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 --> 00:00:17.390 align:middle line:84% hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:17.390 --> 00:00:18.640 align:middle line:90% ocw.mit.edu. 00:00:18.640 --> 00:00:22.710 align:middle line:90% 00:00:22.710 --> 00:00:26.630 align:middle line:84% PROFESSOR: So by now you have seen pretty much every 00:00:26.630 --> 00:00:30.273 align:middle line:84% possible trick there is in basic probability theory, 00:00:30.273 --> 00:00:33.100 align:middle line:84% about how to calculate distributions, and so on. 00:00:33.100 --> 00:00:37.230 align:middle line:84% You have the basic tools to do pretty much anything. 00:00:37.230 --> 00:00:40.610 align:middle line:90% So what's coming after this? 00:00:40.610 --> 00:00:45.370 align:middle line:84% Well, probability is useful for developing the science of 00:00:45.370 --> 00:00:48.120 align:middle line:84% inference, and this is a subject to which we're going 00:00:48.120 --> 00:00:51.280 align:middle line:84% to come back at the end of the semester. 00:00:51.280 --> 00:00:55.240 align:middle line:84% Another chapter, which is what we will be doing over the next 00:00:55.240 --> 00:00:59.210 align:middle line:84% few weeks, is to deal with phenomena that evolve in time. 00:00:59.210 --> 00:01:03.300 align:middle line:84% So so-called random processes or stochastic processes. 00:01:03.300 --> 00:01:05.069 align:middle line:90% So what is this about? 00:01:05.069 --> 00:01:08.100 align:middle line:84% So in the real world, you don't just throw two random 00:01:08.100 --> 00:01:09.520 align:middle line:90% variables and go home. 00:01:09.520 --> 00:01:11.410 align:middle line:90% Rather the world goes on. 00:01:11.410 --> 00:01:14.880 align:middle line:84% So you generate the random variable, then you get more 00:01:14.880 --> 00:01:18.100 align:middle line:84% random variables, and things evolve in time. 00:01:18.100 --> 00:01:21.560 align:middle line:84% And random processes are supposed to be models that 00:01:21.560 --> 00:01:25.680 align:middle line:84% capture the evolution of random phenomena over time. 00:01:25.680 --> 00:01:27.620 align:middle line:84% So that's what we will be doing. 00:01:27.620 --> 00:01:31.230 align:middle line:84% Now when we have evolution in time, mathematically speaking, 00:01:31.230 --> 00:01:34.840 align:middle line:84% you can use discrete time or continuous time. 00:01:34.840 --> 00:01:36.800 align:middle line:84% Of course, discrete time is easier. 00:01:36.800 --> 00:01:39.280 align:middle line:84% And that's where we're going to start from. 00:01:39.280 --> 00:01:43.000 align:middle line:84% And we're going to start from the easiest, simplest random 00:01:43.000 --> 00:01:46.740 align:middle line:84% process, which is the so-called Bernoulli process, 00:01:46.740 --> 00:01:50.250 align:middle line:84% which is nothing but just a sequence of coin flips. 00:01:50.250 --> 00:01:54.650 align:middle line:84% You keep flipping a coin and keep going forever. 00:01:54.650 --> 00:01:56.380 align:middle line:84% That's what the Bernoulli process is. 00:01:56.380 --> 00:01:58.290 align:middle line:84% So in some sense it's something that you have 00:01:58.290 --> 00:01:59.160 align:middle line:90% already seen. 00:01:59.160 --> 00:02:03.180 align:middle line:84% But we're going to introduce a few additional ideas here that 00:02:03.180 --> 00:02:07.460 align:middle line:84% will be useful and relevant as we go along and we move on to 00:02:07.460 --> 00:02:09.949 align:middle line:90% continuous time processes. 00:02:09.949 --> 00:02:12.640 align:middle line:84% So we're going to define the Bernoulli process, talk about 00:02:12.640 --> 00:02:17.800 align:middle line:84% some basic properties that the process has, and derive a few 00:02:17.800 --> 00:02:21.440 align:middle line:84% formulas, and exploit the special structure that it has 00:02:21.440 --> 00:02:24.930 align:middle line:84% to do a few quite interesting things. 00:02:24.930 --> 00:02:29.550 align:middle line:84% By the way, where does the word Bernoulli come from? 00:02:29.550 --> 00:02:33.010 align:middle line:84% Well the Bernoulli's were a family of mathematicians, 00:02:33.010 --> 00:02:37.200 align:middle line:84% Swiss mathematicians and scientists around the 1700s. 00:02:37.200 --> 00:02:39.680 align:middle line:84% There were so many of them that actually-- 00:02:39.680 --> 00:02:42.440 align:middle line:84% and some of them had the same first name-- 00:02:42.440 --> 00:02:46.920 align:middle line:84% historians even have difficulty of figuring out who 00:02:46.920 --> 00:02:48.660 align:middle line:90% exactly did what. 00:02:48.660 --> 00:02:51.430 align:middle line:84% But in any case, you can imagine that at the dinner 00:02:51.430 --> 00:02:53.710 align:middle line:84% table they were probably flipping coins and doing 00:02:53.710 --> 00:02:55.570 align:middle line:90% Bernoulli trials. 00:02:55.570 --> 00:02:58.290 align:middle line:84% So maybe that was their pass-time. 00:02:58.290 --> 00:02:58.610 align:middle line:90% OK. 00:02:58.610 --> 00:03:02.160 align:middle line:84% So what is the Bernoulli process? 00:03:02.160 --> 00:03:05.750 align:middle line:84% The Bernoulli process is nothing but a sequence of 00:03:05.750 --> 00:03:08.700 align:middle line:84% independent Bernoulli trials that you can 00:03:08.700 --> 00:03:11.120 align:middle line:90% think of as coin flips. 00:03:11.120 --> 00:03:13.530 align:middle line:84% So you can think the result of each trial 00:03:13.530 --> 00:03:15.350 align:middle line:90% being heads or tails. 00:03:15.350 --> 00:03:19.370 align:middle line:84% It's a little more convenient maybe to talk about successes 00:03:19.370 --> 00:03:21.690 align:middle line:84% and failures instead of heads or tails. 00:03:21.690 --> 00:03:24.470 align:middle line:84% Or if you wish numerical values, to use a 1 for a 00:03:24.470 --> 00:03:27.330 align:middle line:90% success and 0 for a failure. 00:03:27.330 --> 00:03:30.820 align:middle line:84% So the model is that each one of these trials has the same 00:03:30.820 --> 00:03:34.240 align:middle line:90% probability of success, p. 00:03:34.240 --> 00:03:37.050 align:middle line:84% And the other assumption is that these trials are 00:03:37.050 --> 00:03:40.880 align:middle line:84% statistically independent of each other. 00:03:40.880 --> 00:03:43.520 align:middle line:84% So what could be some examples of Bernoulli trials? 00:03:43.520 --> 00:03:48.010 align:middle line:84% You buy a lottery ticket every week and you win or lose. 00:03:48.010 --> 00:03:50.080 align:middle line:84% Presumably, these are independent of each other. 00:03:50.080 --> 00:03:53.240 align:middle line:84% And if it's the same kind of lottery, the probability of 00:03:53.240 --> 00:03:55.930 align:middle line:84% winning should be the same during every week. 00:03:55.930 --> 00:03:58.840 align:middle line:84% Maybe you want to model the financial markets. 00:03:58.840 --> 00:04:02.810 align:middle line:84% And a crude model could be that on any given day the Dow 00:04:02.810 --> 00:04:05.860 align:middle line:84% Jones is going to go up or down with a certain 00:04:05.860 --> 00:04:07.300 align:middle line:90% probability. 00:04:07.300 --> 00:04:11.770 align:middle line:84% Well that probability must be somewhere around 0.5, or so. 00:04:11.770 --> 00:04:14.890 align:middle line:84% This is a crude model of financial markets. 00:04:14.890 --> 00:04:17.910 align:middle line:84% You say, probably there is more into them. 00:04:17.910 --> 00:04:19.670 align:middle line:90% Life is not that simple. 00:04:19.670 --> 00:04:23.770 align:middle line:84% But actually it's a pretty reasonable model. 00:04:23.770 --> 00:04:26.260 align:middle line:84% It takes quite a bit of work to come up with more 00:04:26.260 --> 00:04:29.590 align:middle line:84% sophisticated models that can do better predictions than 00:04:29.590 --> 00:04:32.150 align:middle line:90% just pure heads and tails. 00:04:32.150 --> 00:04:36.270 align:middle line:84% Now more interesting, perhaps to the examples we will be 00:04:36.270 --> 00:04:37.975 align:middle line:90% dealing with in this class-- 00:04:37.975 --> 00:04:43.760 align:middle line:84% a Bernoulli process is a good model for streams of arrivals 00:04:43.760 --> 00:04:45.940 align:middle line:90% of any kind to a facility. 00:04:45.940 --> 00:04:49.790 align:middle line:84% So it could be a bank, and you are sitting at 00:04:49.790 --> 00:04:50.710 align:middle line:90% the door of the bank. 00:04:50.710 --> 00:04:54.180 align:middle line:84% And at every second, you check whether a customer came in 00:04:54.180 --> 00:04:56.210 align:middle line:90% during that second or not. 00:04:56.210 --> 00:05:00.670 align:middle line:84% Or you can think about arrivals of jobs to a server. 00:05:00.670 --> 00:05:04.890 align:middle line:84% Or any other kind of requests to a service system. 00:05:04.890 --> 00:05:08.560 align:middle line:84% So requests, or jobs, arrive at random times. 00:05:08.560 --> 00:05:12.360 align:middle line:84% You split the time into time slots. 00:05:12.360 --> 00:05:15.180 align:middle line:84% And during each time slot something comes or something 00:05:15.180 --> 00:05:16.610 align:middle line:90% does not come. 00:05:16.610 --> 00:05:20.110 align:middle line:84% And for many applications, it's a reasonable assumption 00:05:20.110 --> 00:05:24.510 align:middle line:84% to make that arrivals on any given slot are independent of 00:05:24.510 --> 00:05:27.530 align:middle line:84% arrivals in any other time slot. 00:05:27.530 --> 00:05:30.910 align:middle line:84% So each time slot can be viewed as a trial, where 00:05:30.910 --> 00:05:32.810 align:middle line:84% either something comes or doesn't come. 00:05:32.810 --> 00:05:35.830 align:middle line:84% And different trials are independent of each other. 00:05:35.830 --> 00:05:38.000 align:middle line:84% Now there's two assumptions that we're making here. 00:05:38.000 --> 00:05:40.000 align:middle line:84% One is the independence assumption. 00:05:40.000 --> 00:05:42.190 align:middle line:84% The other is that this number, p, probability 00:05:42.190 --> 00:05:44.520 align:middle line:90% of success, is constant. 00:05:44.520 --> 00:05:47.840 align:middle line:84% Now if you think about the bank example, if you stand 00:05:47.840 --> 00:05:53.330 align:middle line:84% outside the bank at 9:30 in the morning, you'll see 00:05:53.330 --> 00:05:55.830 align:middle line:84% arrivals happening at a certain rate. 00:05:55.830 --> 00:05:59.370 align:middle line:84% If you stand outside the bank at 12:00 noon, probably 00:05:59.370 --> 00:06:01.240 align:middle line:90% arrivals are more frequent. 