23:37.830 --> 00:23:40.490 align:middle line:90% second order in delta-- 00:23:40.490 --> 00:23:42.650 align:middle line:84% terms that are second order in delta. 00:23:42.650 --> 00:23:46.130 align:middle line:84% So the merged process is a Poisson process whose arrival 00:23:46.130 --> 00:23:48.760 align:middle line:84% rate is the sum of the arrival rates of 00:23:48.760 --> 00:23:52.080 align:middle line:90% the individual processes. 00:23:52.080 --> 00:23:55.290 align:middle line:84% And the calculation we did at the end of the last lecture-- 00:23:55.290 --> 00:23:59.240 align:middle line:84% If I tell you that the new arrival happened here, where 00:23:59.240 --> 00:24:00.610 align:middle line:90% did that arrival come from? 00:24:00.610 --> 00:24:02.910 align:middle line:84% Did it come from here or from there? 00:24:02.910 --> 00:24:06.720 align:middle line:84% If the lambda1 is equal to lambda2, then by symmetry you 00:24:06.720 --> 00:24:09.240 align:middle line:84% would say that it's equally likely to have come from here 00:24:09.240 --> 00:24:10.660 align:middle line:90% or to come from there. 00:24:10.660 --> 00:24:13.720 align:middle line:84% But if this lambda is much bigger than that lambda, the 00:24:13.720 --> 00:24:16.850 align:middle line:84% fact that they saw an arrival is more likely to have come 00:24:16.850 --> 00:24:17.850 align:middle line:90% from there. 00:24:17.850 --> 00:24:22.410 align:middle line:84% And the formula that captures this is the following. 00:24:22.410 --> 00:24:27.300 align:middle line:84% This is the probability that my arrival has come from this 00:24:27.300 --> 00:24:32.360 align:middle line:84% particular stream rather than that particular stream. 00:24:32.360 --> 00:24:38.900 align:middle line:84% So when an arrival comes and you ask, what is the origin of 00:24:38.900 --> 00:24:39.690 align:middle line:90% that arrival? 00:24:39.690 --> 00:24:43.760 align:middle line:84% It's as if I'm flipping a coin with these odds. 00:24:43.760 --> 00:24:46.910 align:middle line:84% And depending on outcome of that coin, I'm going to tell 00:24:46.910 --> 00:24:49.790 align:middle line:84% you came from there or it came from there. 00:24:49.790 --> 00:24:53.850 align:middle line:84% So the origin of an arrival is either this 00:24:53.850 --> 00:24:55.610 align:middle line:90% stream or that stream. 00:24:55.610 --> 00:24:58.190 align:middle line:84% And this is the probability that the origin of the arrival 00:24:58.190 --> 00:24:59.510 align:middle line:90% is that one. 00:24:59.510 --> 00:25:04.160 align:middle line:84% Now if we look at 2 different arrivals, and we ask about 00:25:04.160 --> 00:25:05.570 align:middle line:90% their origins-- 00:25:05.570 --> 00:25:08.130 align:middle line:84% So let's think about the origin of this arrival and 00:25:08.130 --> 00:25:12.060 align:middle line:84% compare it with the origin that arrival. 00:25:12.060 --> 00:25:14.010 align:middle line:84% The origin of this arrival is random. 00:25:14.010 --> 00:25:16.720 align:middle line:84% It could be right be either this or that. 00:25:16.720 --> 00:25:18.840 align:middle line:84% And this is the relevant probability. 00:25:18.840 --> 00:25:20.750 align:middle line:84% The origin of that arrival is random. 00:25:20.750 --> 00:25:24.360 align:middle line:84% It could be either here or is there, and again, with the 00:25:24.360 --> 00:25:26.880 align:middle line:90% same relevant probability. 00:25:26.880 --> 00:25:27.730 align:middle line:90% Question. 00:25:27.730 --> 00:25:31.780 align:middle line:84% The origin of this arrival, is it dependent or independent 00:25:31.780 --> 00:25:34.710 align:middle line:90% from the origin that arrival? 00:25:34.710 --> 00:25:37.500 align:middle line:84% And here's how the argument goes. 00:25:37.500 --> 00:25:40.740 align:middle line:84% Separate times are independent. 00:25:40.740 --> 00:25:45.050 align:middle line:84% Whatever has happened in the process during this set of 00:25:45.050 --> 00:25:48.040 align:middle line:84% times is independent from whatever happened in the 00:25:48.040 --> 00:25:50.980 align:middle line:84% process during that set of times. 00:25:50.980 --> 00:25:55.040 align:middle line:84% Because different times have nothing to do with each other, 00:25:55.040 --> 00:25:59.650 align:middle line:84% the origin of this, of an arrival here, has nothing to 00:25:59.650 --> 00:26:02.480 align:middle line:84% do with the origin of an arrival there. 00:26:02.480 --> 00:26:06.890 align:middle line:84% So the origins of different arrivals are also independent 00:26:06.890 --> 00:26:08.850 align:middle line:90% random variables. 00:26:08.850 --> 00:26:12.710 align:middle line:90% So if I tell you that-- 00:26:12.710 --> 00:26:14.150 align:middle line:90% yeah. 00:26:14.150 --> 00:26:15.310 align:middle line:90% OK. 00:26:15.310 --> 00:26:19.600 align:middle line:84% So it as if that each time that you have an arrival in 00:26:19.600 --> 00:26:22.820 align:middle line:84% the merge process, it's as if you're flipping a coin to 00:26:22.820 --> 00:26:26.410 align:middle line:84% determine where did that arrival came from and these 00:26:26.410 --> 00:26:31.516 align:middle line:84% coins are independent of each other. 00:26:31.516 --> 00:26:32.766 align:middle line:90% OK. 00:26:32.766 --> 00:26:35.550 align:middle line:90% 00:26:35.550 --> 00:26:35.920 align:middle line:90% OK. 00:26:35.920 --> 00:26:37.770 align:middle line:90% Now we're going to use this-- 00:26:37.770 --> 00:26:42.970 align:middle line:84% what we know about merged processes to solve the problem 00:26:42.970 --> 00:26:48.240 align:middle line:84% that would be harder to do, if you were not using ideas from 00:26:48.240 --> 00:26:49.720 align:middle line:90% Poisson processes. 