WEBVTT 00:00:00.000 --> 00:00:02.470 align:middle line:90% [SQUEAKING] 00:00:02.470 --> 00:00:04.446 align:middle line:90% [RUSTLING] 00:00:04.446 --> 00:00:07.410 align:middle line:90% [CLICKING] 00:00:07.410 --> 00:00:10.582 align:middle line:90% 00:00:10.582 --> 00:00:11.540 align:middle line:90% BRYNMOR CHAPMAN: Hello? 00:00:11.540 --> 00:00:14.000 align:middle line:90% Can people hear me in the back? 00:00:14.000 --> 00:00:16.590 align:middle line:90% Are we good? 00:00:16.590 --> 00:00:19.440 align:middle line:84% So welcome to the last lecture of 604-- 00:00:19.440 --> 00:00:22.220 align:middle line:90% no, 6.1200. 00:00:22.220 --> 00:00:23.510 align:middle line:90% I'm totally sane. 00:00:23.510 --> 00:00:26.070 align:middle line:84% I can remember which class I'm teaching. 00:00:26.070 --> 00:00:26.570 align:middle line:90% Welcome. 00:00:26.570 --> 00:00:27.630 align:middle line:84% You made it through the semester. 00:00:27.630 --> 00:00:28.130 align:middle line:90% Well done. 00:00:28.130 --> 00:00:30.840 align:middle line:90% 00:00:30.840 --> 00:00:39.590 align:middle line:84% So for today's material, we are going to wrap up 00:00:39.590 --> 00:00:41.310 align:middle line:90% our discussion of probability. 00:00:41.310 --> 00:00:45.030 align:middle line:84% And in particular, we will be discussing tail bounds. 00:00:45.030 --> 00:00:49.368 align:middle line:84% So what happens if you have a random variable? 00:00:49.368 --> 00:00:51.410 align:middle line:84% What is the probability that this random variable 00:00:51.410 --> 00:00:54.590 align:middle line:84% is very large or very small, very different 00:00:54.590 --> 00:00:58.262 align:middle line:90% from its expectation? 00:00:58.262 --> 00:01:00.685 align:middle line:84% So let's start with a bit of a review. 00:01:00.685 --> 00:01:05.580 align:middle line:90% 00:01:05.580 --> 00:01:06.080 align:middle line:90% Variance. 00:01:06.080 --> 00:01:08.870 align:middle line:90% 00:01:08.870 --> 00:01:12.860 align:middle line:84% So I think we saw from the last-- 00:01:12.860 --> 00:01:14.510 align:middle line:84% not last lecture, two lectures ago, 00:01:14.510 --> 00:01:21.530 align:middle line:84% possibly, so the variance of a random variable R 00:01:21.530 --> 00:01:28.460 align:middle line:84% is defined to be the expectation of R 00:01:28.460 --> 00:01:30.770 align:middle line:84% minus the expectation of R squared. 00:01:30.770 --> 00:01:39.860 align:middle line:90% 00:01:39.860 --> 00:01:42.680 align:middle line:84% And the standard deviation is just the square root of that. 00:01:42.680 --> 00:01:45.570 align:middle line:84% If you've taken stats or anything like that, 00:01:45.570 --> 00:01:47.930 align:middle line:84% that's probably something that you've 00:01:47.930 --> 00:01:51.290 align:middle line:90% seen more often than variance. 00:01:51.290 --> 00:01:54.780 align:middle line:84% Another, not a definition, but another useful formula, 00:01:54.780 --> 00:01:59.660 align:middle line:84% an equivalent formulation of this, is that the variance of R 00:01:59.660 --> 00:02:05.720 align:middle line:84% is also equal to the expectation of R squared 00:02:05.720 --> 00:02:13.460 align:middle line:84% minus the square of the expectation of R. So 00:02:13.460 --> 00:02:15.240 align:middle line:84% not the definition, but it's equivalent. 00:02:15.240 --> 00:02:19.730 align:middle line:84% And I think in recitation perhaps, you proved this. 00:02:19.730 --> 00:02:22.520 align:middle line:84% This formula is a particularly useful one 00:02:22.520 --> 00:02:24.145 align:middle line:84% for what we're going to be doing today. 00:02:24.145 --> 00:02:27.450 align:middle line:90% 00:02:27.450 --> 00:02:30.410 align:middle line:84% We also saw that variance is linear, 00:02:30.410 --> 00:02:33.090 align:middle line:90% subject to certain constraints. 00:02:33.090 --> 00:02:37.060 align:middle line:84% Does anybody remember what you need for variance to be linear? 00:02:37.060 --> 00:02:39.800 align:middle line:90% 00:02:39.800 --> 00:02:40.882 align:middle line:90% Yeah? 00:02:40.882 --> 00:02:41.840 align:middle line:90% AUDIENCE: Independence. 00:02:41.840 --> 00:02:44.492 align:middle line:84% BRYNMOR CHAPMAN: Independence, OK. 00:02:44.492 --> 00:02:55.920 align:middle line:84% So if you have R1, R2, Rn are independent, 00:02:55.920 --> 00:03:01.356 align:middle line:84% then the variance of the sum is the sum of the variances-- 00:03:01.356 --> 00:03:08.790 align:middle line:90% 00:03:08.790 --> 00:03:09.290 align:middle line:90% oops. 00:03:09.290 --> 00:03:19.340 align:middle line:90% 00:03:19.340 --> 00:03:20.626 align:middle line:90% Is that the right constraint? 00:03:20.626 --> 00:03:25.130 align:middle line:90% 00:03:25.130 --> 00:03:28.310 align:middle line:90% Is it correct? 00:03:28.310 --> 00:03:31.970 align:middle line:90% Who believes the statement? 00:03:31.970 --> 00:03:33.016 align:middle line:90% Who disbelieves? 00:03:33.016 --> 00:03:35.690 align:middle line:90% 00:03:35.690 --> 00:03:37.940 align:middle line:90% Who's still asleep? 00:03:37.940 --> 00:03:40.790 align:middle line:84% Oh, OK, maybe we'll give it a few minutes 00:03:40.790 --> 00:03:41.710 align:middle line:90% for people to wake up. 00:03:41.710 --> 00:03:45.260 align:middle line:90% 00:03:45.260 --> 00:03:48.740 align:middle line:84% Well, this is a correct statement. 00:03:48.740 --> 00:03:53.030 align:middle line:84% However, there is something slightly stronger 00:03:53.030 --> 00:03:54.030 align:middle line:90% that we can say. 00:03:54.030 --> 00:03:56.230 align:middle line:90% We can weaken this condition. 00:03:56.230 --> 00:03:57.868 align:middle line:84% We don't need mutual independence. 