00:06:01.240 --> 00:06:03.680 align:middle line:84% Which means that the given time slot has a higher 00:06:03.680 --> 00:06:08.030 align:middle line:84% probability of seeing an arrival around noon time. 00:06:08.030 --> 00:06:11.780 align:middle line:84% This means that the assumption of a constant p is probably 00:06:11.780 --> 00:06:15.080 align:middle line:84% not correct in that setting, if you're talking 00:06:15.080 --> 00:06:16.630 align:middle line:90% about the whole day. 00:06:16.630 --> 00:06:19.670 align:middle line:84% So the probability of successes or arrivals in the 00:06:19.670 --> 00:06:24.340 align:middle line:84% morning is going to be smaller than what it would be at noon. 00:06:24.340 --> 00:06:27.775 align:middle line:84% But if you're talking about a time period, let's say 10:00 00:06:27.775 --> 00:06:31.980 align:middle line:84% to 10:15, probably all slots have the same probability of 00:06:31.980 --> 00:06:34.880 align:middle line:84% seeing an arrival and it's a good approximation. 00:06:34.880 --> 00:06:37.450 align:middle line:84% So we're going to stick with the assumption that p is 00:06:37.450 --> 00:06:41.110 align:middle line:84% constant, doesn't change with time. 00:06:41.110 --> 00:06:44.480 align:middle line:84% Now that we have our model what do we do with it? 00:06:44.480 --> 00:06:47.200 align:middle line:84% Well, we start talking about the statistical properties 00:06:47.200 --> 00:06:48.510 align:middle line:90% that it has. 00:06:48.510 --> 00:06:52.400 align:middle line:84% And here there's two slightly different perspectives of 00:06:52.400 --> 00:06:55.680 align:middle line:84% thinking about what a random process is. 00:06:55.680 --> 00:06:59.060 align:middle line:84% The simplest version is to think about the random process 00:06:59.060 --> 00:07:01.605 align:middle line:84% as being just a sequence of random variables. 00:07:01.605 --> 00:07:04.210 align:middle line:90% 00:07:04.210 --> 00:07:06.900 align:middle line:84% We know what random variables are. 00:07:06.900 --> 00:07:09.440 align:middle line:84% We know what multiple random variables are. 00:07:09.440 --> 00:07:12.760 align:middle line:84% So it's just an experiment that has associated with it a 00:07:12.760 --> 00:07:14.730 align:middle line:90% bunch of random variables. 00:07:14.730 --> 00:07:17.410 align:middle line:84% So once you have random variables, what do you do 00:07:17.410 --> 00:07:18.480 align:middle line:90% instinctively? 00:07:18.480 --> 00:07:20.140 align:middle line:84% You talk about the distribution of 00:07:20.140 --> 00:07:21.610 align:middle line:90% these random variables. 00:07:21.610 --> 00:07:25.690 align:middle line:84% We already specified for the Bernoulli process that each Xi 00:07:25.690 --> 00:07:27.830 align:middle line:84% is a Bernoulli random variable, with probability of 00:07:27.830 --> 00:07:29.480 align:middle line:90% success equal to p. 00:07:29.480 --> 00:07:31.640 align:middle line:84% That specifies the distribution of the random 00:07:31.640 --> 00:07:35.500 align:middle line:84% variable X, or Xt, for general time t. 00:07:35.500 --> 00:07:37.780 align:middle line:84% Then you can calculate expected values and 00:07:37.780 --> 00:07:39.420 align:middle line:90% variances, and so on. 00:07:39.420 --> 00:07:44.050 align:middle line:84% So the expected value is, with probability p, you get a 1. 00:07:44.050 --> 00:07:46.650 align:middle line:84% And with probability 1 - p, you get a 0. 00:07:46.650 --> 00:07:49.510 align:middle line:84% So the expected value is equal to p. 00:07:49.510 --> 00:07:52.890 align:middle line:84% And then we have seen before a formula for the variance of 00:07:52.890 --> 00:07:57.040 align:middle line:84% the Bernoulli random variable, which is p times 1-p. 00:07:57.040 --> 00:08:00.350 align:middle line:84% So this way we basically now have all the statistical 00:08:00.350 --> 00:08:04.680 align:middle line:84% properties of the random variable Xt, and we have those 00:08:04.680 --> 00:08:06.670 align:middle line:90% properties for every t. 00:08:06.670 --> 00:08:10.140 align:middle line:84% Is this enough of a probabilistic description of a 00:08:10.140 --> 00:08:11.580 align:middle line:90% random process? 00:08:11.580 --> 00:08:12.240 align:middle line:90% Well, no. 00:08:12.240 --> 00:08:15.070 align:middle line:84% You need to know how the different random variables 00:08:15.070 --> 00:08:16.820 align:middle line:90% relate to each other. 00:08:16.820 --> 00:08:20.870 align:middle line:84% If you're talking about a general random process, you 00:08:20.870 --> 00:08:23.490 align:middle line:90% would like to know things. 00:08:23.490 --> 00:08:26.370 align:middle line:84% For example, the joint distribution of X2, 00:08:26.370 --> 00:08:29.220 align:middle line:90% with X5, and X7. 00:08:29.220 --> 00:08:31.680 align:middle line:84% For example, that might be something that you're 00:08:31.680 --> 00:08:32.809 align:middle line:90% interested in. 00:08:32.809 --> 00:08:38.820 align:middle line:84% And the way you specify it is by giving the joint PMF of 00:08:38.820 --> 00:08:40.530 align:middle line:90% these random variables. 00:08:40.530 --> 00:08:44.440 align:middle line:84% And you have to do that for every collection, or any 00:08:44.440 --> 00:08:46.150 align:middle line:84% subset, of the random variables you 00:08:46.150 --> 00:08:47.300 align:middle line:90% are interested in. 00:08:47.300 --> 00:08:49.860 align:middle line:84% So to have a complete description of a random 00:08:49.860 --> 00:08:54.770 align:middle line:84% processes, you need to specify for me all the possible joint 00:08:54.770 --> 00:08:55.960 align:middle line:90% distributions. 00:08:55.960 --> 00:08:58.780 align:middle line:84% And once you have all the possible joint distributions, 00:08:58.780 --> 00:09:01.580 align:middle line:84% then you can answer, in principle, any questions you 00:09:01.580 --> 00:09:03.280 align:middle line:90% might be interested in. 00:09:03.280 --> 00:09:05.670 align:middle line:84% How did we get around this issue for 00:09:05.670 --> 00:09:06.690 align:middle line:90% the Bernoulli process? 00:09:06.690 --> 00:09:10.030 align:middle line:84% I didn't give you the joint distributions explicitly. 00:09:10.030 --> 00:09:12.160 align:middle line:84% But I gave them to you implicitly. 00:09:12.160 --> 00:09:15.250 align:middle line:84% And this is because I told you that the different random 00:09:15.250 --> 00:09:18.390 align:middle line:84% variables are independent of each other. 00:09:18.390 --> 00:09:21.490 align:middle line:84% So at least for the Bernoulli process, where we make the 00:09:21.490 --> 00:09:24.320 align:middle line:84% independence assumption, we know that this is going to be 00:09:24.320 --> 00:09:25.970 align:middle line:90% the product of the PMFs. 00:09:25.970 --> 00:09:29.000 align:middle line:90% 00:09:29.000 --> 00:09:33.850 align:middle line:84% And since I have told you what the individual PMFs are, this 00:09:33.850 --> 00:09:37.090 align:middle line:84% means that you automatically know all the joint PMFs. 00:09:37.090 --> 00:09:40.940 align:middle line:84% And we can go to business based on that. 00:09:40.940 --> 00:09:41.310 align:middle line:90% All right. 00:09:41.310 --> 00:09:45.160 align:middle line:84% So this is one view of what a random process is, just a 00:09:45.160 --> 00:09:47.180 align:middle line:84% collection of random variables. 00:09:47.180 --> 00:09:50.660 align:middle line:84% There's another view that's a little more abstract, which is 00:09:50.660 --> 00:09:53.170 align:middle line:90% the following. 00:09:53.170 --> 00:09:57.100 align:middle line:84% The entire process is to be thought of as one long 00:09:57.100 --> 00:09:58.520 align:middle line:90% experiment. 00:09:58.520 --> 00:10:01.970 align:middle line:84% So we go back to the chapter one view of 00:10:01.970 --> 00:10:03.550 align:middle line:90% probabilistic models. 00:10:03.550 --> 00:10:06.240 align:middle line:84% So there must be a sample space involved. 00:10:06.240 --> 00:10:07.660 align:middle line:90% What is the sample space? 00:10:07.660 --> 00:10:11.760 align:middle line:84% If I do my infinite, long experiment of flipping an 00:10:11.760 --> 00:10:15.330 align:middle line:84% infinite number of coins, a typical outcome of the 00:10:15.330 --> 00:10:21.140 align:middle line:84% experiment would be a sequence of 0's and 1's. 00:10:21.140 --> 00:10:25.640 align:middle line:84% So this could be one possible outcome of the experiment, 00:10:25.640 --> 00:10:28.630 align:middle line:84% just an infinite sequence of 0's and 1's. 00:10:28.630 --> 00:10:33.020 align:middle line:84% My sample space is the set of all possible 00:10:33.020 --> 00:10:35.050 align:middle line:90% outcomes of this kind. 00:10:35.050 --> 00:10:40.660 align:middle line:84% Here's another possible outcome, and so on. 00:10:40.660 --> 00:10:44.060 align:middle line:84% And essentially we're dealing with a sample space, which is 00:10:44.060 --> 00:10:46.980 align:middle line:84% the space of all sequences of 0's and 1's. 00:10:46.980 --> 00:10:50.470 align:middle line:84% And we're making some sort of probabilistic assumption about 00:10:50.470 --> 00:10:53.330 align:middle line:84% what may happen in that experiment. 00:10:53.330 --> 00:10:56.350 align:middle line:84% So one particular sequence that we may be interested in 00:10:56.350 --> 00:10:59.660 align:middle line:84% is the sequence of obtaining all 1's. 00:10:59.660 --> 00:11:05.510 align:middle line:84% So this is the sequence that gives you 1's forever. 00:11:05.510 --> 00:11:08.330 align:middle line:84% Once you take the point of view that this is our sample 00:11:08.330 --> 00:11:10.760 align:middle line:84% space-- its the space of all infinite sequences-- 00:11:10.760 --> 00:11:13.770 align:middle line:84% you can start asking questions that have to do 00:11:13.770 --> 00:11:15.470 align:middle line:90% with infinite sequences. 00:11:15.470 --> 00:11:19.120 align:middle line:84% Such as the question, what's the probability of obtaining 00:11:19.120 --> 00:11:23.180 align:middle line:84% the infinite sequence that consists of all 1's? 00:11:23.180 --> 00:11:24.690 align:middle line:90% So what is this probability? 00:11:24.690 --> 00:11:27.240 align:middle line:84% Let's see how we could calculate it. 00:11:27.240 --> 00:11:34.000 align:middle line:84% So the probability of obtaining all 1's is certainly 00:11:34.000 --> 00:11:39.890 align:middle line:84% less than or equal to the probability of obtaining 1's, 00:11:39.890 --> 00:11:42.335 align:middle line:90% just in the first 10 tosses. 