00:26:49.720 --> 00:26:52.250 align:middle line:84% So the formulation of the problem has nothing to do with 00:26:52.250 --> 00:26:54.370 align:middle line:90% the Poisson process. 00:26:54.370 --> 00:26:57.450 align:middle line:84% The formulation is the following. 00:26:57.450 --> 00:26:59.870 align:middle line:90% We have 3 light-bulbs. 00:26:59.870 --> 00:27:03.490 align:middle line:84% And each light bulb is independent and is going to 00:27:03.490 --> 00:27:07.920 align:middle line:84% die out at the time that's exponentially distributed. 00:27:07.920 --> 00:27:11.170 align:middle line:90% So 3 light bulbs. 00:27:11.170 --> 00:27:16.630 align:middle line:84% They start their lives and then at some point 00:27:16.630 --> 00:27:21.260 align:middle line:90% they die or burn out. 00:27:21.260 --> 00:27:26.150 align:middle line:84% So let's think of this as X, this as Y, and this as Z. 00:27:26.150 --> 00:27:31.220 align:middle line:84% And we're interested in the time until the last 00:27:31.220 --> 00:27:33.200 align:middle line:90% light-bulb burns out. 00:27:33.200 --> 00:27:36.930 align:middle line:84% So we're interested in the maximum of the 3 random 00:27:36.930 --> 00:27:41.480 align:middle line:84% variables, X, Y, and Z. And in particular, we want to find 00:27:41.480 --> 00:27:43.170 align:middle line:84% the expected value of this maximum. 00:27:43.170 --> 00:27:45.770 align:middle line:90% 00:27:45.770 --> 00:27:47.490 align:middle line:90% OK. 00:27:47.490 --> 00:27:50.760 align:middle line:84% So you can do derived distribution, use the expected 00:27:50.760 --> 00:27:52.880 align:middle line:90% value rule, anything you want. 00:27:52.880 --> 00:27:56.230 align:middle line:84% You can get this answer using the tools that you already 00:27:56.230 --> 00:27:58.180 align:middle line:90% have in your hands. 00:27:58.180 --> 00:28:02.070 align:middle line:84% But now let us see how we can connect to this picture with a 00:28:02.070 --> 00:28:05.550 align:middle line:84% Poisson picture and come up with the answer in a very 00:28:05.550 --> 00:28:07.240 align:middle line:90% simple way. 00:28:07.240 --> 00:28:09.630 align:middle line:84% What is an exponential random variable? 00:28:09.630 --> 00:28:14.450 align:middle line:84% An exponential random variable is the first act in the long 00:28:14.450 --> 00:28:19.570 align:middle line:84% play that involves a whole Poisson process. 00:28:19.570 --> 00:28:23.020 align:middle line:84% So an exponential random variable is the first act of a 00:28:23.020 --> 00:28:24.650 align:middle line:90% Poisson movie. 00:28:24.650 --> 00:28:25.660 align:middle line:90% Same thing here. 00:28:25.660 --> 00:28:29.700 align:middle line:84% You can think of this random variable as being part of some 00:28:29.700 --> 00:28:31.850 align:middle line:84% Poisson process that has been running. 00:28:31.850 --> 00:28:35.360 align:middle line:90% 00:28:35.360 --> 00:28:38.040 align:middle line:84% So it's part of this bigger picture. 00:28:38.040 --> 00:28:42.370 align:middle line:84% We're still interested in the maximum of the 3. 00:28:42.370 --> 00:28:45.780 align:middle line:84% The other arrivals are not going to affect our answers. 00:28:45.780 --> 00:28:49.640 align:middle line:84% It's just, conceptually speaking, we can think of the 00:28:49.640 --> 00:28:52.840 align:middle line:84% exponential random variable as being embedded in a bigger 00:28:52.840 --> 00:28:55.110 align:middle line:90% Poisson picture. 00:28:55.110 --> 00:29:00.980 align:middle line:84% So we have 3 Poisson process that are running in parallel. 00:29:00.980 --> 00:29:06.150 align:middle line:84% Let us split the expected time until the last burnout into 00:29:06.150 --> 00:29:09.800 align:middle line:84% pieces, which is time until the first burnout, time from 00:29:09.800 --> 00:29:11.810 align:middle line:84% the first until the second, and time from the 00:29:11.810 --> 00:29:13.690 align:middle line:90% second until the third. 00:29:13.690 --> 00:29:16.780 align:middle line:90% 00:29:16.780 --> 00:29:20.570 align:middle line:84% And find the expected values of each one of these pieces. 00:29:20.570 --> 00:29:24.620 align:middle line:84% What can we say about the expected value of this? 00:29:24.620 --> 00:29:29.310 align:middle line:84% This is the first arrival out of all of 00:29:29.310 --> 00:29:31.540 align:middle line:90% these 3 Poisson processes. 00:29:31.540 --> 00:29:34.070 align:middle line:84% It's the first event that happens when you look at all 00:29:34.070 --> 00:29:36.080 align:middle line:84% of these processes simultaneously. 00:29:36.080 --> 00:29:39.660 align:middle line:84% So 3 Poisson processes running in parallel. 00:29:39.660 --> 00:29:43.750 align:middle line:84% We're interested in the time until one of them, any one of 00:29:43.750 --> 00:29:46.380 align:middle line:90% them, gets in arrival. 00:29:46.380 --> 00:29:47.690 align:middle line:90% Rephrase. 00:29:47.690 --> 00:29:51.330 align:middle line:84% We merged the 3 Poisson processes, and we ask for the 00:29:51.330 --> 00:29:56.820 align:middle line:84% time until we observe an arrival in the merged process. 00:29:56.820 --> 00:30:01.250 align:middle line:84% When 1 of the 3 gets an arrival for the first time, 00:30:01.250 --> 00:30:03.880 align:middle line:84% the merged process gets its first arrival. 00:30:03.880 --> 00:30:06.300 align:middle line:84% So what's the expected value of this time 00:30:06.300 --> 00:30:08.820 align:middle line:90% until the first burnout? 