00:03:57.868 --> 00:03:58.910 align:middle line:90% What do we actually need? 00:03:58.910 --> 00:04:01.770 align:middle line:90% 00:04:01.770 --> 00:04:03.395 align:middle line:90% Pairwise independence, yeah. 00:04:03.395 --> 00:04:10.823 align:middle line:90% 00:04:10.823 --> 00:04:12.990 align:middle line:84% So if we have a bunch of pairwise independent random 00:04:12.990 --> 00:04:15.640 align:middle line:84% variables, we want to find the variance of their sum. 00:04:15.640 --> 00:04:17.579 align:middle line:84% We can instead just sum the variances. 00:04:17.579 --> 00:04:21.899 align:middle line:90% So another very useful fact-- 00:04:21.899 --> 00:04:23.950 align:middle line:84% hopefully, you proved it already in recitation. 00:04:23.950 --> 00:04:25.533 align:middle line:84% So we don't need to go over the proof. 00:04:25.533 --> 00:04:27.780 align:middle line:90% 00:04:27.780 --> 00:04:31.997 align:middle line:84% But these-- well, these two formulas 00:04:31.997 --> 00:04:33.455 align:middle line:84% are going to be quite useful today. 00:04:33.455 --> 00:04:39.390 align:middle line:90% 00:04:39.390 --> 00:04:41.315 align:middle line:84% Now, time for the first of our tail bounds. 00:04:41.315 --> 00:04:51.300 align:middle line:90% 00:04:51.300 --> 00:04:56.920 align:middle line:84% So before I give you the bound, let's think about an example. 00:04:56.920 --> 00:05:01.590 align:middle line:84% So looking at this room, suppose I pick a random student. 00:05:01.590 --> 00:05:07.410 align:middle line:84% And let the random variable R be the row 00:05:07.410 --> 00:05:11.454 align:middle line:84% that the randomly picked student is sitting in. 00:05:11.454 --> 00:05:15.180 align:middle line:84% So guesstimate that this is about-- 00:05:15.180 --> 00:05:17.050 align:middle line:90% the expectation of R is about 8. 00:05:17.050 --> 00:05:20.450 align:middle line:90% So pick random student. 00:05:20.450 --> 00:05:25.770 align:middle line:90% 00:05:25.770 --> 00:05:27.320 align:middle line:90% R is the row. 00:05:27.320 --> 00:05:31.692 align:middle line:90% 00:05:31.692 --> 00:05:36.270 align:middle line:84% So for those of you who aren't in the room, those of you who 00:05:36.270 --> 00:05:38.850 align:middle line:84% are watching the video, what does that 00:05:38.850 --> 00:05:43.880 align:middle line:84% tell them about the probability that R is at least 16? 00:05:43.880 --> 00:05:56.880 align:middle line:90% 00:05:56.880 --> 00:05:59.580 align:middle line:84% So what's the probability that a randomly selected student is 00:05:59.580 --> 00:06:01.630 align:middle line:90% sitting in the last five rows? 00:06:01.630 --> 00:06:03.370 align:middle line:90% I think there are 21 rows here. 00:06:03.370 --> 00:06:03.870 align:middle line:90% Yeah. 00:06:03.870 --> 00:06:07.955 align:middle line:90% 00:06:07.955 --> 00:06:09.080 align:middle line:90% Does anybody have any idea? 00:06:09.080 --> 00:06:13.258 align:middle line:90% 00:06:13.258 --> 00:06:15.050 align:middle line:84% So maybe I should label this as an example. 00:06:15.050 --> 00:06:18.330 align:middle line:90% 00:06:18.330 --> 00:06:19.920 align:middle line:90% Yeah? 00:06:19.920 --> 00:06:22.200 align:middle line:84% AUDIENCE: 1 minus the CDF [INAUDIBLE]. 00:06:22.200 --> 00:06:23.740 align:middle line:84% BRYNMOR CHAPMAN: 1 minus the CDF. 00:06:23.740 --> 00:06:27.180 align:middle line:84% So the people watching the video, 00:06:27.180 --> 00:06:28.560 align:middle line:84% I assume they can't see the room. 00:06:28.560 --> 00:06:31.510 align:middle line:84% So they don't know what the CDF is. 00:06:31.510 --> 00:06:35.870 align:middle line:84% The only information they know is that the expectation is 8. 00:06:35.870 --> 00:06:41.190 align:middle line:90% 00:06:41.190 --> 00:06:41.690 align:middle line:90% Yeah? 00:06:41.690 --> 00:06:43.482 align:middle line:84% AUDIENCE: Can we find the variance for that 00:06:43.482 --> 00:06:45.210 align:middle line:84% and see how far away you are from there? 00:06:45.210 --> 00:06:47.877 align:middle line:84% BRYNMOR CHAPMAN: Oh, can we find the variance from that and then 00:06:47.877 --> 00:06:48.930 align:middle line:90% see how far away we are? 00:06:48.930 --> 00:06:50.010 align:middle line:90% No. 00:06:50.010 --> 00:06:54.010 align:middle line:84% [LAUGHS] That requires more information, right? 00:06:54.010 --> 00:06:58.590 align:middle line:84% We need to know a little bit more about what's going on. 00:06:58.590 --> 00:07:03.870 align:middle line:84% We need to somehow figure out the expectation of R minus 8 00:07:03.870 --> 00:07:05.130 align:middle line:90% squared. 00:07:05.130 --> 00:07:07.020 align:middle line:84% So, we need to know something more about R 00:07:07.020 --> 00:07:09.630 align:middle line:90% than just its expectation. 00:07:09.630 --> 00:07:10.350 align:middle line:90% Yeah? 00:07:10.350 --> 00:07:12.810 align:middle line:84% AUDIENCE: Do you know how many [INAUDIBLE]? 00:07:12.810 --> 00:07:15.550 align:middle line:84% BRYNMOR CHAPMAN: Yes, there are 21, so 0 to 20. 00:07:15.550 --> 00:07:16.290 align:middle line:90% AUDIENCE: Wait. 00:07:16.290 --> 00:07:19.990 align:middle line:84% Can we assume that it's uniformly distributed 00:07:19.990 --> 00:07:22.170 align:middle line:90% and [INAUDIBLE]. 00:07:22.170 --> 00:07:25.500 align:middle line:84% BRYNMOR CHAPMAN: Can we assume that it's uniformly distributed? 00:07:25.500 --> 00:07:26.910 align:middle line:90% No. 00:07:26.910 --> 00:07:28.860 align:middle line:90% And in fact, it is not. 00:07:28.860 --> 00:07:31.710 align:middle line:84% There seem to be more students towards the front 00:07:31.710 --> 00:07:33.190 align:middle line:90% than towards the back. 00:07:33.190 --> 00:07:33.840 align:middle line:90% Yeah? 00:07:33.840 --> 00:07:36.340 align:middle line:84% AUDIENCE: [INAUDIBLE] that there are 20 rows. 00:07:36.340 --> 00:07:38.650 align:middle line:84% If you just think there are 16 row, 00:07:38.650 --> 00:07:41.250 align:middle line:84% we know that the way we find the expected value 00:07:41.250 --> 00:07:45.240 align:middle line:84% is multiplying each possible outcome by its probability. 