00:11:42.335 --> 00:11:45.075 align:middle line:90% 00:11:45.075 --> 00:11:47.030 align:middle line:90% OK. 00:11:47.030 --> 00:11:50.810 align:middle line:84% This is asking for more things to happen than this. 00:11:50.810 --> 00:11:55.780 align:middle line:84% If this event is true, then this is also true. 00:11:55.780 --> 00:11:58.660 align:middle line:84% Therefore the probability of this is smaller than the 00:11:58.660 --> 00:11:59.540 align:middle line:90% probability of that. 00:11:59.540 --> 00:12:03.410 align:middle line:84% This event is contained in that event. 00:12:03.410 --> 00:12:04.950 align:middle line:90% This implies this. 00:12:04.950 --> 00:12:06.880 align:middle line:90% So we have this inequality. 00:12:06.880 --> 00:12:12.360 align:middle line:84% Now what's the probability of obtaining 1's in 10 trials? 00:12:12.360 --> 00:12:15.980 align:middle line:84% This is just p to the 10th because the trials are 00:12:15.980 --> 00:12:18.530 align:middle line:90% independent. 00:12:18.530 --> 00:12:22.780 align:middle line:84% Now of course there's no reason why I chose 10 here. 00:12:22.780 --> 00:12:26.160 align:middle line:84% The same argument goes through if I use an 00:12:26.160 --> 00:12:29.850 align:middle line:90% arbitrary number, k. 00:12:29.850 --> 00:12:34.250 align:middle line:84% And this has to be true for all k. 00:12:34.250 --> 00:12:38.690 align:middle line:84% So this probability is less than p to the k, no matter 00:12:38.690 --> 00:12:41.670 align:middle line:90% what k I choose. 00:12:41.670 --> 00:12:46.350 align:middle line:84% Therefore, this must be less than or equal to the limit of 00:12:46.350 --> 00:12:48.660 align:middle line:90% this, as k goes to infinity. 00:12:48.660 --> 00:12:51.210 align:middle line:84% This is smaller than that for all k's. 00:12:51.210 --> 00:12:55.860 align:middle line:84% Let k go to infinity, take k arbitrarily large, this number 00:12:55.860 --> 00:12:57.770 align:middle line:84% is going to become arbitrarily small. 00:12:57.770 --> 00:12:59.190 align:middle line:90% It goes to 0. 00:12:59.190 --> 00:13:02.480 align:middle line:84% And that proves that the probability of an infinite 00:13:02.480 --> 00:13:06.080 align:middle line:90% sequence of 1's is equal to 0. 00:13:06.080 --> 00:13:09.800 align:middle line:90% So take limits of both sides. 00:13:09.800 --> 00:13:13.220 align:middle line:90% 00:13:13.220 --> 00:13:16.217 align:middle line:84% It's going to be less than or equal to the limit-- 00:13:16.217 --> 00:13:18.380 align:middle line:90% I shouldn't take a limit here. 00:13:18.380 --> 00:13:21.745 align:middle line:84% The probability is less than or equal to the limit of p to 00:13:21.745 --> 00:13:26.000 align:middle line:84% the k, as k goes to infinity, which is 0. 00:13:26.000 --> 00:13:30.880 align:middle line:84% So this proves in a formal way that the sequence of all 1's 00:13:30.880 --> 00:13:32.650 align:middle line:90% has 0 probability. 00:13:32.650 --> 00:13:35.770 align:middle line:84% If you have an infinite number of coin flips, what's the 00:13:35.770 --> 00:13:40.610 align:middle line:84% probability that all of the coin flips result in heads? 00:13:40.610 --> 00:13:43.690 align:middle line:84% The probability of this happening is equal to zero. 00:13:43.690 --> 00:13:48.310 align:middle line:84% So this particular sequence has 0 probability. 00:13:48.310 --> 00:13:51.480 align:middle line:84% Of course, I'm assuming here that p is less than 1, 00:13:51.480 --> 00:13:53.420 align:middle line:90% strictly less than 1. 00:13:53.420 --> 00:13:56.280 align:middle line:84% Now the interesting thing is that if you look at any other 00:13:56.280 --> 00:13:59.690 align:middle line:84% infinite sequence, and you try to calculate the probability 00:13:59.690 --> 00:14:03.526 align:middle line:84% of that infinite sequence, you would get a product of (1-p) 00:14:03.526 --> 00:14:07.600 align:middle line:84% times 1, 1-p times 1, 1-p, times p times p, 00:14:07.600 --> 00:14:09.570 align:middle line:90% times 1-p and so on. 00:14:09.570 --> 00:14:13.560 align:middle line:84% You keep multiplying numbers that are less than 1. 00:14:13.560 --> 00:14:16.760 align:middle line:84% Again, I'm making the assumption that p is 00:14:16.760 --> 00:14:17.940 align:middle line:90% between 0 and 1. 00:14:17.940 --> 00:14:21.140 align:middle line:84% So 1-p is less than 1, p is less than 1. 00:14:21.140 --> 00:14:23.600 align:middle line:84% You keep multiplying numbers less than 1. 00:14:23.600 --> 00:14:26.190 align:middle line:84% If you multiply infinitely many such numbers, the 00:14:26.190 --> 00:14:28.470 align:middle line:90% infinite product becomes 0. 00:14:28.470 --> 00:14:33.310 align:middle line:84% So any individual sequence in this sample space actually has 00:14:33.310 --> 00:14:35.230 align:middle line:90% 0 probability. 00:14:35.230 --> 00:14:39.560 align:middle line:84% And that is a little bit counter-intuitive perhaps. 00:14:39.560 --> 00:14:42.670 align:middle line:84% But the situation is more like the situation where we deal 00:14:42.670 --> 00:14:44.680 align:middle line:84% with continuous random variables. 00:14:44.680 --> 00:14:47.430 align:middle line:84% So if you could draw a continuous random variable, 00:14:47.430 --> 00:14:50.810 align:middle line:84% every possible outcome has 0 probability. 00:14:50.810 --> 00:14:52.320 align:middle line:90% And that's fine. 00:14:52.320 --> 00:14:54.860 align:middle line:84% But all of the outcomes collectively still have 00:14:54.860 --> 00:14:56.280 align:middle line:90% positive probability. 00:14:56.280 --> 00:14:59.580 align:middle line:84% So the situation here is very much similar. 00:14:59.580 --> 00:15:03.940 align:middle line:84% So the space of infinite sequences of 0's and 1's, that 00:15:03.940 --> 00:15:07.340 align:middle line:84% sample space is very much like a continuous space. 00:15:07.340 --> 00:15:10.340 align:middle line:84% If you want to push that analogy further, you could 00:15:10.340 --> 00:15:15.830 align:middle line:84% think of this as the expansion of a real number. 00:15:15.830 --> 00:15:18.950 align:middle line:84% Or the representation of a real number in binary. 00:15:18.950 --> 00:15:22.540 align:middle line:84% Take a real number, write it down in binary, you are going 00:15:22.540 --> 00:15:25.580 align:middle line:84% to get an infinite sequence of 0's and 1's. 00:15:25.580 --> 00:15:28.780 align:middle line:84% So you can think of each possible outcome here 00:15:28.780 --> 00:15:30.920 align:middle line:90% essentially as a real number. 00:15:30.920 --> 00:15:36.060 align:middle line:84% So the experiment of doing an infinite number of coin flips 00:15:36.060 --> 00:15:39.670 align:middle line:84% is sort of similar to the experiment of picking a real 00:15:39.670 --> 00:15:41.060 align:middle line:90% number at random. 00:15:41.060 --> 00:15:44.990 align:middle line:84% When you pick real numbers at random, any particular real 00:15:44.990 --> 00:15:46.500 align:middle line:90% number has 0 probability. 00:15:46.500 --> 00:15:49.780 align:middle line:84% So similarly here, any particular infinite sequence 00:15:49.780 --> 00:15:52.440 align:middle line:90% has 0 probability. 00:15:52.440 --> 00:15:55.260 align:middle line:84% So if we were to push that analogy further, there would 00:15:55.260 --> 00:15:57.290 align:middle line:84% be a few interesting things we could do. 00:15:57.290 --> 00:15:59.880 align:middle line:84% But we will not push it further. 00:15:59.880 --> 00:16:05.400 align:middle line:84% This is just to give you an indication that things can get 00:16:05.400 --> 00:16:08.710 align:middle line:84% pretty subtle and interesting once you start talking about 00:16:08.710 --> 00:16:12.640 align:middle line:84% random processes that involve forever, over the infinite 00:16:12.640 --> 00:16:13.900 align:middle line:90% time horizon. 00:16:13.900 --> 00:16:17.960 align:middle line:84% So things get interesting even in this context of the simple 00:16:17.960 --> 00:16:19.750 align:middle line:90% Bernoulli process. 00:16:19.750 --> 00:16:23.130 align:middle line:84% Just to give you a preview of what's coming further, today 00:16:23.130 --> 00:16:26.170 align:middle line:84% we're going to talk just about the Bernoulli process. 00:16:26.170 --> 00:16:30.810 align:middle line:84% And you should make sure before the next lecture-- 00:16:30.810 --> 00:16:34.590 align:middle line:84% I guess between the exam and the next lecture-- 00:16:34.590 --> 00:16:36.740 align:middle line:84% to understand everything we do today. 00:16:36.740 --> 00:16:39.930 align:middle line:84% Because next time we're going to do everything once more, 00:16:39.930 --> 00:16:41.640 align:middle line:90% but in continuous time. 00:16:41.640 --> 00:16:46.360 align:middle line:84% And in continuous time, things become more subtle and a 00:16:46.360 --> 00:16:47.700 align:middle line:90% little more difficult. 00:16:47.700 --> 00:16:50.580 align:middle line:84% But we are going to build on what we understand for the 00:16:50.580 --> 00:16:52.030 align:middle line:90% discrete time case. 00:16:52.030 --> 00:16:55.370 align:middle line:84% Now both the Bernoulli process and its continuous time analog 00:16:55.370 --> 00:16:58.470 align:middle line:84% has a property that we call memorylessness, whatever 00:16:58.470 --> 00:17:01.590 align:middle line:84% happened in the past does not affect the future. 00:17:01.590 --> 00:17:03.920 align:middle line:84% Later on in this class we're going to talk about more 00:17:03.920 --> 00:17:07.310 align:middle line:84% general random processes, so-called Markov chains, in 00:17:07.310 --> 00:17:10.890 align:middle line:84% which there are certain dependences across time. 00:17:10.890 --> 00:17:15.349 align:middle line:84% That is, what has happened in the past will have some 00:17:15.349 --> 00:17:18.390 align:middle line:84% bearing on what may happen in the future. 00:17:18.390 --> 00:17:22.440 align:middle line:84% So it's like having coin flips where the outcome of the next 00:17:22.440 --> 00:17:25.720 align:middle line:84% coin flip has some dependence on the previous coin flip. 00:17:25.720 --> 00:17:28.400 align:middle line:84% And that gives us a richer class of models. 00:17:28.400 --> 00:17:31.670 align:middle line:84% And once we get there, essentially we will have 00:17:31.670 --> 00:17:34.400 align:middle line:90% covered all possible models. 