00:30:08.820 --> 00:30:11.940 align:middle line:84% It's going to be the expected value of a 00:30:11.940 --> 00:30:13.720 align:middle line:90% Poisson random variable. 00:30:13.720 --> 00:30:17.050 align:middle line:84% So the first burnout is going to have an expected 00:30:17.050 --> 00:30:20.430 align:middle line:90% value, which is-- 00:30:20.430 --> 00:30:21.540 align:middle line:90% OK. 00:30:21.540 --> 00:30:23.690 align:middle line:90% It's a Poisson process. 00:30:23.690 --> 00:30:28.530 align:middle line:84% The merged process of the 3 has a collective arrival rate, 00:30:28.530 --> 00:30:32.750 align:middle line:90% which is 3 times lambda. 00:30:32.750 --> 00:30:36.250 align:middle line:84% So this is the parameter over the exponential distribution 00:30:36.250 --> 00:30:39.870 align:middle line:84% that describes the time until the first arrival in the 00:30:39.870 --> 00:30:41.220 align:middle line:90% merged process. 00:30:41.220 --> 00:30:42.990 align:middle line:84% And the expected value of this random 00:30:42.990 --> 00:30:45.670 align:middle line:90% variable is 1 over that. 00:30:45.670 --> 00:30:48.190 align:middle line:84% When you have an exponential random variable with parameter 00:30:48.190 --> 00:30:50.150 align:middle line:84% lambda, the expected value of that random 00:30:50.150 --> 00:30:52.330 align:middle line:90% variable is 1 over lambda. 00:30:52.330 --> 00:30:56.660 align:middle line:84% Here we're talking about the first arrival time in a 00:30:56.660 --> 00:30:58.720 align:middle line:90% process with rate 3 lambda. 00:30:58.720 --> 00:31:00.680 align:middle line:84% The expected time until the first arrival 00:31:00.680 --> 00:31:03.000 align:middle line:90% is 1 over (3 lambda). 00:31:03.000 --> 00:31:03.870 align:middle line:90% Alright. 00:31:03.870 --> 00:31:08.710 align:middle line:84% So at this time, this bulb, this arrival happened, this 00:31:08.710 --> 00:31:11.490 align:middle line:90% bulb has been burned. 00:31:11.490 --> 00:31:15.760 align:middle line:84% So we don't care about that bulb anymore. 00:31:15.760 --> 00:31:21.610 align:middle line:84% We start at this time, and we look forward. 00:31:21.610 --> 00:31:23.640 align:middle line:90% This bulb has been burned. 00:31:23.640 --> 00:31:27.810 align:middle line:84% So let's just look forward from now on. 00:31:27.810 --> 00:31:28.900 align:middle line:90% What have we got? 00:31:28.900 --> 00:31:34.030 align:middle line:84% We have two bulbs that are burning. 00:31:34.030 --> 00:31:37.320 align:middle line:84% We have a Poisson process that's the bigger picture of 00:31:37.320 --> 00:31:40.270 align:middle line:84% what could happen to that light bulb, if we were to keep 00:31:40.270 --> 00:31:41.190 align:middle line:90% replacing it. 00:31:41.190 --> 00:31:42.880 align:middle line:90% Another Poisson process. 00:31:42.880 --> 00:31:45.610 align:middle line:84% These two processes are, again, independent. 00:31:45.610 --> 00:31:50.850 align:middle line:84% From this time until that time, how long does it take? 00:31:50.850 --> 00:31:53.930 align:middle line:84% It's the time until either this process records an 00:31:53.930 --> 00:31:57.090 align:middle line:84% arrival or that process records and arrival. 00:31:57.090 --> 00:32:01.210 align:middle line:84% That's the same as the time that the merged process of 00:32:01.210 --> 00:32:03.810 align:middle line:90% these two records an arrival. 00:32:03.810 --> 00:32:06.430 align:middle line:84% So we're talking about the expected time until the first 00:32:06.430 --> 00:32:08.710 align:middle line:90% arrival in a merged process. 00:32:08.710 --> 00:32:11.030 align:middle line:90% The merged process is Poisson. 00:32:11.030 --> 00:32:14.240 align:middle line:84% It's Poisson with rate 2 lambda. 00:32:14.240 --> 00:32:17.690 align:middle line:84% So that extra time is going to take-- 00:32:17.690 --> 00:32:21.390 align:middle line:84% the expected value is going to be 1 over the (rate of that 00:32:21.390 --> 00:32:22.580 align:middle line:90% Poisson process). 00:32:22.580 --> 00:32:25.170 align:middle line:84% So 1 over (2 lambda) is the expected value 00:32:25.170 --> 00:32:26.980 align:middle line:90% of this random variable. 00:32:26.980 --> 00:32:30.870 align:middle line:84% So at this point, this bulb now is also burned. 00:32:30.870 --> 00:32:33.620 align:middle line:84% So we start looking from this time on. 00:32:33.620 --> 00:32:37.110 align:middle line:84% That part of the picture disappears. 00:32:37.110 --> 00:32:40.150 align:middle line:84% Starting from this time, what's the expected value 00:32:40.150 --> 00:32:43.650 align:middle line:84% until that remaining light-bulb burns out? 00:32:43.650 --> 00:32:47.130 align:middle line:84% Well, as we said before, in a Poisson process or with 00:32:47.130 --> 00:32:50.090 align:middle line:84% exponential random variables, we have memorylessness. 00:32:50.090 --> 00:32:53.120 align:middle line:84% A used bulb is as good as a new one. 00:32:53.120 --> 00:32:55.990 align:middle line:84% So it's as if we're starting from scratch here. 00:32:55.990 --> 00:32:58.700 align:middle line:84% So this is going to be an exponential random variable 00:32:58.700 --> 00:33:00.690 align:middle line:90% with parameter lambda. 00:33:00.690 --> 00:33:05.540 align:middle line:84% And the expected value of it is going to be 1 over lambda. 