00:07:45.240 --> 00:07:49.140 align:middle line:84% Since a is 60 multiplied by 1/2, the probability is generally 00:07:49.140 --> 00:07:51.690 align:middle line:90% [INAUDIBLE] to be less than 1/2. 00:07:51.690 --> 00:07:52.750 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:07:52.750 --> 00:07:54.910 align:middle line:84% AUDIENCE: If you factor in 16 or 20, 00:07:54.910 --> 00:07:59.310 align:middle line:84% it gets to be more fun math [INAUDIBLE]. 00:07:59.310 --> 00:08:02.350 align:middle line:84% BRYNMOR CHAPMAN: Yeah, so your intuition is great, though. 00:08:02.350 --> 00:08:10.650 align:middle line:84% So, well, we're using the observation that 16 is twice 8. 00:08:10.650 --> 00:08:15.120 align:middle line:84% We're also using the observation that the rows are numbered 0 00:08:15.120 --> 00:08:15.820 align:middle line:90% through 20. 00:08:15.820 --> 00:08:18.450 align:middle line:90% Like they're non-negative. 00:08:18.450 --> 00:08:27.210 align:middle line:84% So what happens if more than half of the students 00:08:27.210 --> 00:08:28.850 align:middle line:84% are sitting in the back five rows? 00:08:28.850 --> 00:08:33.750 align:middle line:90% 00:08:33.750 --> 00:08:35.980 align:middle line:84% What does that tell us about the expectation? 00:08:35.980 --> 00:08:42.630 align:middle line:90% 00:08:42.630 --> 00:08:45.630 align:middle line:84% Well, if more than half of the students 00:08:45.630 --> 00:08:51.810 align:middle line:84% give us R is at least 16, that tells us 00:08:51.810 --> 00:08:56.040 align:middle line:84% that the expectation should be at least 8-- 00:08:56.040 --> 00:09:00.210 align:middle line:84% or sorry, more than 8, if more than half of the students 00:09:00.210 --> 00:09:01.710 align:middle line:90% are in the back. 00:09:01.710 --> 00:09:06.270 align:middle line:84% So the converse of that is that there 00:09:06.270 --> 00:09:10.650 align:middle line:84% are at most, half of the students in the back five rows. 00:09:10.650 --> 00:09:13.560 align:middle line:84% So let's try and formalize this intuition. 00:09:13.560 --> 00:09:16.390 align:middle line:84% So this is what's known as Markov's inequality. 00:09:16.390 --> 00:09:25.170 align:middle line:90% 00:09:25.170 --> 00:09:33.590 align:middle line:84% So let R be a non-negative random variable. 00:09:33.590 --> 00:09:36.390 align:middle line:90% 00:09:36.390 --> 00:09:40.890 align:middle line:84% Then the probability that R is greater than 00:09:40.890 --> 00:09:48.340 align:middle line:84% or equal to x is less than or equal to the expectation of R 00:09:48.340 --> 00:09:50.250 align:middle line:90% divided by x. 00:09:50.250 --> 00:09:54.676 align:middle line:90% 00:09:54.676 --> 00:09:56.990 align:middle line:84% So for the example that we just saw, 00:09:56.990 --> 00:10:00.700 align:middle line:84% the probability that R is at least 16, 00:10:00.700 --> 00:10:05.570 align:middle line:84% it's at most the expectation, which is 8 divided by 16. 00:10:05.570 --> 00:10:06.460 align:middle line:90% So that gives us 1/2. 00:10:06.460 --> 00:10:10.546 align:middle line:90% 00:10:10.546 --> 00:10:12.330 align:middle line:84% Does the statement make sense to people? 00:10:12.330 --> 00:10:25.390 align:middle line:90% 00:10:25.390 --> 00:10:27.363 align:middle line:84% So basically, it's saying that you 00:10:27.363 --> 00:10:29.530 align:middle line:84% can't have too many super large values, because that 00:10:29.530 --> 00:10:33.880 align:middle line:84% would push the expectation up too high 00:10:33.880 --> 00:10:36.015 align:middle line:90% So how could we prove this? 00:10:36.015 --> 00:10:41.767 align:middle line:90% 00:10:41.767 --> 00:10:44.350 align:middle line:84% And it is, how do we formalize the intuition that we were just 00:10:44.350 --> 00:10:44.933 align:middle line:90% talking about? 00:10:44.933 --> 00:10:57.680 align:middle line:90% 00:10:57.680 --> 00:11:00.250 align:middle line:84% Let's try using the law of total expectation. 00:11:00.250 --> 00:11:04.510 align:middle line:84% So how can we write the expectation of R? 00:11:04.510 --> 00:11:05.955 align:middle line:90% What might we condition on? 00:11:05.955 --> 00:11:14.253 align:middle line:90% 00:11:14.253 --> 00:11:16.420 align:middle line:84% We don't have too many events that we can work with. 00:11:16.420 --> 00:11:20.785 align:middle line:90% 00:11:20.785 --> 00:11:22.035 align:middle line:90% Which of them might be useful? 00:11:22.035 --> 00:11:32.230 align:middle line:90% 00:11:32.230 --> 00:11:32.950 align:middle line:90% Nobody? 00:11:32.950 --> 00:11:41.040 align:middle line:84% So let's try conditioning on x greater than-- oh wait, x-- 00:11:41.040 --> 00:11:44.020 align:middle line:90% 00:11:44.020 --> 00:11:45.810 align:middle line:84% yeah, or greater than or equal to x. 00:11:45.810 --> 00:11:48.994 align:middle line:90% 00:11:48.994 --> 00:11:52.270 align:middle line:84% So what does the law of total expectation tell us? 00:11:52.270 --> 00:11:57.760 align:middle line:84% Well, this is going to be the expectation of R conditioned 00:11:57.760 --> 00:12:03.550 align:middle line:84% on R being greater than x multiplied by the probability 00:12:03.550 --> 00:12:15.430 align:middle line:84% that R is greater than x plus the expectation R conditioned 00:12:15.430 --> 00:12:21.680 align:middle line:84% on R less than x times that probability. 00:12:21.680 --> 00:12:27.970 align:middle line:90% 00:12:27.970 --> 00:12:29.210 align:middle line:90% Do people remember that? 00:12:29.210 --> 00:12:30.910 align:middle line:90% Is that coming back? 00:12:30.910 --> 00:12:31.510 align:middle line:90% Hopefully. 00:12:31.510 --> 00:12:32.010 align:middle line:90% Yes. 00:12:32.010 --> 00:12:32.590 align:middle line:90% No Maybe. 