00:17:34.400 --> 00:17:38.070 align:middle line:84% So for random processes that are practically useful and 00:17:38.070 --> 00:17:41.480 align:middle line:84% which you can manipulate, Markov chains are a pretty 00:17:41.480 --> 00:17:43.250 align:middle line:90% general class of models. 00:17:43.250 --> 00:17:47.260 align:middle line:84% And almost any real world phenomenon that evolves in 00:17:47.260 --> 00:17:52.190 align:middle line:84% time can be approximately modeled using Markov chains. 00:17:52.190 --> 00:17:55.580 align:middle line:84% So even though this is a first class in probability, we will 00:17:55.580 --> 00:17:59.560 align:middle line:84% get pretty far in that direction. 00:17:59.560 --> 00:17:59.950 align:middle line:90% All right. 00:17:59.950 --> 00:18:04.300 align:middle line:84% So now let's start doing a few calculations and answer some 00:18:04.300 --> 00:18:06.690 align:middle line:84% questions about the Bernoulli process. 00:18:06.690 --> 00:18:11.010 align:middle line:84% So again, the best way to think in terms of models that 00:18:11.010 --> 00:18:13.350 align:middle line:84% correspond to the Bernoulli process is in terms of 00:18:13.350 --> 00:18:15.870 align:middle line:84% arrivals of jobs to a facility. 00:18:15.870 --> 00:18:18.230 align:middle line:84% And there's two types of questions that you can ask. 00:18:18.230 --> 00:18:21.990 align:middle line:84% In a given amount of time, how many jobs arrived? 00:18:21.990 --> 00:18:26.250 align:middle line:84% Or conversely, for a given number of jobs, how much time 00:18:26.250 --> 00:18:28.720 align:middle line:84% did it take for them to arrive? 00:18:28.720 --> 00:18:31.420 align:middle line:84% So we're going to deal with these two questions, starting 00:18:31.420 --> 00:18:32.450 align:middle line:90% with the first. 00:18:32.450 --> 00:18:34.370 align:middle line:90% For a given amount of time-- 00:18:34.370 --> 00:18:37.870 align:middle line:84% that is, for a given number of time periods-- 00:18:37.870 --> 00:18:40.150 align:middle line:90% how many arrivals have we had? 00:18:40.150 --> 00:18:44.180 align:middle line:84% How many of those Xi's happen to be 1's? 00:18:44.180 --> 00:18:46.270 align:middle line:84% We fix the number of time slots-- 00:18:46.270 --> 00:18:47.970 align:middle line:90% let's say n time slots-- 00:18:47.970 --> 00:18:50.920 align:middle line:84% and you measure the number of successes. 00:18:50.920 --> 00:18:54.320 align:middle line:84% Well this is a very familiar random variable. 00:18:54.320 --> 00:18:58.990 align:middle line:84% The number of successes in n independent coin flips-- 00:18:58.990 --> 00:19:01.430 align:middle line:90% or in n independent trials-- 00:19:01.430 --> 00:19:03.660 align:middle line:90% is a binomial random variable. 00:19:03.660 --> 00:19:10.380 align:middle line:84% So we know its distribution is given by the binomial PMF, and 00:19:10.380 --> 00:19:15.820 align:middle line:84% it's just this, for k going from 0 up to n. 00:19:15.820 --> 00:19:18.940 align:middle line:84% And we know everything by now about this random variable. 00:19:18.940 --> 00:19:21.990 align:middle line:84% We know its expected value is n times p. 00:19:21.990 --> 00:19:27.980 align:middle line:84% And we know the variance, which is n times p, times 1-p. 00:19:27.980 --> 00:19:31.740 align:middle line:90% So there's nothing new here. 00:19:31.740 --> 00:19:34.290 align:middle line:90% That's the easy part. 00:19:34.290 --> 00:19:37.590 align:middle line:84% So now let's look at the opposite kind of question. 00:19:37.590 --> 00:19:42.210 align:middle line:84% Instead of fixing the time and asking how many arrivals, now 00:19:42.210 --> 00:19:46.160 align:middle line:84% let us fix the number of arrivals and ask how much time 00:19:46.160 --> 00:19:47.810 align:middle line:90% did it take. 00:19:47.810 --> 00:19:52.780 align:middle line:84% And let's start with the time until the first arrival. 00:19:52.780 --> 00:19:59.470 align:middle line:90% So the process starts. 00:19:59.470 --> 00:20:00.720 align:middle line:90% We got our slots. 00:20:00.720 --> 00:20:04.070 align:middle line:90% 00:20:04.070 --> 00:20:08.250 align:middle line:84% And we see, perhaps, a sequence of 0's and then at 00:20:08.250 --> 00:20:10.660 align:middle line:90% some point we get a 1. 00:20:10.660 --> 00:20:14.810 align:middle line:84% The number of trials it took until we get a 1, we're going 00:20:14.810 --> 00:20:16.500 align:middle line:90% to call it T1. 00:20:16.500 --> 00:20:19.020 align:middle line:84% And it's the time of the first arrival. 00:20:19.020 --> 00:20:23.020 align:middle line:90% 00:20:23.020 --> 00:20:23.680 align:middle line:90% OK. 00:20:23.680 --> 00:20:27.130 align:middle line:84% What is the probability distribution of T1? 00:20:27.130 --> 00:20:30.430 align:middle line:84% What kind of random variable is it? 00:20:30.430 --> 00:20:31.835 align:middle line:84% We've gone through this before. 00:20:31.835 --> 00:20:34.360 align:middle line:90% 00:20:34.360 --> 00:20:40.560 align:middle line:84% The event that the first arrival happens at time little 00:20:40.560 --> 00:20:48.400 align:middle line:84% t is the event that the first t-1 trials were failures, and 00:20:48.400 --> 00:20:52.850 align:middle line:84% the trial number t happens to be a success. 00:20:52.850 --> 00:20:57.960 align:middle line:84% So for the first success to happen at time slot number 5, 00:20:57.960 --> 00:21:02.390 align:middle line:84% it means that the first 4 slots had failures and the 5th 00:21:02.390 --> 00:21:04.800 align:middle line:90% slot had a success. 00:21:04.800 --> 00:21:08.440 align:middle line:84% So the probability of this happening is the probability 00:21:08.440 --> 00:21:13.580 align:middle line:84% of having failures in the first t -1 trials, and having 00:21:13.580 --> 00:21:16.300 align:middle line:90% a success at trial number 1. 00:21:16.300 --> 00:21:20.060 align:middle line:84% And this is the formula for t equal 1,2, and so on. 00:21:20.060 --> 00:21:22.460 align:middle line:84% So we know what this distribution is. 00:21:22.460 --> 00:21:25.200 align:middle line:84% It's the so-called geometric distribution. 00:21:25.200 --> 00:21:30.900 align:middle line:90% 00:21:30.900 --> 00:21:35.010 align:middle line:84% Let me jump this through this for a minute. 00:21:35.010 --> 00:21:38.360 align:middle line:84% In the past, we did calculate the expected value of the 00:21:38.360 --> 00:21:42.240 align:middle line:84% geometric distribution, and it's 1/p. 00:21:42.240 --> 00:21:46.010 align:middle line:84% Which means that if p is small, you expect to take a 00:21:46.010 --> 00:21:48.810 align:middle line:84% long time until the first success. 00:21:48.810 --> 00:21:52.540 align:middle line:84% And then there's a formula also for the variance of T1, 00:21:52.540 --> 00:21:56.410 align:middle line:84% which we never formally derived in class, but it was 00:21:56.410 --> 00:22:01.600 align:middle line:84% in your textbook and it just happens to be this. 00:22:01.600 --> 00:22:02.310 align:middle line:90% All right. 00:22:02.310 --> 00:22:05.920 align:middle line:84% So nothing new until this point. 00:22:05.920 --> 00:22:08.700 align:middle line:84% Now, let's talk about this property, the 00:22:08.700 --> 00:22:10.210 align:middle line:90% memorylessness property. 00:22:10.210 --> 00:22:13.240 align:middle line:84% We kind of touched on this property when we discussed-- 00:22:13.240 --> 00:22:15.760 align:middle line:84% when we did the derivation in class of the 00:22:15.760 --> 00:22:18.010 align:middle line:90% expected value of T1. 00:22:18.010 --> 00:22:20.040 align:middle line:84% Now what is the memoryless property? 00:22:20.040 --> 00:22:22.980 align:middle line:84% It's essentially a consequence of independence. 00:22:22.980 --> 00:22:26.740 align:middle line:84% If I tell you the results of my coin flips up to a certain 00:22:26.740 --> 00:22:30.550 align:middle line:84% time, this, because of independence, doesn't give you 00:22:30.550 --> 00:22:34.410 align:middle line:84% any information about the coin flips after that time. 00:22:34.410 --> 00:22:37.180 align:middle line:90% 00:22:37.180 --> 00:22:41.180 align:middle line:84% So knowing that we had lots of 0's here does not change what 00:22:41.180 --> 00:22:44.920 align:middle line:84% I believe about the future coin flips, because the future 00:22:44.920 --> 00:22:47.580 align:middle line:84% coin flips are going to be just independent coin flips 00:22:47.580 --> 00:22:53.170 align:middle line:84% with a given probability, p, for obtaining tails. 00:22:53.170 --> 00:22:58.240 align:middle line:84% So this is a statement that I made about a specific time. 00:22:58.240 --> 00:23:02.270 align:middle line:84% That is, you do coin flips until 12 o'clock. 00:23:02.270 --> 00:23:05.210 align:middle line:84% And then at 12 o'clock, you start watching. 00:23:05.210 --> 00:23:09.900 align:middle line:84% No matter what happens before 12 o'clock, after 12:00, what 00:23:09.900 --> 00:23:12.890 align:middle line:84% you're going to see is just a sequence of independent 00:23:12.890 --> 00:23:15.610 align:middle line:84% Bernoulli trials with the same probability, p. 00:23:15.610 --> 00:23:18.450 align:middle line:84% Whatever happened in the past is irrelevant. 00:23:18.450 --> 00:23:21.590 align:middle line:84% Now instead of talking about the fixed time at which you 00:23:21.590 --> 00:23:26.940 align:middle line:84% start watching, let's think about a situation where your 00:23:26.940 --> 00:23:31.240 align:middle line:84% sister sits in the next room, flips the coins until she 00:23:31.240 --> 00:23:35.760 align:middle line:84% observes the first success, and then calls you inside. 00:23:35.760 --> 00:23:38.700 align:middle line:84% And you start watching after this time. 00:23:38.700 --> 00:23:40.970 align:middle line:90% What are you're going to see? 00:23:40.970 --> 00:23:45.160 align:middle line:84% Well, you're going to see a coin flip with probability p 00:23:45.160 --> 00:23:46.850 align:middle line:90% of success. 00:23:46.850 --> 00:23:49.870 align:middle line:84% You're going to see another trial that has probability p 00:23:49.870 --> 00:23:53.250 align:middle line:84% as a success, and these are all independent of each other. 00:23:53.250 --> 00:23:56.800 align:middle line:84% So what you're going to see starting at that time is going 00:23:56.800 --> 00:24:02.610 align:middle line:84% to be just a sequence of independent Bernoulli trials, 00:24:02.610 --> 00:24:06.190 align:middle line:84% as if the process was starting at this time. 00:24:06.190 --> 00:24:10.880 align:middle line:84% How long it took for the first success to occur doesn't have 00:24:10.880 --> 00:24:15.850 align:middle line:84% any bearing on what is going to happen afterwards. 