00:33:05.540 --> 00:33:07.990 align:middle line:84% So the beauty of approaching this problem in this 00:33:07.990 --> 00:33:10.930 align:middle line:84% particular way is, of course, that we manage to do 00:33:10.930 --> 00:33:14.100 align:middle line:84% everything without any calculus at all, without 00:33:14.100 --> 00:33:16.990 align:middle line:84% striking an integral, without trying to calculate 00:33:16.990 --> 00:33:19.220 align:middle line:90% expectations in any form. 00:33:19.220 --> 00:33:23.150 align:middle line:84% Most of the non-trivial problems that you encounter in 00:33:23.150 --> 00:33:28.540 align:middle line:84% the Poisson world basically involve tricks of these kind. 00:33:28.540 --> 00:33:31.830 align:middle line:84% You have a question and you try to rephrase it, trying to 00:33:31.830 --> 00:33:35.240 align:middle line:84% think in terms of what might happen in the Poisson setting, 00:33:35.240 --> 00:33:39.200 align:middle line:84% use memorylessness, use merging, et cetera, et cetera. 00:33:39.200 --> 00:33:43.360 align:middle line:90% 00:33:43.360 --> 00:33:46.080 align:middle line:90% Now we talked about merging. 00:33:46.080 --> 00:33:49.480 align:middle line:84% It turns out that the splitting of Poisson processes 00:33:49.480 --> 00:33:53.400 align:middle line:90% also works in a nice way. 00:33:53.400 --> 00:33:57.160 align:middle line:84% The story here is exactly the same as for 00:33:57.160 --> 00:33:58.820 align:middle line:90% the Bernoulli process. 00:33:58.820 --> 00:34:01.870 align:middle line:84% So I'm having a Poisson process. 00:34:01.870 --> 00:34:06.060 align:middle line:84% And each time, with some rate lambda, and each time that an 00:34:06.060 --> 00:34:09.790 align:middle line:84% arrival comes, I'm going to send it to that stream and the 00:34:09.790 --> 00:34:13.179 align:middle line:84% record an arrival here with some probability P. And I'm 00:34:13.179 --> 00:34:16.120 align:middle line:84% going to send it to the other stream with some probability 1 00:34:16.120 --> 00:34:19.469 align:middle line:84% minus P. So either of this will happen or that will 00:34:19.469 --> 00:34:21.940 align:middle line:84% happen, depending on the outcome of the 00:34:21.940 --> 00:34:23.550 align:middle line:90% coin flip that I do. 00:34:23.550 --> 00:34:27.449 align:middle line:84% Each time that then arrival occurs, I flip a coin and I 00:34:27.449 --> 00:34:30.929 align:middle line:84% decide whether to record it here or there. 00:34:30.929 --> 00:34:32.620 align:middle line:84% This is called splitting a Poisson 00:34:32.620 --> 00:34:34.719 align:middle line:90% process into two pieces. 00:34:34.719 --> 00:34:37.120 align:middle line:84% What kind of process do we get here? 00:34:37.120 --> 00:34:40.250 align:middle line:84% If you look at the little interval for length delta, 00:34:40.250 --> 00:34:41.810 align:middle line:84% what's the probability that this little 00:34:41.810 --> 00:34:44.090 align:middle line:90% interval gets an arrival? 00:34:44.090 --> 00:34:47.739 align:middle line:84% It's the probability that this one gets an arrival, which is 00:34:47.739 --> 00:34:51.260 align:middle line:84% lambda delta times the probability that after I get 00:34:51.260 --> 00:34:55.210 align:middle line:84% an arrival my coin flip came out to be that way, so that it 00:34:55.210 --> 00:34:56.270 align:middle line:90% sends me there. 00:34:56.270 --> 00:34:58.740 align:middle line:84% So this means that this little interval is going to have 00:34:58.740 --> 00:35:03.620 align:middle line:84% probability lambda delta P. Or maybe more suggestively, I 00:35:03.620 --> 00:35:09.480 align:middle line:84% should write it as lambda P times delta. 00:35:09.480 --> 00:35:12.350 align:middle line:84% So every little interval has a probability of an arrival 00:35:12.350 --> 00:35:13.470 align:middle line:90% proportional to delta. 00:35:13.470 --> 00:35:16.780 align:middle line:84% The proportionality factor is lambda P. So lambda P is the 00:35:16.780 --> 00:35:18.590 align:middle line:90% rate of that process. 00:35:18.590 --> 00:35:22.500 align:middle line:84% And then you go through the mental exercise that you went 00:35:22.500 --> 00:35:25.170 align:middle line:84% through for the Bernoulli process to argue that a 00:35:25.170 --> 00:35:28.520 align:middle line:84% different intervals here are independent and so on. 00:35:28.520 --> 00:35:31.710 align:middle line:84% And that completes checking that this process is going to 00:35:31.710 --> 00:35:33.360 align:middle line:90% be a Poisson process. 00:35:33.360 --> 00:35:38.060 align:middle line:84% So when you split a Poisson process by doing independent 00:35:38.060 --> 00:35:41.040 align:middle line:84% coin flips each time that something happens, the 00:35:41.040 --> 00:35:44.330 align:middle line:84% processes that you get is again a Poisson process, but 00:35:44.330 --> 00:35:46.490 align:middle line:90% of course with a reduced rate. 00:35:46.490 --> 00:35:50.040 align:middle line:84% So instead of the word splitting, sometimes people 00:35:50.040 --> 00:35:54.330 align:middle line:84% also use the words thinning-out. 00:35:54.330 --> 00:35:57.650 align:middle line:84% That is, out of the arrivals that came, you keep a few but 00:35:57.650 --> 00:35:59.000 align:middle line:90% throw away a few. 00:35:59.000 --> 00:36:01.820 align:middle line:90% 00:36:01.820 --> 00:36:02.730 align:middle line:90% OK. 00:36:02.730 --> 00:36:08.570 align:middle line:84% So now the last topic over this lecture is a quite 00:36:08.570 --> 00:36:11.270 align:middle line:84% curious phenomenon that goes under the 00:36:11.270 --> 00:36:12.595 align:middle line:90% name of random incidents. 00:36:12.595 --> 00:36:15.550 align:middle line:90% 00:36:15.550 --> 00:36:18.950 align:middle line:90% So here's the story. 00:36:18.950 --> 00:36:22.550 align:middle line:84% Buses have been running on Mass Ave. from time 00:36:22.550 --> 00:36:24.070 align:middle line:90% immemorial. 