00:12:32.590 --> 00:12:33.255 align:middle line:90% A few nods. 00:12:33.255 --> 00:12:36.550 align:middle line:90% 00:12:36.550 --> 00:12:40.420 align:middle line:84% So what do we know about each of these values? 00:12:40.420 --> 00:13:00.927 align:middle line:90% 00:13:00.927 --> 00:13:02.010 align:middle line:90% Let's start with this one. 00:13:02.010 --> 00:13:03.295 align:middle line:90% Can anybody bound this? 00:13:03.295 --> 00:13:14.330 align:middle line:90% 00:13:14.330 --> 00:13:15.170 align:middle line:90% Yeah? 00:13:15.170 --> 00:13:16.640 align:middle line:90% AUDIENCE: It's at least x. 00:13:16.640 --> 00:13:17.580 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:13:17.580 --> 00:13:18.920 align:middle line:90% It's at least x. 00:13:18.920 --> 00:13:23.270 align:middle line:84% If you conditioned on R being at least x, the expectation of R 00:13:23.270 --> 00:13:26.240 align:middle line:90% is also going to be at least x. 00:13:26.240 --> 00:13:30.140 align:middle line:90% Every outcome is at least x. 00:13:30.140 --> 00:13:35.750 align:middle line:84% So this is going to be at least x times-- 00:13:35.750 --> 00:13:38.340 align:middle line:84% well, this is the probability that we're looking for. 00:13:38.340 --> 00:13:40.095 align:middle line:84% So let's just write it out as it is. 00:13:40.095 --> 00:13:49.412 align:middle line:90% 00:13:49.412 --> 00:13:50.790 align:middle line:90% Now, what about the bottom line? 00:13:50.790 --> 00:13:53.920 align:middle line:90% 00:13:53.920 --> 00:13:56.000 align:middle line:84% What do we know about this expectation? 00:13:56.000 --> 00:14:01.120 align:middle line:90% 00:14:01.120 --> 00:14:01.880 align:middle line:90% Yeah? 00:14:01.880 --> 00:14:05.500 align:middle line:84% AUDIENCE: Is it 1 minus the expectation of the other one? 00:14:05.500 --> 00:14:07.480 align:middle line:84% BRYNMOR CHAPMAN: Is it 1 minus the expectation 00:14:07.480 --> 00:14:09.100 align:middle line:90% of the other one? 00:14:09.100 --> 00:14:09.620 align:middle line:90% No. 00:14:09.620 --> 00:14:13.600 align:middle line:84% So the probability is 1 minus the other probability. 00:14:13.600 --> 00:14:16.010 align:middle line:84% The expectation does not work like that, though. 00:14:16.010 --> 00:14:16.510 align:middle line:90% Yeah? 00:14:16.510 --> 00:14:17.605 align:middle line:90% AUDIENCE: At most, x. 00:14:17.605 --> 00:14:18.980 align:middle line:84% BRYNMOR CHAPMAN: It is at most x. 00:14:18.980 --> 00:14:20.320 align:middle line:90% That is true. 00:14:20.320 --> 00:14:24.755 align:middle line:84% We're actually looking for a lower bound, though. 00:14:24.755 --> 00:14:25.255 align:middle line:90% Yeah? 00:14:25.255 --> 00:14:27.755 align:middle line:90% 00:14:27.755 --> 00:14:28.630 align:middle line:90% AUDIENCE: At least 0. 00:14:28.630 --> 00:14:30.255 align:middle line:84% BRYNMOR CHAPMAN: It's at least 0, yeah. 00:14:30.255 --> 00:14:41.170 align:middle line:90% 00:14:41.170 --> 00:14:44.530 align:middle line:84% We know that R is always non-negative. 00:14:44.530 --> 00:14:49.060 align:middle line:84% So conditioning on anything, R is still always going 00:14:49.060 --> 00:14:50.530 align:middle line:90% to be non-negative. 00:14:50.530 --> 00:14:52.970 align:middle line:84% So its expectation has to be non-negative. 00:14:52.970 --> 00:14:58.543 align:middle line:90% 00:14:58.543 --> 00:15:00.050 align:middle line:90% So we can drop this. 00:15:00.050 --> 00:15:02.890 align:middle line:90% This is just going to be 0. 00:15:02.890 --> 00:15:05.230 align:middle line:84% Now we've got the expectation of R 00:15:05.230 --> 00:15:12.160 align:middle line:84% is at least x times the probability 00:15:12.160 --> 00:15:13.135 align:middle line:90% that R is at least x. 00:15:13.135 --> 00:15:21.940 align:middle line:90% 00:15:21.940 --> 00:15:38.110 align:middle line:84% Now we can just move this x to the other side, 00:15:38.110 --> 00:15:39.560 align:middle line:90% assuming x is non-negative. 00:15:39.560 --> 00:15:44.260 align:middle line:84% If x is negative, then it's a very silly question-- or sorry, 00:15:44.260 --> 00:15:45.710 align:middle line:90% assuming x is positive. 00:15:45.710 --> 00:15:48.560 align:middle line:90% 00:15:48.560 --> 00:15:49.935 align:middle line:84% Are people happy with this proof? 00:15:49.935 --> 00:15:52.870 align:middle line:90% 00:15:52.870 --> 00:15:54.590 align:middle line:90% Does anybody have any questions? 00:15:54.590 --> 00:16:08.260 align:middle line:90% 00:16:08.260 --> 00:16:13.240 align:middle line:84% So another form of Markov which you may often encounter 00:16:13.240 --> 00:16:14.480 align:middle line:90% is the following. 00:16:14.480 --> 00:16:25.690 align:middle line:90% 00:16:25.690 --> 00:16:35.630 align:middle line:84% So the probability that R is at least C times its expectation 00:16:35.630 --> 00:16:44.560 align:middle line:84% is at most 1 over C for any positive C-- 00:16:44.560 --> 00:16:48.050 align:middle line:90% 00:16:48.050 --> 00:16:51.097 align:middle line:84% so, essentially, equivalent statement, just you 00:16:51.097 --> 00:16:52.930 align:middle line:84% might sometimes see it in this form instead. 00:16:52.930 --> 00:16:56.835 align:middle line:90% 00:16:56.835 --> 00:16:58.793 align:middle line:84% So let's take a look at a couple more examples. 00:16:58.793 --> 00:17:07.520 align:middle line:90% 00:17:07.520 --> 00:17:18.637 align:middle line:84% So suppose R is the test score of a randomly chosen student 00:17:18.637 --> 00:17:20.470 align:middle line:84% instead of the roll that they're sitting in. 00:17:20.470 --> 00:17:38.906 align:middle line:90% 00:17:38.906 --> 00:17:47.760 align:middle line:84% And suppose that these are all in the range, say, 30% to 100%. 