00:24:15.850 --> 00:24:19.170 align:middle line:84% What happens afterwards is still a sequence of 00:24:19.170 --> 00:24:21.230 align:middle line:90% independent coin flips. 00:24:21.230 --> 00:24:24.680 align:middle line:84% And this story is actually even more general. 00:24:24.680 --> 00:24:28.980 align:middle line:84% So your sister watches the coin flips and at some point 00:24:28.980 --> 00:24:31.690 align:middle line:84% tells you, oh, something really interesting is 00:24:31.690 --> 00:24:32.440 align:middle line:90% happening here. 00:24:32.440 --> 00:24:35.250 align:middle line:84% I got this string of a hundred 1's in a row. 00:24:35.250 --> 00:24:37.250 align:middle line:90% Come and watch. 00:24:37.250 --> 00:24:40.260 align:middle line:84% Now when you go in there and you start watching, do you 00:24:40.260 --> 00:24:43.890 align:middle line:84% expect to see something unusual? 00:24:43.890 --> 00:24:46.830 align:middle line:84% There were unusual things that happened before 00:24:46.830 --> 00:24:48.180 align:middle line:90% you were called in. 00:24:48.180 --> 00:24:50.620 align:middle line:84% Does this means that you're going to see unusual things 00:24:50.620 --> 00:24:51.780 align:middle line:90% afterwards? 00:24:51.780 --> 00:24:52.180 align:middle line:90% No. 00:24:52.180 --> 00:24:55.060 align:middle line:84% Afterwards, what you're going to see is, again, just a 00:24:55.060 --> 00:24:57.700 align:middle line:84% sequence of independent coin flips. 00:24:57.700 --> 00:25:00.640 align:middle line:84% The fact that some strange things happened before doesn't 00:25:00.640 --> 00:25:03.940 align:middle line:84% have any bearing as to what is going to happen in the future. 00:25:03.940 --> 00:25:08.560 align:middle line:84% So if the roulettes in the casino are properly made, the 00:25:08.560 --> 00:25:12.610 align:middle line:84% fact that there were 3 reds in a row doesn't affect the odds 00:25:12.610 --> 00:25:16.430 align:middle line:84% of whether in the next roll it's going to 00:25:16.430 --> 00:25:19.570 align:middle line:90% be a red or a black. 00:25:19.570 --> 00:25:22.850 align:middle line:84% So whatever happens in the past-- no matter 00:25:22.850 --> 00:25:25.010 align:middle line:90% how unusual it is-- 00:25:25.010 --> 00:25:28.910 align:middle line:84% at the time when you're called in, what's going to happen in 00:25:28.910 --> 00:25:32.510 align:middle line:84% the future is going to be just independent Bernoulli trials, 00:25:32.510 --> 00:25:34.170 align:middle line:90% with the same probability, p. 00:25:34.170 --> 00:25:36.730 align:middle line:90% 00:25:36.730 --> 00:25:41.900 align:middle line:84% The only case where this story changes is if your sister has 00:25:41.900 --> 00:25:43.850 align:middle line:90% a little bit of foresight. 00:25:43.850 --> 00:25:48.500 align:middle line:84% So your sister can look ahead into the future and knows that 00:25:48.500 --> 00:25:54.230 align:middle line:84% the next 10 coin flips will be heads, and calls you before 00:25:54.230 --> 00:25:56.430 align:middle line:90% those 10 flips will happen. 00:25:56.430 --> 00:25:59.660 align:middle line:84% If she calls you in, then what are you going to see? 00:25:59.660 --> 00:26:02.310 align:middle line:84% You're not going to see independent Bernoulli trials, 00:26:02.310 --> 00:26:05.510 align:middle line:84% since she has psychic powers and she knows that the next 00:26:05.510 --> 00:26:06.910 align:middle line:90% ones would be 1's. 00:26:06.910 --> 00:26:12.770 align:middle line:84% She called you in and you will see a sequence of 1's. 00:26:12.770 --> 00:26:15.470 align:middle line:84% So it's no more independent Bernoulli trials. 00:26:15.470 --> 00:26:19.420 align:middle line:84% So what's the subtle difference here? 00:26:19.420 --> 00:26:24.010 align:middle line:84% The future is independent from the past, provided that the 00:26:24.010 --> 00:26:28.310 align:middle line:84% time that you are called and asked to start watching is 00:26:28.310 --> 00:26:31.460 align:middle line:84% determined by someone who doesn't have any foresight, 00:26:31.460 --> 00:26:33.470 align:middle line:90% who cannot see the future. 00:26:33.470 --> 00:26:36.960 align:middle line:84% If you are called in, just on the basis of what has happened 00:26:36.960 --> 00:26:39.630 align:middle line:90% so far, then you don't have any 00:26:39.630 --> 00:26:41.430 align:middle line:90% information about the future. 00:26:41.430 --> 00:26:44.870 align:middle line:84% And one special case is the picture here. 00:26:44.870 --> 00:26:47.240 align:middle line:90% You have your coin flips. 00:26:47.240 --> 00:26:51.060 align:middle line:84% Once you see a one that happens, once you see a 00:26:51.060 --> 00:26:53.310 align:middle line:90% success, you are called in. 00:26:53.310 --> 00:26:57.690 align:middle line:84% You are called in on the basis of what happened in the past, 00:26:57.690 --> 00:26:59.070 align:middle line:90% but without any foresight. 00:26:59.070 --> 00:27:02.208 align:middle line:90% 00:27:02.208 --> 00:27:03.140 align:middle line:90% OK. 00:27:03.140 --> 00:27:07.020 align:middle line:84% And this subtle distinction is what's going to make our next 00:27:07.020 --> 00:27:10.420 align:middle line:84% example interesting and subtle. 00:27:10.420 --> 00:27:13.380 align:middle line:90% So here's the question. 00:27:13.380 --> 00:27:17.790 align:middle line:84% You buy a lottery ticket every day, so we have a Bernoulli 00:27:17.790 --> 00:27:21.880 align:middle line:84% process that's running in time. 00:27:21.880 --> 00:27:26.050 align:middle line:84% And you're interested in the length of the first string of 00:27:26.050 --> 00:27:26.860 align:middle line:90% losing days. 00:27:26.860 --> 00:27:28.410 align:middle line:90% What does that mean? 00:27:28.410 --> 00:27:33.700 align:middle line:84% So suppose that a typical sequence of events 00:27:33.700 --> 00:27:35.040 align:middle line:90% could be this one. 00:27:35.040 --> 00:27:40.970 align:middle line:90% 00:27:40.970 --> 00:27:43.470 align:middle line:84% So what are we discussing here? 00:27:43.470 --> 00:27:47.180 align:middle line:84% We're looking at the first string of losing days, where 00:27:47.180 --> 00:27:49.140 align:middle line:90% losing days means 0's. 00:27:49.140 --> 00:27:51.940 align:middle line:90% 00:27:51.940 --> 00:27:56.910 align:middle line:84% So the string of losing days is this string here. 00:27:56.910 --> 00:28:02.030 align:middle line:84% Let's call the length of that string, L. We're interested in 00:28:02.030 --> 00:28:06.520 align:middle line:84% the random variable, which is the length of this interval. 00:28:06.520 --> 00:28:08.670 align:middle line:84% What kind of random variable is it? 00:28:08.670 --> 00:28:11.230 align:middle line:90% 00:28:11.230 --> 00:28:11.600 align:middle line:90% OK. 00:28:11.600 --> 00:28:16.190 align:middle line:84% Here's one possible way you might think about the problem. 00:28:16.190 --> 00:28:16.550 align:middle line:90% OK. 00:28:16.550 --> 00:28:24.140 align:middle line:84% Starting from this time, and looking until this time here, 00:28:24.140 --> 00:28:27.420 align:middle line:90% what are we looking at? 00:28:27.420 --> 00:28:31.640 align:middle line:84% We're looking at the time, starting from here, until the 00:28:31.640 --> 00:28:35.040 align:middle line:90% first success. 00:28:35.040 --> 00:28:40.530 align:middle line:90% So the past doesn't matter. 00:28:40.530 --> 00:28:43.550 align:middle line:84% Starting from here we have coin flips 00:28:43.550 --> 00:28:45.870 align:middle line:90% until the first success. 00:28:45.870 --> 00:28:48.160 align:middle line:84% The time until the first success 00:28:48.160 --> 00:28:50.280 align:middle line:90% in a Bernoulli process-- 00:28:50.280 --> 00:28:54.700 align:middle line:84% we just discussed that it's a geometric random variable. 00:28:54.700 --> 00:28:58.090 align:middle line:84% So your first conjecture would be that this random variable 00:28:58.090 --> 00:29:02.960 align:middle line:84% here, which is 1 longer than the one we are interested in, 00:29:02.960 --> 00:29:06.310 align:middle line:84% that perhaps is a geometric random variable. 00:29:06.310 --> 00:29:14.040 align:middle line:90% 00:29:14.040 --> 00:29:18.460 align:middle line:84% And if this were so, then you could say that the random 00:29:18.460 --> 00:29:23.300 align:middle line:84% variable, L, is a geometric, minus 1. 00:29:23.300 --> 00:29:26.160 align:middle line:84% Can that be the correct answer? 00:29:26.160 --> 00:29:29.590 align:middle line:84% A geometric random variable, what values does it take? 00:29:29.590 --> 00:29:33.160 align:middle line:84% It takes values 1, 2, 3, and so on. 00:29:33.160 --> 00:29:37.576 align:middle line:84% 1 minus a geometric would take values from 0, 00:29:37.576 --> 00:29:40.940 align:middle line:90% 1, 2, and so on. 00:29:40.940 --> 00:29:45.200 align:middle line:84% Can the random variable L be 0? 00:29:45.200 --> 00:29:45.920 align:middle line:90% No. 00:29:45.920 --> 00:29:48.500 align:middle line:84% The random variable L is the length of a 00:29:48.500 --> 00:29:50.390 align:middle line:90% string of losing days. 00:29:50.390 --> 00:29:56.190 align:middle line:84% So the shortest that L could be, would be just 1. 00:29:56.190 --> 00:29:59.770 align:middle line:84% If you get just one losing day and then you start winning, L 00:29:59.770 --> 00:30:01.520 align:middle line:90% would be equal to 1. 00:30:01.520 --> 00:30:05.338 align:middle line:84% So L cannot be 0 by definition, which means that L 00:30:05.338 --> 00:30:09.820 align:middle line:84% + 1 cannot be 1, by definition. 00:30:09.820 --> 00:30:14.310 align:middle line:84% But if L +1 were geometric, it could be equal to 1. 00:30:14.310 --> 00:30:16.226 align:middle line:84% Therefore this random variable, L 00:30:16.226 --> 00:30:18.830 align:middle line:90% + 1, is not a geometric. 00:30:18.830 --> 00:30:23.550 align:middle line:90% 00:30:23.550 --> 00:30:23.920 align:middle line:90% OK. 00:30:23.920 --> 00:30:26.720 align:middle line:90% Why is it not geometric? 00:30:26.720 --> 00:30:29.100 align:middle line:84% I started watching at this time. 00:30:29.100 --> 00:30:33.810 align:middle line:84% From this time until the first success, that should be a 00:30:33.810 --> 00:30:35.690 align:middle line:90% geometric random variable. 00:30:35.690 --> 00:30:38.180 align:middle line:90% Where's the catch? 