00:36:24.070 --> 00:36:29.060 align:middle line:84% And the bus company that runs the buses claims that they 00:36:29.060 --> 00:36:33.150 align:middle line:84% come as a Poisson process with some rate, let's say, of 4 00:36:33.150 --> 00:36:34.970 align:middle line:90% buses per hour. 00:36:34.970 --> 00:36:39.250 align:middle line:84% So that the expected time between bus arrivals is going 00:36:39.250 --> 00:36:42.500 align:middle line:90% to be 15 minutes. 00:36:42.500 --> 00:36:45.180 align:middle line:90% OK. 00:36:45.180 --> 00:36:45.840 align:middle line:90% Alright. 00:36:45.840 --> 00:36:48.130 align:middle line:84% So people have been complaining that they have 00:36:48.130 --> 00:36:49.150 align:middle line:90% been showing up there. 00:36:49.150 --> 00:36:51.500 align:middle line:84% They think the buses are taking too long. 00:36:51.500 --> 00:36:54.270 align:middle line:84% So you are asked to investigate. 00:36:54.270 --> 00:36:56.840 align:middle line:90% Is the company-- 00:36:56.840 --> 00:37:00.730 align:middle line:84% Does it operate according to its promises or not. 00:37:00.730 --> 00:37:05.880 align:middle line:84% So you send an undercover agent to go and check the 00:37:05.880 --> 00:37:07.940 align:middle line:84% interarrival times of the buses. 00:37:07.940 --> 00:37:09.660 align:middle line:90% Are they 15 minutes? 00:37:09.660 --> 00:37:11.690 align:middle line:90% Or are they longer? 00:37:11.690 --> 00:37:17.660 align:middle line:84% So you put your dark glasses and you show up at the bus 00:37:17.660 --> 00:37:21.110 align:middle line:90% stop at some random time. 00:37:21.110 --> 00:37:25.530 align:middle line:84% And you go and ask the guy in the falafel truck, how long 00:37:25.530 --> 00:37:28.370 align:middle line:84% has it been since the last arrival? 00:37:28.370 --> 00:37:31.310 align:middle line:84% So of course that guy works for the FBI, right? 00:37:31.310 --> 00:37:36.900 align:middle line:84% So they tell you, well, it's been, let's say, 12 minutes 00:37:36.900 --> 00:37:39.360 align:middle line:90% since the last bus arrival. 00:37:39.360 --> 00:37:40.960 align:middle line:84% And then you say, "Oh, 12 minutes. 00:37:40.960 --> 00:37:42.780 align:middle line:90% Average time is 15. 00:37:42.780 --> 00:37:47.000 align:middle line:84% So a bus should be coming any time now." 00:37:47.000 --> 00:37:48.230 align:middle line:90% Is that correct? 00:37:48.230 --> 00:37:49.660 align:middle line:84% No, you wouldn't think that way. 00:37:49.660 --> 00:37:51.010 align:middle line:90% It's a Poisson process. 00:37:51.010 --> 00:37:53.810 align:middle line:84% It doesn't matter how long it has been since 00:37:53.810 --> 00:37:55.270 align:middle line:90% the last bus arrival. 00:37:55.270 --> 00:37:56.920 align:middle line:84% So you don't go through that fallacy. 00:37:56.920 --> 00:37:59.970 align:middle line:84% Instead of predicting how long it's going to be, you just sit 00:37:59.970 --> 00:38:03.300 align:middle line:84% down there and wait and measure the time. 00:38:03.300 --> 00:38:08.820 align:middle line:84% And you find that this is, let's say, 11 minutes. 00:38:08.820 --> 00:38:13.260 align:middle line:84% And you go to your boss and report, "Well, it took-- 00:38:13.260 --> 00:38:16.410 align:middle line:84% I went there and the time from the previous bus to the next 00:38:16.410 --> 00:38:18.310 align:middle line:90% one was 23 minutes. 00:38:18.310 --> 00:38:20.360 align:middle line:84% It's more than the 15 that they said." 00:38:20.360 --> 00:38:21.830 align:middle line:90% So go and do that again. 00:38:21.830 --> 00:38:23.590 align:middle line:90% You go day after day. 00:38:23.590 --> 00:38:28.350 align:middle line:84% You keep these statistics of the length of this interval. 00:38:28.350 --> 00:38:32.160 align:middle line:84% And you tell your boss it's a lot more than 15. 00:38:32.160 --> 00:38:36.720 align:middle line:84% It tends to be more like 30 or so. 00:38:36.720 --> 00:38:39.170 align:middle line:84% So the bus company is cheating us. 00:38:39.170 --> 00:38:43.490 align:middle line:84% Does the bus company really run Poisson buses at the rate 00:38:43.490 --> 00:38:46.490 align:middle line:90% that they have promised? 00:38:46.490 --> 00:38:51.270 align:middle line:84% Well let's analyze the situation here and figure out 00:38:51.270 --> 00:38:55.010 align:middle line:84% what the length of this interval 00:38:55.010 --> 00:38:57.900 align:middle line:90% should be, on the average. 00:38:57.900 --> 00:39:01.120 align:middle line:84% The naive argument is that this interval is an 00:39:01.120 --> 00:39:02.590 align:middle line:90% interarrival time. 00:39:02.590 --> 00:39:06.410 align:middle line:84% And interarrival times, on the average, are 15 minutes, if 00:39:06.410 --> 00:39:10.610 align:middle line:84% the company runs indeed Poisson processes with these 00:39:10.610 --> 00:39:11.850 align:middle line:90% interarrival times. 00:39:11.850 --> 00:39:14.970 align:middle line:84% But actually the situation is a little more subtle because 00:39:14.970 --> 00:39:19.940 align:middle line:84% this is not a typical interarrival interval. 00:39:19.940 --> 00:39:23.440 align:middle line:84% This interarrival interval consists of two pieces. 00:39:23.440 --> 00:39:28.810 align:middle line:84% Let's call them T1 and T1 prime. 00:39:28.810 --> 00:39:32.250 align:middle line:84% What can you tell me about those two random variables? 00:39:32.250 --> 00:39:35.940 align:middle line:84% What kind of random variable is T1? 00:39:35.940 --> 00:39:39.950 align:middle line:84% Starting from this time, with the Poisson process, the past 00:39:39.950 --> 00:39:41.290 align:middle line:90% doesn't matter. 