00:17:47.760 --> 00:18:00.180 align:middle line:90% 00:18:00.180 --> 00:18:11.360 align:middle line:84% Can we use Markov to bound the probability that R is at least, 00:18:11.360 --> 00:18:12.223 align:middle line:90% say, 90%? 00:18:12.223 --> 00:18:17.300 align:middle line:90% 00:18:17.300 --> 00:18:18.250 align:middle line:90% How might we do that? 00:18:18.250 --> 00:18:29.150 align:middle line:90% 00:18:29.150 --> 00:18:31.622 align:middle line:84% Sorry, I completely forgot to give you the expectation. 00:18:31.622 --> 00:18:32.330 align:middle line:90% The answer is no. 00:18:32.330 --> 00:18:41.100 align:middle line:90% 00:18:41.100 --> 00:18:42.426 align:middle line:90% OK. 00:18:42.426 --> 00:18:44.703 align:middle line:90% Now can you do it? 00:18:44.703 --> 00:18:46.370 align:middle line:84% Hopefully, it's a little bit easier now. 00:18:46.370 --> 00:18:58.260 align:middle line:90% 00:18:58.260 --> 00:18:59.454 align:middle line:90% Mm-hmm? 00:18:59.454 --> 00:19:06.703 align:middle line:84% AUDIENCE: What if [INAUDIBLE] value of C such that C times 75 00:19:06.703 --> 00:19:07.363 align:middle line:90% is 90. 00:19:07.363 --> 00:19:08.280 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:19:08.280 --> 00:19:14.040 align:middle line:84% So find the value of C such that C times 75 is 90. 00:19:14.040 --> 00:19:17.740 align:middle line:90% So I guess 75-- 00:19:17.740 --> 00:19:26.722 align:middle line:84% or 90 divided by 75, that's, what's, 1.2. 00:19:26.722 --> 00:19:27.305 align:middle line:90% Is that right? 00:19:27.305 --> 00:19:30.410 align:middle line:90% 00:19:30.410 --> 00:19:34.070 align:middle line:90% 75 is 5 by 15, 96 by 15. 00:19:34.070 --> 00:19:38.406 align:middle line:90% OK, nice, arithmetic, score. 00:19:38.406 --> 00:19:44.450 align:middle line:84% So then using this formulation here, 00:19:44.450 --> 00:19:49.850 align:middle line:84% we have the probability that R is 00:19:49.850 --> 00:20:02.180 align:middle line:84% at least 1.2 times expectation of R is at most 1 00:20:02.180 --> 00:20:05.130 align:middle line:90% divided by 1.2. 00:20:05.130 --> 00:20:07.673 align:middle line:84% So this is going to be something like 83%. 00:20:07.673 --> 00:20:17.190 align:middle line:90% 00:20:17.190 --> 00:20:19.970 align:middle line:84% People reasonably happy with that? 00:20:19.970 --> 00:20:20.810 align:middle line:90% Yeah? 00:20:20.810 --> 00:20:23.965 align:middle line:84% AUDIENCE: Can you please expand a little bit [INAUDIBLE]? 00:20:23.965 --> 00:20:25.340 align:middle line:84% BRYNMOR CHAPMAN: Oh, the question 00:20:25.340 --> 00:20:28.310 align:middle line:84% is, can we explain the difference between these two 00:20:28.310 --> 00:20:29.580 align:middle line:90% formulations? 00:20:29.580 --> 00:20:30.757 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:20:30.757 --> 00:20:31.715 align:middle line:90% BRYNMOR CHAPMAN: Sorry? 00:20:31.715 --> 00:20:33.827 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:20:33.827 --> 00:20:35.660 align:middle line:84% BRYNMOR CHAPMAN: So yeah, essentially, there 00:20:35.660 --> 00:20:36.710 align:middle line:90% is no difference. 00:20:36.710 --> 00:20:40.980 align:middle line:84% They're just different ways of thinking about the bound. 00:20:40.980 --> 00:20:43.010 align:middle line:84% You can either think about bounding it in terms 00:20:43.010 --> 00:20:45.150 align:middle line:90% of some absolute threshold. 00:20:45.150 --> 00:20:47.360 align:middle line:84% Or you can think about it as in terms 00:20:47.360 --> 00:20:51.860 align:middle line:84% of some relative threshold, multiply the expectation 00:20:51.860 --> 00:20:55.224 align:middle line:90% by some scaling factor. 00:20:55.224 --> 00:20:58.760 align:middle line:84% So, depending on which of them feels more natural to you, 00:20:58.760 --> 00:21:01.160 align:middle line:84% we could also use the first formulation to say, 00:21:01.160 --> 00:21:06.995 align:middle line:84% probability that R is greater than 90 greater than or equal to 00:21:06.995 --> 00:21:16.730 align:middle line:84% is going to be at most, 75 over 90, which same thing, 83%. 00:21:16.730 --> 00:21:18.298 align:middle line:90% So either of them works. 00:21:18.298 --> 00:21:19.590 align:middle line:90% They both give the same answer. 00:21:19.590 --> 00:21:20.923 align:middle line:90% They're equivalent formulations. 00:21:20.923 --> 00:21:23.630 align:middle line:84% It's just whichever one you find easier to think about. 00:21:23.630 --> 00:21:25.010 align:middle line:90% Question? 00:21:25.010 --> 00:21:28.200 align:middle line:84% AUDIENCE: Based on the fact that we know the range is 30 to 100, 00:21:28.200 --> 00:21:34.120 align:middle line:84% not 0 to 100, is there a way to make the final probability more 00:21:34.120 --> 00:21:34.620 align:middle line:90% accurate? 00:21:34.620 --> 00:21:36.330 align:middle line:90% BRYNMOR CHAPMAN: Yes, exactly. 00:21:36.330 --> 00:21:39.350 align:middle line:84% So the question was, we don't just 00:21:39.350 --> 00:21:41.560 align:middle line:84% have a non-negative random variable R. Right. 00:21:41.560 --> 00:21:43.310 align:middle line:84% We've actually got an even stronger bound. 00:21:43.310 --> 00:21:46.040 align:middle line:84% R is at least 30, not just at least 0. 00:21:46.040 --> 00:21:50.020 align:middle line:84% So can we leverage that to get an even stronger bound? 00:21:50.020 --> 00:21:52.760 align:middle line:90% 00:21:52.760 --> 00:21:54.055 align:middle line:90% So how could we do that? 00:21:54.055 --> 00:21:57.337 align:middle line:90% 00:21:57.337 --> 00:21:59.545 align:middle line:84% Well, let's try defining a different random variable. 00:21:59.545 --> 00:22:06.530 align:middle line:90% 00:22:06.530 --> 00:22:13.525 align:middle line:90% So let R prime be R negative 30. 00:22:13.525 --> 00:22:20.410 align:middle line:90% 00:22:20.410 --> 00:22:22.063 align:middle line:84% So this is another random variable. 