00:30:38.180 --> 00:30:42.670 align:middle line:84% If I'm asked to start watching at this time, it's because my 00:30:42.670 --> 00:30:48.260 align:middle line:84% sister knows that the next one was a failure. 00:30:48.260 --> 00:30:52.360 align:middle line:84% This is the time where the string of failures starts. 00:30:52.360 --> 00:30:56.050 align:middle line:84% In order to know that they should start watching here, 00:30:56.050 --> 00:30:58.770 align:middle line:84% it's the same as if I'm told that the 00:30:58.770 --> 00:31:01.240 align:middle line:90% next one is a failure. 00:31:01.240 --> 00:31:05.180 align:middle line:84% So to be asked to start watching at this time requires 00:31:05.180 --> 00:31:08.050 align:middle line:84% that someone looked in the future. 00:31:08.050 --> 00:31:13.210 align:middle line:84% And in that case, it's no longer true that these will be 00:31:13.210 --> 00:31:14.850 align:middle line:90% independent Bernoulli trials. 00:31:14.850 --> 00:31:16.000 align:middle line:90% In fact, they're not. 00:31:16.000 --> 00:31:18.990 align:middle line:84% If you start watching here, you're certain that the next 00:31:18.990 --> 00:31:20.210 align:middle line:90% one is a failure. 00:31:20.210 --> 00:31:23.510 align:middle line:84% The next one is not an independent Bernoulli trial. 00:31:23.510 --> 00:31:26.860 align:middle line:84% That's why the argument that would claim that this L + 1 is 00:31:26.860 --> 00:31:30.400 align:middle line:90% geometric would be incorrect. 00:31:30.400 --> 00:31:33.680 align:middle line:84% So if this is not the correct answer, which 00:31:33.680 --> 00:31:35.150 align:middle line:90% is the correct answer? 00:31:35.150 --> 00:31:37.700 align:middle line:84% The correct answer goes as follows. 00:31:37.700 --> 00:31:39.400 align:middle line:90% Your sister is watching. 00:31:39.400 --> 00:31:44.080 align:middle line:84% Your sister sees the first failure, and then tells you, 00:31:44.080 --> 00:31:45.530 align:middle line:90% OK, the failures-- 00:31:45.530 --> 00:31:46.670 align:middle line:90% or losing days-- 00:31:46.670 --> 00:31:47.790 align:middle line:90% have started. 00:31:47.790 --> 00:31:49.260 align:middle line:90% Come in and watch. 00:31:49.260 --> 00:31:51.550 align:middle line:84% So you start to watching at this time. 00:31:51.550 --> 00:31:56.060 align:middle line:84% And you start watching until the first success comes. 00:31:56.060 --> 00:31:59.290 align:middle line:84% This will be a geometric random variable. 00:31:59.290 --> 00:32:05.005 align:middle line:84% So from here to here, this will be geometric. 00:32:05.005 --> 00:32:09.430 align:middle line:90% 00:32:09.430 --> 00:32:11.560 align:middle line:90% So things happen. 00:32:11.560 --> 00:32:14.270 align:middle line:84% You are asked to start watching. 00:32:14.270 --> 00:32:18.660 align:middle line:84% After you start watching, the future is just a sequence of 00:32:18.660 --> 00:32:20.540 align:middle line:90% independent Bernoulli trials. 00:32:20.540 --> 00:32:23.930 align:middle line:84% And the time until the first failure occurs, this is going 00:32:23.930 --> 00:32:27.470 align:middle line:84% to be a geometric random variable with parameter p. 00:32:27.470 --> 00:32:31.830 align:middle line:84% And then you notice that the interval of interest is 00:32:31.830 --> 00:32:35.030 align:middle line:84% exactly the same as the length of this interval. 00:32:35.030 --> 00:32:37.550 align:middle line:84% This starts one time step later, and ends 00:32:37.550 --> 00:32:39.260 align:middle line:90% one time step later. 00:32:39.260 --> 00:32:43.370 align:middle line:84% So conclusion is that L is actually geometric, with 00:32:43.370 --> 00:32:44.620 align:middle line:90% parameter p. 00:32:44.620 --> 00:33:33.090 align:middle line:90% 00:33:33.090 --> 00:33:36.160 align:middle line:84% OK, it looks like I'm missing one slide. 00:33:36.160 --> 00:33:38.540 align:middle line:84% Can I cheat a little from here? 00:33:38.540 --> 00:33:46.830 align:middle line:90% 00:33:46.830 --> 00:33:48.080 align:middle line:90% OK. 00:33:48.080 --> 00:33:52.550 align:middle line:84% So now that we dealt with the time until the first arrival, 00:33:52.550 --> 00:33:56.110 align:middle line:84% we can start talking about the time until the second 00:33:56.110 --> 00:33:57.190 align:middle line:90% arrival, and so on. 00:33:57.190 --> 00:33:59.550 align:middle line:90% How do we define these? 00:33:59.550 --> 00:34:02.920 align:middle line:84% After the first arrival happens, we're going to have a 00:34:02.920 --> 00:34:06.100 align:middle line:84% sequence of time slots with no arrivals, and then the next 00:34:06.100 --> 00:34:08.199 align:middle line:90% arrival is going to happen. 00:34:08.199 --> 00:34:11.080 align:middle line:84% So we call this time that elapses-- 00:34:11.080 --> 00:34:14.760 align:middle line:84% or number of time slots after the first arrival 00:34:14.760 --> 00:34:16.730 align:middle line:90% until the next one-- 00:34:16.730 --> 00:34:18.500 align:middle line:90% we call it T2. 00:34:18.500 --> 00:34:22.070 align:middle line:84% This is the second inter-arrival time, that is, 00:34:22.070 --> 00:34:23.780 align:middle line:90% time between arrivals. 00:34:23.780 --> 00:34:28.260 align:middle line:84% Once this arrival has happened, then we wait and see 00:34:28.260 --> 00:34:31.600 align:middle line:84% how many more it takes until the third arrival. 00:34:31.600 --> 00:34:37.040 align:middle line:84% And we call this time here, T3. 00:34:37.040 --> 00:34:42.429 align:middle line:84% We're interested in the time of the k-th arrival, which is 00:34:42.429 --> 00:34:45.230 align:middle line:84% going to be just the sum of the first k 00:34:45.230 --> 00:34:46.889 align:middle line:90% inter-arrival times. 00:34:46.889 --> 00:34:51.909 align:middle line:84% So for example, let's say Y3 is the time that the third 00:34:51.909 --> 00:34:53.510 align:middle line:90% arrival comes. 00:34:53.510 --> 00:34:58.940 align:middle line:84% Y3 is just the sum of T1, plus T2, plus T3. 00:34:58.940 --> 00:35:01.840 align:middle line:90% 00:35:01.840 --> 00:35:06.310 align:middle line:84% So we're interested in this random variable, Y3, and it's 00:35:06.310 --> 00:35:08.765 align:middle line:84% the sum of inter-arrival times. 00:35:08.765 --> 00:35:12.190 align:middle line:84% To understand what kind of random variable it is, I guess 00:35:12.190 --> 00:35:16.970 align:middle line:84% we should understand what kind of random variables these are 00:35:16.970 --> 00:35:18.230 align:middle line:90% going to be. 00:35:18.230 --> 00:35:22.040 align:middle line:84% So what kind of random variable is T2? 00:35:22.040 --> 00:35:27.440 align:middle line:84% Your sister is doing her coin flips until a success is 00:35:27.440 --> 00:35:29.750 align:middle line:90% observed for the first time. 00:35:29.750 --> 00:35:32.540 align:middle line:84% Based on that information about what has happened so 00:35:32.540 --> 00:35:34.730 align:middle line:84% far, you are called into the room. 00:35:34.730 --> 00:35:39.320 align:middle line:84% And you start watching until a success is observed again. 00:35:39.320 --> 00:35:42.160 align:middle line:84% So after you start watching, what you have is just a 00:35:42.160 --> 00:35:45.190 align:middle line:84% sequence of independent Bernoulli trials. 00:35:45.190 --> 00:35:47.110 align:middle line:84% So each one of these has probability 00:35:47.110 --> 00:35:49.400 align:middle line:90% p of being a success. 00:35:49.400 --> 00:35:52.180 align:middle line:84% The time it's going to take until the first success, this 00:35:52.180 --> 00:35:57.600 align:middle line:84% number, T2, is going to be again just another geometric 00:35:57.600 --> 00:35:58.840 align:middle line:90% random variable. 00:35:58.840 --> 00:36:01.860 align:middle line:84% It's as if the process just started. 00:36:01.860 --> 00:36:06.280 align:middle line:84% After you are called into the room, you have no foresight, 00:36:06.280 --> 00:36:09.540 align:middle line:84% you don't have any information about the future, other than 00:36:09.540 --> 00:36:11.020 align:middle line:84% the fact that these are going to be 00:36:11.020 --> 00:36:13.280 align:middle line:90% independent Bernoulli trials. 00:36:13.280 --> 00:36:18.480 align:middle line:84% So T2 itself is going to be geometric with the same 00:36:18.480 --> 00:36:20.590 align:middle line:90% parameter p. 00:36:20.590 --> 00:36:24.700 align:middle line:84% And then you can continue the arguments and argue that T3 is 00:36:24.700 --> 00:36:27.880 align:middle line:84% also geometric with the same parameter p. 00:36:27.880 --> 00:36:30.850 align:middle line:84% Furthermore, whatever happened, how long it took 00:36:30.850 --> 00:36:34.330 align:middle line:84% until you were called in, it doesn't change the statistics 00:36:34.330 --> 00:36:36.810 align:middle line:84% about what's going to happen in the future. 00:36:36.810 --> 00:36:39.130 align:middle line:84% So whatever happens in the future is 00:36:39.130 --> 00:36:41.460 align:middle line:90% independent from the past. 00:36:41.460 --> 00:36:47.340 align:middle line:84% So T1, T2, and T3 are independent random variables. 00:36:47.340 --> 00:36:54.220 align:middle line:84% So conclusion is that the time until the third arrival is the 00:36:54.220 --> 00:37:00.510 align:middle line:84% sum of 3 independent geometric random variables, with the 00:37:00.510 --> 00:37:02.150 align:middle line:90% same parameter. 00:37:02.150 --> 00:37:04.550 align:middle line:84% And this is true more generally. 00:37:04.550 --> 00:37:08.820 align:middle line:84% The time until the k-th arrival is going to be the sum 00:37:08.820 --> 00:37:14.900 align:middle line:84% of k independent random variables. 00:37:14.900 --> 00:37:21.540 align:middle line:84% So in general, Yk is going to be T1 plus Tk, where the Ti's 00:37:21.540 --> 00:37:26.850 align:middle line:84% are geometric, with the same parameter p, and independent. 00:37:26.850 --> 00:37:30.440 align:middle line:90% 00:37:30.440 --> 00:37:33.520 align:middle line:84% So now what's more natural than trying to find the 00:37:33.520 --> 00:37:37.100 align:middle line:84% distribution of the random variable Yk? 00:37:37.100 --> 00:37:38.350 align:middle line:90% How can we find it? 00:37:38.350 --> 00:37:40.260 align:middle line:90% So I fixed k for you. 00:37:40.260 --> 00:37:41.680 align:middle line:90% Let's say k is 100. 00:37:41.680 --> 00:37:43.580 align:middle line:84% I'm interested in how long it takes until 00:37:43.580 --> 00:37:46.180 align:middle line:90% 100 customers arrive. 