00:39:41.290 --> 00:39:43.870 align:middle line:84% It's the time until an arrival happens. 00:39:43.870 --> 00:39:49.110 align:middle line:84% So T1 is going to be an exponential random variable 00:39:49.110 --> 00:39:50.425 align:middle line:90% with parameter lambda. 00:39:50.425 --> 00:39:53.300 align:middle line:90% 00:39:53.300 --> 00:39:56.620 align:middle line:84% So in particular, the expected value of T1 is 00:39:56.620 --> 00:40:00.260 align:middle line:90% going to be 15 by itself. 00:40:00.260 --> 00:40:02.720 align:middle line:84% How about the random variable T1 prime. 00:40:02.720 --> 00:40:07.130 align:middle line:84% What kind of random variable is it? 00:40:07.130 --> 00:40:14.180 align:middle line:84% This is like the first arrival in a Poisson process that runs 00:40:14.180 --> 00:40:17.650 align:middle line:90% backwards in time. 00:40:17.650 --> 00:40:20.330 align:middle line:84% What kind of process is a Poisson process running 00:40:20.330 --> 00:40:21.200 align:middle line:90% backwards in time? 00:40:21.200 --> 00:40:23.030 align:middle line:90% Let's think of coin flips. 00:40:23.030 --> 00:40:26.130 align:middle line:84% Suppose you have a movie of coin flips. 00:40:26.130 --> 00:40:29.480 align:middle line:84% And for some accident, that fascinating movie, you happen 00:40:29.480 --> 00:40:31.100 align:middle line:90% to watch it backwards. 00:40:31.100 --> 00:40:33.610 align:middle line:84% Will it look any different statistically? 00:40:33.610 --> 00:40:33.780 align:middle line:90% No. 00:40:33.780 --> 00:40:36.940 align:middle line:84% It's going to be just the sequence of random coin flips. 00:40:36.940 --> 00:40:40.770 align:middle line:84% So a Bernoulli process that's runs in reverse time is 00:40:40.770 --> 00:40:42.410 align:middle line:84% statistically identical to a Bernoulli 00:40:42.410 --> 00:40:44.290 align:middle line:90% process in forward time. 00:40:44.290 --> 00:40:46.600 align:middle line:84% The Poisson process is a limit of the Bernoulli. 00:40:46.600 --> 00:40:48.950 align:middle line:84% So, same story with the Poisson process. 00:40:48.950 --> 00:40:51.410 align:middle line:84% If you run it backwards in time it looks the same. 00:40:51.410 --> 00:40:55.190 align:middle line:84% So looking backwards in time, this is a Poisson process. 00:40:55.190 --> 00:40:58.930 align:middle line:84% And T1 prime is the time until the first arrival in this 00:40:58.930 --> 00:41:00.260 align:middle line:90% backward process. 00:41:00.260 --> 00:41:04.910 align:middle line:84% So T1 prime is also going to be an exponential random 00:41:04.910 --> 00:41:07.340 align:middle line:84% variable with the same parameter, lambda. 00:41:07.340 --> 00:41:11.000 align:middle line:84% And the expected value of T1 prime is 15. 00:41:11.000 --> 00:41:15.860 align:middle line:84% Conclusion is that the expected length of this 00:41:15.860 --> 00:41:22.860 align:middle line:84% interval is going to be 30 minutes. 00:41:22.860 --> 00:41:26.690 align:middle line:84% And the fact that this agent found the average to be 00:41:26.690 --> 00:41:31.230 align:middle line:84% something like 30 does not contradict the claims of the 00:41:31.230 --> 00:41:35.010 align:middle line:84% bus company that they're running Poisson buses with a 00:41:35.010 --> 00:41:38.370 align:middle line:90% rate of lambda equal to 4. 00:41:38.370 --> 00:41:38.780 align:middle line:90% OK. 00:41:38.780 --> 00:41:43.390 align:middle line:84% So maybe the company can this way-- they can defend 00:41:43.390 --> 00:41:44.970 align:middle line:90% themselves in court. 00:41:44.970 --> 00:41:47.490 align:middle line:84% But there's something puzzling here. 00:41:47.490 --> 00:41:50.360 align:middle line:84% How long is the interarrival time? 00:41:50.360 --> 00:41:51.910 align:middle line:90% Is it 15? 00:41:51.910 --> 00:41:53.216 align:middle line:90% Or is it 30? 00:41:53.216 --> 00:41:55.750 align:middle line:90% On the average. 00:41:55.750 --> 00:41:59.960 align:middle line:84% The issue is what do we mean by a typical 00:41:59.960 --> 00:42:01.360 align:middle line:90% interarrival time. 00:42:01.360 --> 00:42:04.940 align:middle line:84% When we say typical, we mean some kind of average. 00:42:04.940 --> 00:42:08.690 align:middle line:90% But average over what? 00:42:08.690 --> 00:42:13.280 align:middle line:84% And here's two different ways of thinking about averages. 00:42:13.280 --> 00:42:15.080 align:middle line:90% You number the buses. 00:42:15.080 --> 00:42:17.120 align:middle line:90% And you have bus number 100. 00:42:17.120 --> 00:42:21.120 align:middle line:84% You have bus number 101, bus number 102, bus 00:42:21.120 --> 00:42:24.660 align:middle line:90% number 110, and so on. 00:42:24.660 --> 00:42:29.370 align:middle line:84% One way of thinking about averages is that you pick a 00:42:29.370 --> 00:42:32.150 align:middle line:90% bus number at random. 00:42:32.150 --> 00:42:36.070 align:middle line:84% I pick, let's say, that bus, all buses being sort of 00:42:36.070 --> 00:42:37.760 align:middle line:90% equally likely to be picked. 00:42:37.760 --> 00:42:41.610 align:middle line:84% And I measure this interarrival time. 00:42:41.610 --> 00:42:45.380 align:middle line:90% So for a typical bus. 00:42:45.380 --> 00:42:50.390 align:middle line:84% Then, starting from here until there, the expected time has 00:42:50.390 --> 00:42:56.600 align:middle line:84% to be 1 over lambda, for the Poisson process. 00:42:56.600 --> 00:42:58.370 align:middle line:84% But what we did in this experiment 00:42:58.370 --> 00:42:59.720 align:middle line:90% was something different. 00:42:59.720 --> 00:43:02.040 align:middle line:84% We didn't pick a bus at random. 