1 00:00:00,860 --> 00:00:04,410 In what kind of situations does the Poisson process arise? 2 00:00:04,410 --> 00:00:07,240 In general, it arises whenever we 3 00:00:07,240 --> 00:00:11,560 have events like arrivals that are somewhat rare, 4 00:00:11,560 --> 00:00:15,180 and which happen in a completely uncoordinated manner, 5 00:00:15,180 --> 00:00:18,550 so that they can show up at any particular time. 6 00:00:18,550 --> 00:00:21,250 In such situations, the number of arrivals 7 00:00:21,250 --> 00:00:25,480 will be often described by a certain distribution called 8 00:00:25,480 --> 00:00:27,860 the Poisson distribution, and which 9 00:00:27,860 --> 00:00:31,530 is named after the person who first studied this situation, 10 00:00:31,530 --> 00:00:33,750 who is a famous French mathematician 11 00:00:33,750 --> 00:00:37,350 by the name of Simon Denis Poisson. 12 00:00:37,350 --> 00:00:39,820 An early example where the data seems 13 00:00:39,820 --> 00:00:42,920 to fit the description of the Poisson process 14 00:00:42,920 --> 00:00:44,450 is a curious one. 15 00:00:44,450 --> 00:00:46,690 It had to do with deaths from horse 16 00:00:46,690 --> 00:00:51,010 kicks, that is, accidental deaths, in the Prussian army. 17 00:00:51,010 --> 00:00:54,110 The idea here is that a death by horse kick 18 00:00:54,110 --> 00:00:56,410 can happen pretty much at any time. 19 00:00:56,410 --> 00:01:00,200 And different arrivals, that is, different accidents 20 00:01:00,200 --> 00:01:02,760 are completely uncoordinated from each other. 21 00:01:02,760 --> 00:01:06,450 So the process is sort of completely random. 22 00:01:06,450 --> 00:01:08,240 For more scientific applications, 23 00:01:08,240 --> 00:01:11,260 it was realized that certain physical phenomena 24 00:01:11,260 --> 00:01:13,410 obey the Poisson process. 25 00:01:13,410 --> 00:01:15,590 Examples are the following. 26 00:01:15,590 --> 00:01:20,150 You have some radioactive body which decays, 27 00:01:20,150 --> 00:01:23,660 and the decaying happens once in awhile, 28 00:01:23,660 --> 00:01:25,880 emitting various particles. 29 00:01:25,880 --> 00:01:29,860 Different particles get emitted at completely random times 30 00:01:29,860 --> 00:01:32,390 in a completely uncoordinated manner 31 00:01:32,390 --> 00:01:34,460 and, therefore, this process is actually 32 00:01:34,460 --> 00:01:37,039 described as a Poisson process. 33 00:01:37,039 --> 00:01:39,900 Conversely, if you have a photo detector who 34 00:01:39,900 --> 00:01:42,710 looks at a very weak light source. 35 00:01:42,710 --> 00:01:47,030 So photons arrive from that weak light source one at a time. 36 00:01:47,030 --> 00:01:50,810 And you look at the time at which photons hit the detector. 37 00:01:50,810 --> 00:01:52,880 Then, the process of photon arrivals 38 00:01:52,880 --> 00:01:56,560 is very well-modeled by the Poisson process. 39 00:01:56,560 --> 00:01:59,800 For more modern applications, if you 40 00:01:59,800 --> 00:02:02,910 look at the financial markets and the times at which 41 00:02:02,910 --> 00:02:07,170 certain very unexpected events, like certain market shocks, 42 00:02:07,170 --> 00:02:10,160 occur, a model that is commonly employed 43 00:02:10,160 --> 00:02:13,610 is to use a Poisson process model. 44 00:02:13,610 --> 00:02:17,100 Although this is not an entirely accurate model, 45 00:02:17,100 --> 00:02:22,030 it provides a first approach to situations like this. 46 00:02:22,030 --> 00:02:24,750 But these days, the predominant source 47 00:02:24,750 --> 00:02:27,930 of applications for the Poisson process 48 00:02:27,930 --> 00:02:31,260 is in various service operations. 49 00:02:31,260 --> 00:02:32,940 You are the phone company. 50 00:02:32,940 --> 00:02:36,746 Phone calls get placed at random times. 51 00:02:36,746 --> 00:02:38,620 And because there are several people involved 52 00:02:38,620 --> 00:02:40,900 who are uncoordinated with each other, 53 00:02:40,900 --> 00:02:44,870 those calls get placed at completely random times. 54 00:02:44,870 --> 00:02:47,079 And the same story goes about, let's 55 00:02:47,079 --> 00:02:50,130 say, service requests to a web server, 56 00:02:50,130 --> 00:02:53,020 service requests to any kind of company. 57 00:02:53,020 --> 00:02:56,079 So, many applications that are being studied these days 58 00:02:56,079 --> 00:03:00,540 and which rest on Poisson models involve service operations 59 00:03:00,540 --> 00:03:02,510 of this type.