00:37:46.180 --> 00:37:48.850 align:middle line:84% How can we find the distribution of Yk? 00:37:48.850 --> 00:37:51.980 align:middle line:84% Well one way of doing it is to use this 00:37:51.980 --> 00:37:54.200 align:middle line:90% lovely convolution formula. 00:37:54.200 --> 00:37:57.950 align:middle line:84% Take a geometric, convolve it with another geometric, you 00:37:57.950 --> 00:37:59.430 align:middle line:90% get something. 00:37:59.430 --> 00:38:02.040 align:middle line:84% Take that something that you got, convolve it with a 00:38:02.040 --> 00:38:07.090 align:middle line:84% geometric once more, do this 99 times, and this gives you 00:38:07.090 --> 00:38:09.450 align:middle line:90% the distribution of Yk. 00:38:09.450 --> 00:38:14.300 align:middle line:84% So that's definitely doable, and it's extremely tedious. 00:38:14.300 --> 00:38:16.900 align:middle line:84% Let's try to find the distribution 00:38:16.900 --> 00:38:22.520 align:middle line:90% of Yk using a shortcut. 00:38:22.520 --> 00:38:28.660 align:middle line:84% So the probability that Yk is equal to t. 00:38:28.660 --> 00:38:31.620 align:middle line:84% So we're trying to find the PMF of Yk. 00:38:31.620 --> 00:38:34.030 align:middle line:90% k has been fixed for us. 00:38:34.030 --> 00:38:36.810 align:middle line:84% And we want to calculate this probability for the various 00:38:36.810 --> 00:38:39.580 align:middle line:84% values of t, because this is going to give 00:38:39.580 --> 00:38:43.642 align:middle line:90% us the PMF of Yk. 00:38:43.642 --> 00:38:45.850 align:middle line:90% OK. 00:38:45.850 --> 00:38:47.540 align:middle line:90% What is this event? 00:38:47.540 --> 00:38:53.210 align:middle line:84% What does it take for the k-th arrival to be at time t? 00:38:53.210 --> 00:38:56.580 align:middle line:84% For that to happen, we need two things. 00:38:56.580 --> 00:39:00.960 align:middle line:84% In the first t -1 slots, how many arrivals 00:39:00.960 --> 00:39:03.130 align:middle line:90% should we have gotten? 00:39:03.130 --> 00:39:04.830 align:middle line:90% k - 1. 00:39:04.830 --> 00:39:09.400 align:middle line:84% And then in the last slot, we get one more arrival, and 00:39:09.400 --> 00:39:11.430 align:middle line:90% that's the k-th one. 00:39:11.430 --> 00:39:20.340 align:middle line:84% So this is the probability that we have k - 1 arrivals in 00:39:20.340 --> 00:39:24.250 align:middle line:84% the time interval from 1 up to t. 00:39:24.250 --> 00:39:28.670 align:middle line:90% 00:39:28.670 --> 00:39:34.120 align:middle line:84% And then, an arrival at time t. 00:39:34.120 --> 00:39:39.860 align:middle line:90% 00:39:39.860 --> 00:39:43.210 align:middle line:84% That's the only way that it can happen, that the k-th 00:39:43.210 --> 00:39:45.450 align:middle line:90% arrival happens at time t. 00:39:45.450 --> 00:39:48.470 align:middle line:84% We need to have an arrival at time t. 00:39:48.470 --> 00:39:50.150 align:middle line:84% And before that time, we need to have 00:39:50.150 --> 00:39:53.380 align:middle line:90% exactly k - 1 arrivals. 00:39:53.380 --> 00:39:55.590 align:middle line:84% Now this is an event that refers-- 00:39:55.590 --> 00:39:58.710 align:middle line:90% 00:39:58.710 --> 00:39:59.960 align:middle line:90% t-1. 00:39:59.960 --> 00:40:02.830 align:middle line:90% 00:40:02.830 --> 00:40:07.460 align:middle line:84% In the previous time slots we had exactly k -1 arrivals. 00:40:07.460 --> 00:40:10.680 align:middle line:84% And then at the last time slot we get one more arrival. 00:40:10.680 --> 00:40:14.560 align:middle line:84% Now the interesting thing is that this event here has to do 00:40:14.560 --> 00:40:18.620 align:middle line:84% with what happened from time 1 up to time t -1. 00:40:18.620 --> 00:40:22.350 align:middle line:84% This event has to do with what happened at time t. 00:40:22.350 --> 00:40:25.510 align:middle line:84% Different time slots are independent of each other. 00:40:25.510 --> 00:40:31.130 align:middle line:84% So this event and that event are independent. 00:40:31.130 --> 00:40:34.930 align:middle line:84% So this means that we can multiply their probabilities. 00:40:34.930 --> 00:40:37.280 align:middle line:84% So take the probability of this. 00:40:37.280 --> 00:40:38.590 align:middle line:90% What is that? 00:40:38.590 --> 00:40:41.600 align:middle line:84% Well probability of having a certain number of arrivals in 00:40:41.600 --> 00:40:44.650 align:middle line:84% a certain number of time slots, these are just the 00:40:44.650 --> 00:40:46.530 align:middle line:90% binomial probabilities. 00:40:46.530 --> 00:40:51.250 align:middle line:84% So this is, out of t - 1 slots, to get exactly k - 1 00:40:51.250 --> 00:41:02.670 align:middle line:84% arrivals, p to the k-1, (1-p) to the t-1 - (k-1), 00:41:02.670 --> 00:41:05.480 align:middle line:90% this gives us t-k. 00:41:05.480 --> 00:41:07.960 align:middle line:84% And then we multiply with this probability, the probability 00:41:07.960 --> 00:41:14.300 align:middle line:84% of an arrival, at time t is equal to p. 00:41:14.300 --> 00:41:21.310 align:middle line:84% And so this is the formula for the PMF of the number-- 00:41:21.310 --> 00:41:27.410 align:middle line:84% of the time it takes until the k-th arrival happens. 00:41:27.410 --> 00:41:32.760 align:middle line:90% 00:41:32.760 --> 00:41:35.730 align:middle line:84% Does it agree with the formula in your handout? 00:41:35.730 --> 00:41:37.850 align:middle line:90% Or its not there? 00:41:37.850 --> 00:41:38.710 align:middle line:90% It's not there. 00:41:38.710 --> 00:41:39.960 align:middle line:90% OK. 00:41:39.960 --> 00:41:48.018 align:middle line:90% 00:41:48.018 --> 00:41:49.010 align:middle line:90% Yeah. 00:41:49.010 --> 00:41:50.020 align:middle line:90% OK. 00:41:50.020 --> 00:41:57.338 align:middle line:84% So that's the formula and it is true for what values of t? 00:41:57.338 --> 00:41:58.588 align:middle line:90% [INAUDIBLE]. 00:41:58.588 --> 00:42:03.182 align:middle line:90% 00:42:03.182 --> 00:42:08.350 align:middle line:84% It takes at least k time slots in order to get k arrivals, so 00:42:08.350 --> 00:42:12.370 align:middle line:84% this formula should be true for k larger 00:42:12.370 --> 00:42:13.990 align:middle line:90% than or equal to t. 00:42:13.990 --> 00:42:20.330 align:middle line:90% 00:42:20.330 --> 00:42:23.391 align:middle line:84% For t larger than or equal to k. 00:42:23.391 --> 00:42:30.130 align:middle line:90% 00:42:30.130 --> 00:42:31.040 align:middle line:90% All right. 00:42:31.040 --> 00:42:34.950 align:middle line:84% So this gives us the PMF of the random variable Yk. 00:42:34.950 --> 00:42:37.430 align:middle line:84% Of course, we may also be interested in the mean and 00:42:37.430 --> 00:42:39.260 align:middle line:90% variance of Yk. 00:42:39.260 --> 00:42:42.150 align:middle line:90% But this is a lot easier. 00:42:42.150 --> 00:42:46.350 align:middle line:84% Since Yk is the sum of independent random variables, 00:42:46.350 --> 00:42:50.310 align:middle line:84% the expected value of Yk is going to be just k times the 00:42:50.310 --> 00:42:52.980 align:middle line:84% expected value of your typical t. 00:42:52.980 --> 00:43:03.080 align:middle line:84% So the expected value of Yk is going to be just k times 1/p, 00:43:03.080 --> 00:43:06.600 align:middle line:84% which is the mean of the geometric. 00:43:06.600 --> 00:43:09.470 align:middle line:84% And similarly for the variance, it's going to be k 00:43:09.470 --> 00:43:12.960 align:middle line:84% times the variance of a geometric. 00:43:12.960 --> 00:43:16.950 align:middle line:84% So we have everything there is to know about the distribution 00:43:16.950 --> 00:43:19.965 align:middle line:84% of how long it takes until the first arrival comes. 00:43:19.965 --> 00:43:23.760 align:middle line:90% 00:43:23.760 --> 00:43:25.420 align:middle line:90% OK. 00:43:25.420 --> 00:43:27.810 align:middle line:84% Finally, let's do a few more things about 00:43:27.810 --> 00:43:30.680 align:middle line:90% the Bernoulli process. 00:43:30.680 --> 00:43:34.970 align:middle line:84% It's interesting to talk about several processes at the time. 00:43:34.970 --> 00:43:39.600 align:middle line:84% So in the situation here of splitting a Bernoulli process 00:43:39.600 --> 00:43:43.730 align:middle line:84% is where you have arrivals that come to a server. 00:43:43.730 --> 00:43:46.370 align:middle line:84% And that's a picture of which slots get arrivals. 00:43:46.370 --> 00:43:48.900 align:middle line:84% But actually maybe you have two servers. 00:43:48.900 --> 00:43:53.110 align:middle line:84% And whenever an arrival comes to the system, you flip a coin 00:43:53.110 --> 00:43:56.620 align:middle line:84% and with some probability, q, you send it to one server. 00:43:56.620 --> 00:44:00.390 align:middle line:84% And with probability 1-q, you send it to another server. 00:44:00.390 --> 00:44:03.520 align:middle line:84% So there is a single arrival stream, but 00:44:03.520 --> 00:44:05.040 align:middle line:90% two possible servers. 00:44:05.040 --> 00:44:07.280 align:middle line:84% And whenever there's an arrival, you either send it 00:44:07.280 --> 00:44:09.270 align:middle line:90% here or you send it there. 00:44:09.270 --> 00:44:13.780 align:middle line:84% And each time you decide where you send it by flipping an 00:44:13.780 --> 00:44:17.950 align:middle line:84% independent coin that has its own bias q. 00:44:17.950 --> 00:44:22.450 align:middle line:84% The coin flips that decide where do you send it are 00:44:22.450 --> 00:44:27.460 align:middle line:84% assumed to be independent from the arrival process itself. 00:44:27.460 --> 00:44:30.480 align:middle line:84% So there's two coin flips that are happening. 00:44:30.480 --> 00:44:33.850 align:middle line:84% At each time slot, there's a coin flip that decides whether 00:44:33.850 --> 00:44:37.150 align:middle line:84% you have an arrival in this process here, and that coin 00:44:37.150 --> 00:44:39.630 align:middle line:90% flip is with parameter p. 00:44:39.630 --> 00:44:43.000 align:middle line:84% And if you have something that arrives, you flip another coin 00:44:43.000 --> 00:44:47.050 align:middle line:84% with probabilities q, and 1-q, that decides whether you send 00:44:47.050 --> 00:44:49.770 align:middle line:84% it up there or you send it down there. 00:44:49.770 --> 00:44:55.460 align:middle line:84% So what kind of arrival process does this server see? 00:44:55.460 --> 00:44:59.510 align:middle line:84% At any given time slot, there's probability p that 00:44:59.510 --> 00:45:01.480 align:middle line:90% there's an arrival here. 00:45:01.480 --> 00:45:04.300 align:middle line:84% And there's a further probability q that this 00:45:04.300 --> 00:45:07.320 align:middle line:90% arrival gets sent up there. 