00:43:02.040 --> 00:43:05.090 align:middle line:90% We picked a time at random. 00:43:05.090 --> 00:43:08.870 align:middle line:84% And if the picture is, let's say, this way, I'm much more 00:43:08.870 --> 00:43:12.770 align:middle line:84% likely to pick this interval and therefore this 00:43:12.770 --> 00:43:16.290 align:middle line:84% interarrival time, rather than that interval. 00:43:16.290 --> 00:43:20.480 align:middle line:84% Because, this interval corresponds to very few times. 00:43:20.480 --> 00:43:23.680 align:middle line:84% So if I'm picking a time at random and, in some sense, 00:43:23.680 --> 00:43:27.430 align:middle line:84% let's say, uniform, so that all times are equally likely, 00:43:27.430 --> 00:43:31.190 align:middle line:84% I'm much more likely to fall inside a big interval rather 00:43:31.190 --> 00:43:32.710 align:middle line:90% than a small interval. 00:43:32.710 --> 00:43:37.140 align:middle line:84% So a person who shows up at the bus stop at a random time. 00:43:37.140 --> 00:43:42.040 align:middle line:84% They're selecting an interval in a biased way, with the bias 00:43:42.040 --> 00:43:44.350 align:middle line:90% favor of longer intervals. 00:43:44.350 --> 00:43:47.850 align:middle line:84% And that's why what they observe is a random variable 00:43:47.850 --> 00:43:51.830 align:middle line:84% that has a larger expected value then the ordinary 00:43:51.830 --> 00:43:53.080 align:middle line:90% expected value. 00:43:53.080 --> 00:43:56.780 align:middle line:84% So the subtlety here is to realize that we're talking 00:43:56.780 --> 00:43:59.590 align:middle line:84% between two different kinds of experiments. 00:43:59.590 --> 00:44:05.250 align:middle line:84% Picking a bus number at random verses picking an interval at 00:44:05.250 --> 00:44:11.500 align:middle line:84% random with a bias in favor of longer intervals. 00:44:11.500 --> 00:44:14.840 align:middle line:84% Lots of paradoxes that one can cook up using Poisson 00:44:14.840 --> 00:44:19.190 align:middle line:84% processes and random processes in general often have to do 00:44:19.190 --> 00:44:21.340 align:middle line:90% with the story of this kind. 00:44:21.340 --> 00:44:24.780 align:middle line:84% The phenomenon that we had in this particular example also 00:44:24.780 --> 00:44:28.970 align:middle line:84% shows up in general, whenever you have other kinds of 00:44:28.970 --> 00:44:30.470 align:middle line:90% arrival processes. 00:44:30.470 --> 00:44:34.210 align:middle line:84% So the Poisson process is the simplest arrival process there 00:44:34.210 --> 00:44:36.830 align:middle line:84% is, where the interarrival times are 00:44:36.830 --> 00:44:38.820 align:middle line:90% exponential random variables. 00:44:38.820 --> 00:44:40.280 align:middle line:84% There's a larger class of models. 00:44:40.280 --> 00:44:43.580 align:middle line:84% They're called renewal processes, in which, again, we 00:44:43.580 --> 00:44:46.900 align:middle line:84% have a sequence of successive arrivals, interarrival times 00:44:46.900 --> 00:44:50.100 align:middle line:84% are identically distributed and independent, but they may 00:44:50.100 --> 00:44:52.320 align:middle line:84% come from a general distribution. 00:44:52.320 --> 00:44:55.100 align:middle line:84% So to make the same point of the previous example but in a 00:44:55.100 --> 00:44:59.250 align:middle line:84% much simpler setting, suppose that bus interarrival times 00:44:59.250 --> 00:45:02.830 align:middle line:84% are either 5 or 10 minutes apart. 00:45:02.830 --> 00:45:05.930 align:middle line:84% So you get some intervals that are of length 5. 00:45:05.930 --> 00:45:08.790 align:middle line:84% You get some that are of length 10. 00:45:08.790 --> 00:45:12.810 align:middle line:84% And suppose that these are equally likely. 00:45:12.810 --> 00:45:16.990 align:middle line:90% So we have -- not exactly -- 00:45:16.990 --> 00:45:20.380 align:middle line:84% In the long run, we have as many 5 minute intervals as we 00:45:20.380 --> 00:45:22.490 align:middle line:90% have 10 minute intervals. 00:45:22.490 --> 00:45:30.590 align:middle line:84% So the average interarrival time is 7 and 1/2. 00:45:30.590 --> 00:45:35.850 align:middle line:84% But if a person shows up at a random time, what are they 00:45:35.850 --> 00:45:37.100 align:middle line:90% going to see? 00:45:37.100 --> 00:45:40.520 align:middle line:90% 00:45:40.520 --> 00:45:43.150 align:middle line:90% Do we have as many 5s as 10s? 00:45:43.150 --> 00:45:47.490 align:middle line:84% But every 10 covers twice as much space. 00:45:47.490 --> 00:45:52.640 align:middle line:84% So if I show up at a random time, I have probability 2/3 00:45:52.640 --> 00:45:57.180 align:middle line:84% falling inside an interval of duration 10. 00:45:57.180 --> 00:46:00.990 align:middle line:84% And I have one 1/3 probability of falling inside an interval 00:46:00.990 --> 00:46:02.460 align:middle line:90% of duration 5. 00:46:02.460 --> 00:46:06.710 align:middle line:84% That's because, out of the whole real line, 2/3 of it is 00:46:06.710 --> 00:46:08.810 align:middle line:84% covered by intervals of length 10, just 00:46:08.810 --> 00:46:09.590 align:middle line:90% because they're longer. 00:46:09.590 --> 00:46:12.280 align:middle line:84% 1/3 is covered by the smaller intervals. 00:46:12.280 --> 00:46:19.530 align:middle line:84% Now if I fall inside an interval of length 10 and I 00:46:19.530 --> 00:46:23.260 align:middle line:84% measure the length of the interval that I fell into, 00:46:23.260 --> 00:46:25.030 align:middle line:90% that's going to be 10. 00:46:25.030 --> 00:46:27.780 align:middle line:84% But if I fall inside an interval of length 5 and I 00:46:27.780 --> 00:46:30.320 align:middle line:84% measure how long it is, I'm going to get a 5. 