00:22:22.063 --> 00:22:23.730 align:middle line:84% We're just going to take our original R. 00:22:23.730 --> 00:22:26.107 align:middle line:90% We're going to subtract 30. 00:22:26.107 --> 00:22:28.690 align:middle line:84% We're not going to do this to people's exam scores in reality. 00:22:28.690 --> 00:22:30.600 align:middle line:90% Don't worry. 00:22:30.600 --> 00:22:33.060 align:middle line:84% But that this makes the minimum exam score 0. 00:22:33.060 --> 00:22:35.835 align:middle line:90% 00:22:35.835 --> 00:22:40.110 align:middle line:84% So the hope is that now we're satisfying the condition 00:22:40.110 --> 00:22:44.530 align:middle line:84% to Markov's inequality more tightly. 00:22:44.530 --> 00:22:48.370 align:middle line:84% So hopefully, Markov will give us a tighter bound. 00:22:48.370 --> 00:22:50.070 align:middle line:84% So what bound does it give us now? 00:22:50.070 --> 00:22:54.875 align:middle line:84% What is the probability that R is at least 90? 00:22:54.875 --> 00:23:00.360 align:middle line:90% 00:23:00.360 --> 00:23:03.068 align:middle line:84% Well, what is that in terms of R prime. 00:23:03.068 --> 00:23:03.860 align:middle line:90% What is that event? 00:23:03.860 --> 00:23:09.660 align:middle line:90% 00:23:09.660 --> 00:23:10.380 align:middle line:90% Yeah? 00:23:10.380 --> 00:23:12.030 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:23:12.030 --> 00:23:13.380 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:23:13.380 --> 00:23:17.100 align:middle line:84% So this is the same event, and so the same probability 00:23:17.100 --> 00:23:19.393 align:middle line:90% that r prime is at least 60. 00:23:19.393 --> 00:23:28.152 align:middle line:90% 00:23:28.152 --> 00:23:30.110 align:middle line:84% Now, what happens when we apply Markov to this? 00:23:30.110 --> 00:23:44.880 align:middle line:90% 00:23:44.880 --> 00:23:58.050 align:middle line:84% Well, this is going to be at most, the expectation of R 00:23:58.050 --> 00:24:03.000 align:middle line:90% prime divided by 60. 00:24:03.000 --> 00:24:04.730 align:middle line:84% What is the expectation of R prime? 00:24:04.730 --> 00:24:17.070 align:middle line:90% 00:24:17.070 --> 00:24:18.040 align:middle line:90% Yeah? 00:24:18.040 --> 00:24:18.720 align:middle line:90% AUDIENCE: 45. 00:24:18.720 --> 00:24:22.320 align:middle line:90% BRYNMOR CHAPMAN: 45, yeah. 00:24:22.320 --> 00:24:24.720 align:middle line:84% So this is going to be the expectation 00:24:24.720 --> 00:24:34.120 align:middle line:84% of R minus 30 divided by 60, which is the expectation. 00:24:34.120 --> 00:24:36.750 align:middle line:84% So linearity of expectation, we can pull this-- 00:24:36.750 --> 00:24:40.270 align:middle line:84% or we can split this into two expectations. 00:24:40.270 --> 00:24:46.110 align:middle line:84% So this is the expectation of R minus the expectation of 30. 00:24:46.110 --> 00:24:48.400 align:middle line:84% There's not really any randomness involved there. 00:24:48.400 --> 00:24:52.990 align:middle line:90% It's just 30 over 60. 00:24:52.990 --> 00:24:56.760 align:middle line:90% So this is 45 over 60. 00:24:56.760 --> 00:24:59.340 align:middle line:90% 00:24:59.340 --> 00:25:00.970 align:middle line:84% Does that make sense to everybody? 00:25:00.970 --> 00:25:04.080 align:middle line:90% 00:25:04.080 --> 00:25:08.828 align:middle line:84% So now, instead of this bound of it's at most 83%, 00:25:08.828 --> 00:25:10.120 align:middle line:90% now we've got the better bound. 00:25:10.120 --> 00:25:10.828 align:middle line:90% It's at most 75%. 00:25:10.828 --> 00:25:19.015 align:middle line:90% 00:25:19.015 --> 00:25:21.473 align:middle line:84% So by shifting our random variable, 00:25:21.473 --> 00:25:23.015 align:middle line:84% we've managed to get a tighter bound. 00:25:23.015 --> 00:25:33.960 align:middle line:90% 00:25:33.960 --> 00:25:48.090 align:middle line:84% What about the probability that R is at most 65? 00:25:48.090 --> 00:25:49.140 align:middle line:90% Yeah? 00:25:49.140 --> 00:25:54.078 align:middle line:84% AUDIENCE: For the previous one, where R is from 30 to 100, 00:25:54.078 --> 00:25:58.290 align:middle line:84% is that only the [INAUDIBLE] assumed 00:25:58.290 --> 00:26:03.480 align:middle line:84% that the lower bound for x is 0, and that somehow, 00:26:03.480 --> 00:26:05.320 align:middle line:90% the lower bound for R is 30. 00:26:05.320 --> 00:26:10.563 align:middle line:84% So I [INAUDIBLE] do a more complicated thing. 00:26:10.563 --> 00:26:12.730 align:middle line:84% BRYNMOR CHAPMAN: So that's kind of what we're doing. 00:26:12.730 --> 00:26:15.400 align:middle line:84% The question was, in the proof of Markov, 00:26:15.400 --> 00:26:18.360 align:middle line:84% we're assuming that the lower bound is 0. 00:26:18.360 --> 00:26:20.440 align:middle line:84% Here we've got a lower bound of 30 instead. 00:26:20.440 --> 00:26:22.630 align:middle line:84% So do we need to do something more complicated? 00:26:22.630 --> 00:26:24.670 align:middle line:84% The answer is no, we don't have to. 00:26:24.670 --> 00:26:27.340 align:middle line:84% But if we do, it gives us a stronger result. 00:26:27.340 --> 00:26:29.160 align:middle line:90% So that's what we've done here. 00:26:29.160 --> 00:26:31.030 align:middle line:90% But we don't have to do it. 00:26:31.030 --> 00:26:35.220 align:middle line:84% Because 0 is still a valid lower bound. 00:26:35.220 --> 00:26:37.770 align:middle line:84% We could say that the co-domain of R 00:26:37.770 --> 00:26:40.020 align:middle line:84% is 0 to 100 instead of 30 to 100, 00:26:40.020 --> 00:26:43.960 align:middle line:84% even if the range stays unchanged. 00:26:43.960 --> 00:26:46.240 align:middle line:84% So it's still a non-negative random variable. 