WEBVTT 00:00:00.820 --> 00:00:04.030 We now start our discussion of stochastic processes 00:00:04.030 --> 00:00:07.580 by starting with the simplest stochastic process there is. 00:00:07.580 --> 00:00:10.160 This is the so-called Bernoulli process, 00:00:10.160 --> 00:00:14.020 which is nothing but a sequence of independent Bernoulli 00:00:14.020 --> 00:00:15.310 trials. 00:00:15.310 --> 00:00:19.570 We let Xi stand for the random variable that 00:00:19.570 --> 00:00:22.700 describes the result in the i trial. 00:00:22.700 --> 00:00:26.460 The assumptions that we make are that at each trial 00:00:26.460 --> 00:00:31.360 there is a certain probability, p, that the trial results in 1. 00:00:31.360 --> 00:00:33.840 And in that case, we usually say that there 00:00:33.840 --> 00:00:36.090 is a success at the ith trial. 00:00:36.090 --> 00:00:38.310 And the remaining probability, 1 minus p, 00:00:38.310 --> 00:00:42.070 is assigned to the possibility that the random variable Xi 00:00:42.070 --> 00:00:44.310 takes a value of 0, in which case 00:00:44.310 --> 00:00:46.950 sometimes we say that there was a failure. 00:00:46.950 --> 00:00:49.330 Now, to keep things nontrivial, we 00:00:49.330 --> 00:00:53.390 will always assume that p is a number strictly between 0 00:00:53.390 --> 00:00:57.620 and 1, because otherwise, in the extreme cases of p equal to 0 00:00:57.620 --> 00:01:00.165 or p equal to 1, there isn't really any randomness. 00:01:02.700 --> 00:01:05.930 This process is something that we have already seen. 00:01:05.930 --> 00:01:09.130 We have worked plenty of examples involving 00:01:09.130 --> 00:01:13.210 repeated Bernoulli trials or repeated coin flips. 00:01:13.210 --> 00:01:17.730 We have solved several problems, we have seen several formulas. 00:01:17.730 --> 00:01:21.030 Here we will recapitulate some of them. 00:01:21.030 --> 00:01:23.700 But we will also start looking at the process 00:01:23.700 --> 00:01:26.910 from a new point of view. 00:01:26.910 --> 00:01:30.680 Before continuing, let me emphasize the assumptions that 00:01:30.680 --> 00:01:32.950 come into the Bernoulli process. 00:01:32.950 --> 00:01:36.810 One key assumption is that we have independence. 00:01:36.810 --> 00:01:40.920 The different trials are independent. 00:01:40.920 --> 00:01:43.150 The second assumption that we make 00:01:43.150 --> 00:01:46.420 is that the model is time homogeneous. 00:01:46.420 --> 00:01:49.009 What I mean by this is that this probability 00:01:49.009 --> 00:01:54.170 p of success at a given trial is the same for all trials. 00:01:54.170 --> 00:01:57.229 It does not depend on i. 00:01:57.229 --> 00:02:01.920 So this process is as simple as a stochastic process could be. 00:02:01.920 --> 00:02:05.360 But nevertheless, it can be used as a model 00:02:05.360 --> 00:02:07.080 in various situations. 00:02:07.080 --> 00:02:09.460 Sometimes it's clear that we're dealing with a Bernoulli 00:02:09.460 --> 00:02:13.160 process, but sometimes it also shows up in unexpected ways. 00:02:13.160 --> 00:02:16.400 In any case, the first simple example could be the following. 00:02:16.400 --> 00:02:19.920 Every week you play the lottery, and either you win 00:02:19.920 --> 00:02:21.250 or you do not to win. 00:02:21.250 --> 00:02:23.920 And assuming that it is the same kind of lottery 00:02:23.920 --> 00:02:27.040 that you play each week, the constant p 00:02:27.040 --> 00:02:30.100 would be the same, the probability of success. 00:02:30.100 --> 00:02:33.690 And assuming a lottery that is not rigged in any way, 00:02:33.690 --> 00:02:35.730 whether you win on one week, should 00:02:35.730 --> 00:02:39.570 be independent from what happens in other weeks. 00:02:39.570 --> 00:02:41.350 Quite often, the Bernoulli process 00:02:41.350 --> 00:02:45.720 is used as a model of arrivals, in which case, 00:02:45.720 --> 00:02:47.740 instead of saying probability of success, 00:02:47.740 --> 00:02:50.280 we would say probably of an arrival. 00:02:50.280 --> 00:02:53.170 The idea is that time is slotted, 00:02:53.170 --> 00:02:55.970 let's say, in seconds, for example. 00:02:55.970 --> 00:02:59.420 And each second you're sitting at the entrance of a bank 00:02:59.420 --> 00:03:02.430 and you make a note whether somebody 00:03:02.430 --> 00:03:04.620 came into the bank, in which case 00:03:04.620 --> 00:03:08.420 we have an arrival or success, or whether no one came 00:03:08.420 --> 00:03:11.760 during that time interval. 00:03:11.760 --> 00:03:15.560 If one here believes that different slots, 00:03:15.560 --> 00:03:18.320 different seconds, are independent of each other, 00:03:18.320 --> 00:03:21.140 then you do have a Bernoulli process. 00:03:21.140 --> 00:03:23.890 It might not be an exactly accurate model, 00:03:23.890 --> 00:03:25.780 but it is a good first approximation 00:03:25.780 --> 00:03:29.260 to start working with a model of this situation. 00:03:29.260 --> 00:03:32.540 Similarly, if you have a server, a computer, that 00:03:32.540 --> 00:03:36.579 takes jobs to process and jobs are coming randomly, 00:03:36.579 --> 00:03:38.940 you divide time into slots, and during each slot 00:03:38.940 --> 00:03:42.090 a job might arrive or might not arrive. 00:03:42.090 --> 00:03:45.480 And as a first approach to a model of this kind, 00:03:45.480 --> 00:03:49.950 you might as well employ the Bernoulli process. 00:03:49.950 --> 00:03:53.100 A final note, why is this process called the Bernoulli 00:03:53.100 --> 00:03:54.380 process? 00:03:54.380 --> 00:03:58.340 Well, the name comes from a famous family 00:03:58.340 --> 00:04:01.310 of Swiss mathematicians, the Bernoulli family. 00:04:01.310 --> 00:04:04.420 And one of them, Jacob Bernoulli, 00:04:04.420 --> 00:04:07.260 did many contributions to many branches of mathematics. 00:04:07.260 --> 00:04:09.180 But an important one was in the field 00:04:09.180 --> 00:04:11.190 of probability, where he actually 00:04:11.190 --> 00:04:17.920 derived quite deep results on a sequence of what we now 00:04:17.920 --> 00:04:20.370 call Bernoulli trials.