00:45:07.320 --> 00:45:10.860 align:middle line:84% So the probability that this server sees an arrival at any 00:45:10.860 --> 00:45:14.090 align:middle line:90% given time is p times q. 00:45:14.090 --> 00:45:18.900 align:middle line:84% So this process here is going to be a Bernoulli process, but 00:45:18.900 --> 00:45:21.810 align:middle line:84% with a different parameter, p times q. 00:45:21.810 --> 00:45:24.820 align:middle line:84% And this one down here, with the same argument, is going to 00:45:24.820 --> 00:45:29.860 align:middle line:84% be Bernoulli with parameter p times (1-q). 00:45:29.860 --> 00:45:33.500 align:middle line:84% So by taking a Bernoulli stream of arrivals and 00:45:33.500 --> 00:45:36.460 align:middle line:84% splitting it into two, you get two 00:45:36.460 --> 00:45:39.050 align:middle line:90% separate Bernoulli processes. 00:45:39.050 --> 00:45:40.890 align:middle line:84% This is going to be a Bernoulli process, that's 00:45:40.890 --> 00:45:42.980 align:middle line:84% going to be a Bernoulli process. 00:45:42.980 --> 00:45:45.630 align:middle line:84% Well actually, I'm running a little too fast. 00:45:45.630 --> 00:45:49.330 align:middle line:84% What does it take to verify that it's a Bernoulli process? 00:45:49.330 --> 00:45:52.650 align:middle line:84% At each time slot, it's a 0 or 1. 00:45:52.650 --> 00:45:55.330 align:middle line:84% And it's going to be a 1, you're going to see an arrival 00:45:55.330 --> 00:45:57.450 align:middle line:90% with probability p times q. 00:45:57.450 --> 00:46:00.510 align:middle line:84% What else do we need to verify, to be able to tell-- 00:46:00.510 --> 00:46:02.820 align:middle line:84% to say that it's a Bernoulli process? 00:46:02.820 --> 00:46:05.620 align:middle line:84% We need to make sure that whatever happens in this 00:46:05.620 --> 00:46:09.340 align:middle line:84% process, in different time slots, are statistically 00:46:09.340 --> 00:46:11.240 align:middle line:90% independent from each other. 00:46:11.240 --> 00:46:13.030 align:middle line:90% Is that property true? 00:46:13.030 --> 00:46:16.900 align:middle line:84% For example, what happens in this time slot whether you got 00:46:16.900 --> 00:46:20.100 align:middle line:84% an arrival or not, is it independent from what happened 00:46:20.100 --> 00:46:22.660 align:middle line:90% at that time slot? 00:46:22.660 --> 00:46:26.850 align:middle line:84% The answer is yes for the following reason. 00:46:26.850 --> 00:46:30.760 align:middle line:84% What happens in this time slot has to do with the coin flip 00:46:30.760 --> 00:46:34.840 align:middle line:84% associated with the original process at this time, and the 00:46:34.840 --> 00:46:38.340 align:middle line:84% coin flip that decides where to send things. 00:46:38.340 --> 00:46:41.370 align:middle line:84% What happens at that time slot has to do with the coin flip 00:46:41.370 --> 00:46:45.010 align:middle line:84% here, and the additional coin flip that decides where to 00:46:45.010 --> 00:46:47.130 align:middle line:90% send it if something came. 00:46:47.130 --> 00:46:50.570 align:middle line:84% Now all these coin flips are independent of each other. 00:46:50.570 --> 00:46:53.460 align:middle line:84% The coin flips that determine whether we have an arrival 00:46:53.460 --> 00:46:56.860 align:middle line:84% here is independent from the coin flips that determined 00:46:56.860 --> 00:46:59.280 align:middle line:84% whether we had an arrival there. 00:46:59.280 --> 00:47:02.770 align:middle line:84% And you can generalize this argument and conclude that, 00:47:02.770 --> 00:47:07.390 align:middle line:84% indeed, every time slot here is independent from 00:47:07.390 --> 00:47:09.030 align:middle line:90% any other time slot. 00:47:09.030 --> 00:47:12.020 align:middle line:84% And this does make it a Bernoulli process. 00:47:12.020 --> 00:47:15.590 align:middle line:84% And the reason is that, in the original process, every time 00:47:15.590 --> 00:47:18.390 align:middle line:84% slot is independent from every other time slot. 00:47:18.390 --> 00:47:21.000 align:middle line:84% And the additional assumption that the coin flips that we're 00:47:21.000 --> 00:47:24.370 align:middle line:84% using to decide where to send things, these are also 00:47:24.370 --> 00:47:26.020 align:middle line:90% independent of each other. 00:47:26.020 --> 00:47:30.220 align:middle line:84% So we're using here the basic property that functions of 00:47:30.220 --> 00:47:33.283 align:middle line:84% independent things remain independent. 00:47:33.283 --> 00:47:36.390 align:middle line:90% 00:47:36.390 --> 00:47:38.710 align:middle line:84% There's a converse picture of this. 00:47:38.710 --> 00:47:41.970 align:middle line:84% Instead of taking one stream and splitting it into two 00:47:41.970 --> 00:47:44.720 align:middle line:84% streams, you can do the opposite. 00:47:44.720 --> 00:47:48.040 align:middle line:84% You could start from two streams of arrivals. 00:47:48.040 --> 00:47:51.300 align:middle line:84% Let's say you have arrivals of men and you have arrivals of 00:47:51.300 --> 00:47:54.000 align:middle line:84% women, but you don't care about gender. 00:47:54.000 --> 00:47:57.430 align:middle line:84% And the only thing you record is whether, in a given time 00:47:57.430 --> 00:48:00.450 align:middle line:84% slot, you had an arrival or not. 00:48:00.450 --> 00:48:04.320 align:middle line:84% Notice that here we may have an arrival of a man and the 00:48:04.320 --> 00:48:05.790 align:middle line:90% arrival of a woman. 00:48:05.790 --> 00:48:11.180 align:middle line:84% We just record it with a 1, by saying there was an arrival. 00:48:11.180 --> 00:48:14.260 align:middle line:84% So in the merged process, we're not keeping track of how 00:48:14.260 --> 00:48:16.880 align:middle line:90% many arrivals we had total. 00:48:16.880 --> 00:48:18.840 align:middle line:84% We just record whether there was an 00:48:18.840 --> 00:48:21.400 align:middle line:90% arrival or not an arrival. 00:48:21.400 --> 00:48:25.830 align:middle line:84% So an arrival gets recorded here if, and only if, one or 00:48:25.830 --> 00:48:28.800 align:middle line:84% both of these streams had an arrival. 00:48:28.800 --> 00:48:31.780 align:middle line:84% So that we call a merging of two Bernoull-- of two 00:48:31.780 --> 00:48:34.470 align:middle line:84% processes, of two arrival processes. 00:48:34.470 --> 00:48:37.840 align:middle line:84% So let's make the assumption that this arrival process is 00:48:37.840 --> 00:48:41.440 align:middle line:84% independent from that arrival process. 00:48:41.440 --> 00:48:44.330 align:middle line:84% So what happens at the typical slot here? 00:48:44.330 --> 00:48:49.680 align:middle line:84% I'm going to see an arrival, unless none of 00:48:49.680 --> 00:48:51.950 align:middle line:90% these had an arrival. 00:48:51.950 --> 00:48:56.380 align:middle line:84% So the probability of an arrival in a typical time slot 00:48:56.380 --> 00:49:02.650 align:middle line:84% is going to be 1 minus the probability of no arrival. 00:49:02.650 --> 00:49:07.370 align:middle line:84% And the event of no arrival corresponds to the first 00:49:07.370 --> 00:49:10.330 align:middle line:84% process having no arrival, and the second 00:49:10.330 --> 00:49:14.350 align:middle line:90% process having no arrival. 00:49:14.350 --> 00:49:18.110 align:middle line:84% So there's no arrival in the merged process if, and only 00:49:18.110 --> 00:49:21.270 align:middle line:84% if, there's no arrival in the first process and no arrival 00:49:21.270 --> 00:49:22.710 align:middle line:90% in the second process. 00:49:22.710 --> 00:49:26.080 align:middle line:84% We're assuming that the two processes are independent and 00:49:26.080 --> 00:49:29.160 align:middle line:84% that's why we can multiply probabilities here. 00:49:29.160 --> 00:49:34.270 align:middle line:84% And then you can take this formula and it simplifies to p 00:49:34.270 --> 00:49:38.120 align:middle line:90% + q, minus p times q. 00:49:38.120 --> 00:49:41.360 align:middle line:84% So each time slot of the merged process has a certain 00:49:41.360 --> 00:49:44.620 align:middle line:84% probability of seeing an arrival. 00:49:44.620 --> 00:49:47.360 align:middle line:84% Is the merged process a Bernoulli process? 00:49:47.360 --> 00:49:51.260 align:middle line:84% Yes, it is after you verify the additional property that 00:49:51.260 --> 00:49:54.650 align:middle line:84% different slots are independent of each other. 00:49:54.650 --> 00:49:56.560 align:middle line:90% Why are they independent? 00:49:56.560 --> 00:50:01.070 align:middle line:84% What happens in this slot has to do with that slot, and that 00:50:01.070 --> 00:50:03.160 align:middle line:90% slot down here. 00:50:03.160 --> 00:50:05.790 align:middle line:90% These two slots-- 00:50:05.790 --> 00:50:08.570 align:middle line:84% so what happens here, has to do with what 00:50:08.570 --> 00:50:11.430 align:middle line:90% happens here and there. 00:50:11.430 --> 00:50:16.680 align:middle line:84% What happens in this slot has to do with whatever happened 00:50:16.680 --> 00:50:19.200 align:middle line:90% here and there. 00:50:19.200 --> 00:50:23.190 align:middle line:84% Now, whatever happens here and there is independent from 00:50:23.190 --> 00:50:25.330 align:middle line:84% whatever happens here and there. 00:50:25.330 --> 00:50:29.180 align:middle line:84% Therefore, what happens here is independent from what 00:50:29.180 --> 00:50:30.220 align:middle line:90% happens there. 00:50:30.220 --> 00:50:33.310 align:middle line:84% So the independence property is preserved. 00:50:33.310 --> 00:50:36.640 align:middle line:84% The different slots of this merged process are independent 00:50:36.640 --> 00:50:37.590 align:middle line:90% of each other. 00:50:37.590 --> 00:50:41.970 align:middle line:84% So the merged process is itself a Bernoulli process. 00:50:41.970 --> 00:50:45.900 align:middle line:84% So please digest these two pictures of merging and 00:50:45.900 --> 00:50:48.960 align:middle line:84% splitting, because we're going to revisit them in continuous 00:50:48.960 --> 00:50:52.510 align:middle line:84% time where things are little subtler than that. 00:50:52.510 --> 00:50:53.160 align:middle line:90% OK. 00:50:53.160 --> 00:50:56.240 align:middle line:84% Good luck on the exam and see you in a week. 00:50:56.240 --> 00:50:57.490 align:middle line:90%