00:26:46.240 --> 00:26:48.400 align:middle line:84% So we're still allowed to apply Markov to it. 00:26:48.400 --> 00:26:51.600 align:middle line:84% It's just that doing it on R instead of R prime 00:26:51.600 --> 00:26:56.910 align:middle line:84% gives us a weaker result. Does that make sense to everybody? 00:26:56.910 --> 00:26:58.150 align:middle line:90% Another question? 00:26:58.150 --> 00:27:00.240 align:middle line:84% AUDIENCE: [INAUDIBLE] you mean use a tighter 00:27:00.240 --> 00:27:02.880 align:middle line:90% bound when you shift on. 00:27:02.880 --> 00:27:05.160 align:middle line:84% Does a tighter bound mean that the probability 00:27:05.160 --> 00:27:06.370 align:middle line:90% is lower or higher? 00:27:06.370 --> 00:27:07.870 align:middle line:84% BRYNMOR CHAPMAN: So the question is, 00:27:07.870 --> 00:27:11.460 align:middle line:84% what do I mean by a tighter bound when we shift R? 00:27:11.460 --> 00:27:16.660 align:middle line:84% So tighter in the sense that it is stronger. 00:27:16.660 --> 00:27:22.360 align:middle line:84% So R at least 30 implies that R is at least 0. 00:27:22.360 --> 00:27:29.530 align:middle line:84% So R at least 30 is a stronger condition than R at least 0. 00:27:29.530 --> 00:27:31.980 align:middle line:84% So that's what I mean by tighter. 00:27:31.980 --> 00:27:33.780 align:middle line:84% AUDIENCE: So what's the difference 00:27:33.780 --> 00:27:38.010 align:middle line:84% between the 75% that we got here for R 00:27:38.010 --> 00:27:41.430 align:middle line:84% is greater than or equal to 90 versus the 83%? 00:27:41.430 --> 00:27:44.460 align:middle line:84% BRYNMOR CHAPMAN: So the difference 00:27:44.460 --> 00:27:47.970 align:middle line:90% is how we're applying Markov. 00:27:47.970 --> 00:27:51.240 align:middle line:84% In the first case, we're applying Markov to R directly. 00:27:51.240 --> 00:27:55.230 align:middle line:84% In the second case, we are creating a new random variable. 00:27:55.230 --> 00:27:57.100 align:middle line:84% And we're applying Markov to that. 00:27:57.100 --> 00:28:05.693 align:middle line:84% And because we've got a stronger condition, Markov's-- 00:28:05.693 --> 00:28:07.110 align:middle line:84% what do you call it-- precondition 00:28:07.110 --> 00:28:15.785 align:middle line:84% is tight now, we can get a stronger result. Yeah? 00:28:15.785 --> 00:28:18.120 align:middle line:84% AUDIENCE: I also have a question. 00:28:18.120 --> 00:28:20.790 align:middle line:84% So does that mean it's depending on the situation 00:28:20.790 --> 00:28:26.370 align:middle line:84% to [INAUDIBLE] it could go higher than 83%? 00:28:26.370 --> 00:28:28.346 align:middle line:84% BRYNMOR CHAPMAN: Could it go higher than 83%? 00:28:28.346 --> 00:28:31.203 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:28:31.203 --> 00:28:33.370 align:middle line:84% BRYNMOR CHAPMAN: So we will get to that in a moment. 00:28:33.370 --> 00:28:37.020 align:middle line:84% The question is, if we're in a different situation, 00:28:37.020 --> 00:28:38.670 align:middle line:84% could shifting it give us something 00:28:38.670 --> 00:28:40.510 align:middle line:90% that's higher than 83%? 00:28:40.510 --> 00:28:44.400 align:middle line:90% And the answer is yes-ish. 00:28:44.400 --> 00:28:47.730 align:middle line:84% In the case where we've applied Markov incorrectly 00:28:47.730 --> 00:28:50.670 align:middle line:90% the first time, then yes. 00:28:50.670 --> 00:28:55.080 align:middle line:84% But if we've applied Markov correctly, 00:28:55.080 --> 00:28:59.610 align:middle line:90% then the first is a valid bound. 00:28:59.610 --> 00:29:03.460 align:middle line:84% So we could, in principle, improve it. 00:29:03.460 --> 00:29:06.540 align:middle line:84% But we're not going to do any worse 00:29:06.540 --> 00:29:09.810 align:middle line:90% by imposing tighter conditions. 00:29:09.810 --> 00:29:13.063 align:middle line:84% Does that at least kind of intuitively make sense? 00:29:13.063 --> 00:29:15.105 align:middle line:84% I'll give you a more complete answer in a moment. 00:29:15.105 --> 00:29:18.545 align:middle line:90% 00:29:18.545 --> 00:29:20.170 align:middle line:84% Any more questions about this before we 00:29:20.170 --> 00:29:22.730 align:middle line:90% move on to the next example? 00:29:22.730 --> 00:29:23.230 align:middle line:90% Yeah? 00:29:23.230 --> 00:29:25.480 align:middle line:84% AUDIENCE: So it's a good thing that the 75% 00:29:25.480 --> 00:29:27.590 align:middle line:90% is lower than the 83%? 00:29:27.590 --> 00:29:29.840 align:middle line:84% BRYNMOR CHAPMAN: Yes, it is a good thing. 00:29:29.840 --> 00:29:32.950 align:middle line:84% So the question is, is it a good thing that the 75% 00:29:32.950 --> 00:29:35.020 align:middle line:90% is smaller than the 83%? 00:29:35.020 --> 00:29:35.710 align:middle line:90% Yes. 00:29:35.710 --> 00:29:38.810 align:middle line:84% We are looking for an upper bound on the probability. 00:29:38.810 --> 00:29:42.020 align:middle line:84% So a smaller upper bound is a stronger condition. 00:29:42.020 --> 00:29:43.310 align:middle line:90% So that's better. 00:29:43.310 --> 00:29:46.145 align:middle line:90% It's a stronger bound. 00:29:46.145 --> 00:29:47.770 align:middle line:84% AUDIENCE: We want a smaller upper bound 00:29:47.770 --> 00:29:48.853 align:middle line:90% or a stronger [INAUDIBLE]? 00:29:48.853 --> 00:29:50.395 align:middle line:84% BRYNMOR CHAPMAN: So they're the same. 00:29:50.395 --> 00:29:52.360 align:middle line:84% Smaller upper bounds are stronger upper bounds. 00:29:52.360 --> 00:29:52.860 align:middle line:90% Yeah. 00:29:52.860 --> 00:29:56.650 align:middle line:90% 00:29:56.650 --> 00:30:00.747 align:middle line:84% So less than or equal to 83% implies less than or equal to-- 00:30:00.747 --> 00:30:02.830 align:middle line:84% sorry, less than or equal to 75% implies less than 00:30:02.830 --> 00:30:04.550 align:middle line:90% or equal to 83%. 