00:46:30.320 --> 00:46:37.270 align:middle line:84% And that, of course, is going to be different than 7.5. 00:46:37.270 --> 00:46:38.010 align:middle line:90% OK. 00:46:38.010 --> 00:46:42.310 align:middle line:84% And which number should be bigger? 00:46:42.310 --> 00:46:45.110 align:middle line:84% It's the second number that's bigger because this one is 00:46:45.110 --> 00:46:48.930 align:middle line:84% biased in favor of the longer intervals. 00:46:48.930 --> 00:46:51.380 align:middle line:84% So that's, again, another illustration of the different 00:46:51.380 --> 00:46:54.640 align:middle line:84% results that you get when you have this random incidence 00:46:54.640 --> 00:46:55.990 align:middle line:90% phenomenon. 00:46:55.990 --> 00:46:59.320 align:middle line:84% So the bottom line, again, is that if you talk about a 00:46:59.320 --> 00:47:03.380 align:middle line:84% typical interarrival time, one must be very precise in 00:47:03.380 --> 00:47:05.370 align:middle line:84% specifying what we mean typical. 00:47:05.370 --> 00:47:08.120 align:middle line:84% So typical means sort of random. 00:47:08.120 --> 00:47:11.250 align:middle line:84% But to use the word random, you must specify very 00:47:11.250 --> 00:47:15.070 align:middle line:84% precisely what is the random experiment that you are using. 00:47:15.070 --> 00:47:18.920 align:middle line:84% And if you're not careful, you can get into apparent puzzles, 00:47:18.920 --> 00:47:20.770 align:middle line:90% such as the following. 00:47:20.770 --> 00:47:25.170 align:middle line:84% Suppose somebody tells you the average family size is 4, but 00:47:25.170 --> 00:47:30.340 align:middle line:84% the average person lives in a family of size 6. 00:47:30.340 --> 00:47:33.330 align:middle line:90% Is that compatible? 00:47:33.330 --> 00:47:36.610 align:middle line:84% Family size is 4 on the average, but typical people 00:47:36.610 --> 00:47:40.110 align:middle line:84% live, on the average, in families of size 6. 00:47:40.110 --> 00:47:41.590 align:middle line:90% Well yes. 00:47:41.590 --> 00:47:43.080 align:middle line:90% There's no contradiction here. 00:47:43.080 --> 00:47:45.450 align:middle line:84% We're talking about two different experiments. 00:47:45.450 --> 00:47:50.000 align:middle line:84% In one experiment, I pick a family at random, and I tell 00:47:50.000 --> 00:47:51.960 align:middle line:90% you the average family is 4. 00:47:51.960 --> 00:47:55.910 align:middle line:84% In another experiment, I pick a person at random and I tell 00:47:55.910 --> 00:47:58.310 align:middle line:84% you that this person, on the average, will be in their 00:47:58.310 --> 00:48:00.080 align:middle line:90% family of size 6. 00:48:00.080 --> 00:48:01.140 align:middle line:90% And what is the catch here? 00:48:01.140 --> 00:48:05.440 align:middle line:84% That if I pick a person at random, large families are 00:48:05.440 --> 00:48:08.160 align:middle line:90% more likely to be picked. 00:48:08.160 --> 00:48:11.710 align:middle line:84% So there's a bias in favor of large families. 00:48:11.710 --> 00:48:15.270 align:middle line:84% Or if you want to survey, let's say, are trains crowded 00:48:15.270 --> 00:48:16.495 align:middle line:90% in your city? 00:48:16.495 --> 00:48:19.170 align:middle line:90% Or are buses crowded? 00:48:19.170 --> 00:48:22.040 align:middle line:84% One choice is to pick a bus at random and inspect 00:48:22.040 --> 00:48:23.220 align:middle line:90% how crowded it is. 00:48:23.220 --> 00:48:27.260 align:middle line:84% Another choice is to pick a typical person and ask them, 00:48:27.260 --> 00:48:29.080 align:middle line:90% "Did you ride the bus today? 00:48:29.080 --> 00:48:33.500 align:middle line:84% Was it's crowded?" Well suppose that in this city 00:48:33.500 --> 00:48:36.265 align:middle line:84% there's one bus that's extremely crowded and all the 00:48:36.265 --> 00:48:38.520 align:middle line:84% other buses are completely empty. 00:48:38.520 --> 00:48:42.300 align:middle line:84% If you ask a person. "Was your bus crowded?" They will tell 00:48:42.300 --> 00:48:46.040 align:middle line:84% you, "Yes, my bus was crowded." There's no witness 00:48:46.040 --> 00:48:49.460 align:middle line:84% from the empty buses to testify in their favor. 00:48:49.460 --> 00:48:52.780 align:middle line:84% So by sampling people instead of sampling buses, you're 00:48:52.780 --> 00:48:54.940 align:middle line:90% going to get different result. 00:48:54.940 --> 00:48:58.320 align:middle line:84% And in the process industry, if your job is to inspect and 00:48:58.320 --> 00:49:01.450 align:middle line:84% check cookies, you will be faced with a big dilemma. 00:49:01.450 --> 00:49:05.190 align:middle line:84% Do you want to find out how many chocolate chips there are 00:49:05.190 --> 00:49:06.940 align:middle line:90% on a typical cookie? 00:49:06.940 --> 00:49:09.990 align:middle line:84% Are you going to interview cookies or are you going to 00:49:09.990 --> 00:49:13.880 align:middle line:84% interview chocolate chips and ask them how many other chips 00:49:13.880 --> 00:49:16.520 align:middle line:90% where there on your cookie? 00:49:16.520 --> 00:49:18.020 align:middle line:84% And you're going to get different 00:49:18.020 --> 00:49:19.210 align:middle line:90% answers in these cases. 00:49:19.210 --> 00:49:22.670 align:middle line:84% So moral is, one has to be very precise on how you 00:49:22.670 --> 00:49:26.160 align:middle line:84% formulate the sampling procedure that you have. 00:49:26.160 --> 00:49:28.330 align:middle line:84% And you'll get different answers.