1 00:00:00,820 --> 00:00:04,030 We now start our discussion of stochastic processes 2 00:00:04,030 --> 00:00:07,580 by starting with the simplest stochastic process there is. 3 00:00:07,580 --> 00:00:10,160 This is the so-called Bernoulli process, 4 00:00:10,160 --> 00:00:14,020 which is nothing but a sequence of independent Bernoulli 5 00:00:14,020 --> 00:00:15,310 trials. 6 00:00:15,310 --> 00:00:19,570 We let Xi stand for the random variable that 7 00:00:19,570 --> 00:00:22,700 describes the result in the i trial. 8 00:00:22,700 --> 00:00:26,460 The assumptions that we make are that at each trial 9 00:00:26,460 --> 00:00:31,360 there is a certain probability, p, that the trial results in 1. 10 00:00:31,360 --> 00:00:33,840 And in that case, we usually say that there 11 00:00:33,840 --> 00:00:36,090 is a success at the ith trial. 12 00:00:36,090 --> 00:00:38,310 And the remaining probability, 1 minus p, 13 00:00:38,310 --> 00:00:42,070 is assigned to the possibility that the random variable Xi 14 00:00:42,070 --> 00:00:44,310 takes a value of 0, in which case 15 00:00:44,310 --> 00:00:46,950 sometimes we say that there was a failure. 16 00:00:46,950 --> 00:00:49,330 Now, to keep things nontrivial, we 17 00:00:49,330 --> 00:00:53,390 will always assume that p is a number strictly between 0 18 00:00:53,390 --> 00:00:57,620 and 1, because otherwise, in the extreme cases of p equal to 0 19 00:00:57,620 --> 00:01:00,165 or p equal to 1, there isn't really any randomness. 20 00:01:02,700 --> 00:01:05,930 This process is something that we have already seen. 21 00:01:05,930 --> 00:01:09,130 We have worked plenty of examples involving 22 00:01:09,130 --> 00:01:13,210 repeated Bernoulli trials or repeated coin flips. 23 00:01:13,210 --> 00:01:17,730 We have solved several problems, we have seen several formulas. 24 00:01:17,730 --> 00:01:21,030 Here we will recapitulate some of them. 25 00:01:21,030 --> 00:01:23,700 But we will also start looking at the process 26 00:01:23,700 --> 00:01:26,910 from a new point of view. 27 00:01:26,910 --> 00:01:30,680 Before continuing, let me emphasize the assumptions that 28 00:01:30,680 --> 00:01:32,950 come into the Bernoulli process. 29 00:01:32,950 --> 00:01:36,810 One key assumption is that we have independence. 30 00:01:36,810 --> 00:01:40,920 The different trials are independent. 31 00:01:40,920 --> 00:01:43,150 The second assumption that we make 32 00:01:43,150 --> 00:01:46,420 is that the model is time homogeneous. 33 00:01:46,420 --> 00:01:49,009 What I mean by this is that this probability 34 00:01:49,009 --> 00:01:54,170 p of success at a given trial is the same for all trials. 35 00:01:54,170 --> 00:01:57,229 It does not depend on i. 36 00:01:57,229 --> 00:02:01,920 So this process is as simple as a stochastic process could be. 37 00:02:01,920 --> 00:02:05,360 But nevertheless, it can be used as a model 38 00:02:05,360 --> 00:02:07,080 in various situations. 39 00:02:07,080 --> 00:02:09,460 Sometimes it's clear that we're dealing with a Bernoulli 40 00:02:09,460 --> 00:02:13,160 process, but sometimes it also shows up in unexpected ways. 41 00:02:13,160 --> 00:02:16,400 In any case, the first simple example could be the following. 42 00:02:16,400 --> 00:02:19,920 Every week you play the lottery, and either you win 43 00:02:19,920 --> 00:02:21,250 or you do not to win. 44 00:02:21,250 --> 00:02:23,920 And assuming that it is the same kind of lottery 45 00:02:23,920 --> 00:02:27,040 that you play each week, the constant p 46 00:02:27,040 --> 00:02:30,100 would be the same, the probability of success. 47 00:02:30,100 --> 00:02:33,690 And assuming a lottery that is not rigged in any way, 48 00:02:33,690 --> 00:02:35,730 whether you win on one week, should 49 00:02:35,730 --> 00:02:39,570 be independent from what happens in other weeks. 50 00:02:39,570 --> 00:02:41,350 Quite often, the Bernoulli process 51 00:02:41,350 --> 00:02:45,720 is used as a model of arrivals, in which case, 52 00:02:45,720 --> 00:02:47,740 instead of saying probability of success, 53 00:02:47,740 --> 00:02:50,280 we would say probably of an arrival. 54 00:02:50,280 --> 00:02:53,170 The idea is that time is slotted, 55 00:02:53,170 --> 00:02:55,970 let's say, in seconds, for example. 56 00:02:55,970 --> 00:02:59,420 And each second you're sitting at the entrance of a bank 57 00:02:59,420 --> 00:03:02,430 and you make a note whether somebody 58 00:03:02,430 --> 00:03:04,620 came into the bank, in which case 59 00:03:04,620 --> 00:03:08,420 we have an arrival or success, or whether no one came 60 00:03:08,420 --> 00:03:11,760 during that time interval. 61 00:03:11,760 --> 00:03:15,560 If one here believes that different slots, 62 00:03:15,560 --> 00:03:18,320 different seconds, are independent of each other, 63 00:03:18,320 --> 00:03:21,140 then you do have a Bernoulli process. 64 00:03:21,140 --> 00:03:23,890 It might not be an exactly accurate model, 65 00:03:23,890 --> 00:03:25,780 but it is a good first approximation 66 00:03:25,780 --> 00:03:29,260 to start working with a model of this situation. 67 00:03:29,260 --> 00:03:32,540 Similarly, if you have a server, a computer, that 68 00:03:32,540 --> 00:03:36,579 takes jobs to process and jobs are coming randomly, 69 00:03:36,579 --> 00:03:38,940 you divide time into slots, and during each slot 70 00:03:38,940 --> 00:03:42,090 a job might arrive or might not arrive. 71 00:03:42,090 --> 00:03:45,480 And as a first approach to a model of this kind, 72 00:03:45,480 --> 00:03:49,950 you might as well employ the Bernoulli process. 73 00:03:49,950 --> 00:03:53,100 A final note, why is this process called the Bernoulli 74 00:03:53,100 --> 00:03:54,380 process? 75 00:03:54,380 --> 00:03:58,340 Well, the name comes from a famous family 76 00:03:58,340 --> 00:04:01,310 of Swiss mathematicians, the Bernoulli family. 77 00:04:01,310 --> 00:04:04,420 And one of them, Jacob Bernoulli, 78 00:04:04,420 --> 00:04:07,260 did many contributions to many branches of mathematics. 79 00:04:07,260 --> 00:04:09,180 But an important one was in the field 80 00:04:09,180 --> 00:04:11,190 of probability, where he actually 81 00:04:11,190 --> 00:04:17,920 derived quite deep results on a sequence of what we now 82 00:04:17,920 --> 00:04:20,370 call Bernoulli trials.