00:30:04.550 --> 00:30:06.285 align:middle line:90% So 75 is stronger. 00:30:06.285 --> 00:30:13.613 align:middle line:90% 00:30:13.613 --> 00:30:16.030 align:middle line:84% Have people had enough time to think about the probability 00:30:16.030 --> 00:30:17.143 align:middle line:90% that R is at most 65? 00:30:17.143 --> 00:30:21.490 align:middle line:90% 00:30:21.490 --> 00:30:23.775 align:middle line:84% How might we use Markov to figure this one out? 00:30:23.775 --> 00:30:28.570 align:middle line:90% 00:30:28.570 --> 00:30:29.290 align:middle line:90% Yeah? 00:30:29.290 --> 00:30:31.480 align:middle line:84% AUDIENCE: Can you do 1 minus probability 00:30:31.480 --> 00:30:35.930 align:middle line:90% that R is greater than 65. 00:30:35.930 --> 00:30:37.930 align:middle line:90% So [INAUDIBLE]. 00:30:37.930 --> 00:30:41.238 align:middle line:90% 00:30:41.238 --> 00:30:45.500 align:middle line:84% BRYNMOR CHAPMAN: So R greater than or equal to 66, 00:30:45.500 --> 00:30:48.970 align:middle line:90% so if we assume integer scores. 00:30:48.970 --> 00:30:49.470 align:middle line:90% OK. 00:30:49.470 --> 00:30:52.470 align:middle line:90% 00:30:52.470 --> 00:30:55.245 align:middle line:84% And what does Markov tell us about this? 00:30:55.245 --> 00:31:16.960 align:middle line:90% 00:31:16.960 --> 00:31:23.380 align:middle line:84% Well, the probability that R is at least 66 00:31:23.380 --> 00:31:32.200 align:middle line:84% is going to be at most 75 over 66, which is bigger than 1. 00:31:32.200 --> 00:31:35.422 align:middle line:84% So applying Markov does not tell us anything useful. 00:31:35.422 --> 00:31:37.630 align:middle line:84% It tells us that this is a probability that's bounded 00:31:37.630 --> 00:31:39.420 align:middle line:90% by 1, which we already knew. 00:31:39.420 --> 00:31:43.870 align:middle line:90% 00:31:43.870 --> 00:31:45.965 align:middle line:90% So it is a good idea. 00:31:45.965 --> 00:31:47.340 align:middle line:84% It's not quite what we want here. 00:31:47.340 --> 00:31:59.830 align:middle line:90% 00:31:59.830 --> 00:32:03.880 align:middle line:84% So what if instead of fiddling with the event, 00:32:03.880 --> 00:32:08.085 align:middle line:84% we fiddle with R again, kind of similar to what we did before? 00:32:08.085 --> 00:32:16.480 align:middle line:90% 00:32:16.480 --> 00:32:17.200 align:middle line:90% Yeah? 00:32:17.200 --> 00:32:19.900 align:middle line:84% AUDIENCE: [INAUDIBLE] that R [INAUDIBLE]. 00:32:19.900 --> 00:32:22.450 align:middle line:90% 00:32:22.450 --> 00:32:23.780 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 00:32:23.780 --> 00:32:27.220 align:middle line:84% So we're going to flip R. So we're 00:32:27.220 --> 00:32:29.000 align:middle line:84% going to define a new random variable. 00:32:29.000 --> 00:32:40.510 align:middle line:84% Let's say S equals 100 minus R. So now the probability that R 00:32:40.510 --> 00:32:46.000 align:middle line:84% is at most 65 is the probability that S is at least, 00:32:46.000 --> 00:32:48.385 align:middle line:90% what, 100 minus 65? 00:32:48.385 --> 00:32:48.885 align:middle line:90% 35. 00:32:48.885 --> 00:32:52.690 align:middle line:90% 00:32:52.690 --> 00:32:54.910 align:middle line:90% OK. 00:32:54.910 --> 00:32:56.400 align:middle line:90% Can we apply Markov to that? 00:32:56.400 --> 00:32:59.685 align:middle line:90% 00:32:59.685 --> 00:33:00.560 align:middle line:90% I'm seeing some nods. 00:33:00.560 --> 00:33:02.143 align:middle line:84% Does anybody want to do that and spare 00:33:02.143 --> 00:33:03.636 align:middle line:84% me the trouble of doing arithmetic? 00:33:03.636 --> 00:33:06.550 align:middle line:90% 00:33:06.550 --> 00:33:07.360 align:middle line:90% Nobody? 00:33:07.360 --> 00:33:07.860 align:middle line:90% Sadness. 00:33:07.860 --> 00:33:11.170 align:middle line:90% 00:33:11.170 --> 00:33:13.790 align:middle line:90% Well, what does Markov say? 00:33:13.790 --> 00:33:17.980 align:middle line:84% Well, it's at most the expectation of x-- 00:33:17.980 --> 00:33:21.375 align:middle line:90% oh, sorry S-- divided by 35. 00:33:21.375 --> 00:33:24.010 align:middle line:90% 00:33:24.010 --> 00:33:27.730 align:middle line:90% What is the expectation of S? 00:33:27.730 --> 00:33:29.890 align:middle line:84% Well, using linearity in the same way 00:33:29.890 --> 00:33:32.560 align:middle line:84% that we did a moment ago, this is 00:33:32.560 --> 00:33:38.150 align:middle line:84% going to be equal to, what, 100 minus the expectation of R, 00:33:38.150 --> 00:33:42.050 align:middle line:90% which is 75, over 35. 00:33:42.050 --> 00:33:44.030 align:middle line:90% So that's 25 over 35. 00:33:44.030 --> 00:33:44.733 align:middle line:90% This is 5/7. 00:33:44.733 --> 00:33:54.845 align:middle line:90% 00:33:54.845 --> 00:33:57.310 align:middle line:84% So does that make sense to everybody? 00:33:57.310 --> 00:34:01.223 align:middle line:84% There is at most, a 5/7 chance that you did worse than 65. 00:34:01.223 --> 00:34:25.452 align:middle line:90% 00:34:25.452 --> 00:34:28.179 align:middle line:84% Now, perhaps not terribly realistic in terms of exam 00:34:28.179 --> 00:34:32.500 align:middle line:84% scores, but what if, say, some of the students 00:34:32.500 --> 00:34:36.400 align:middle line:84% did so badly that they got scores that are less than 0? 00:34:36.400 --> 00:34:53.464 align:middle line:84% Maybe you've got R is in the range minus 30 to 100. 00:34:53.464 --> 00:34:59.303 align:middle line:84% Same deal, expectation of R Is 75. 00:34:59.303 --> 00:35:06.002 align:middle line:90% 00:35:06.002 --> 00:35:11.100 align:middle line:90% A question, or-- sorry. 00:35:11.100 --> 00:35:13.850 align:middle line:84% So what can we say about the probability 00:35:13.850 --> 00:35:18.520 align:middle line:90% that R is at least, say, 90? 00:35:18.520 --> 00:35:27.272 align:middle line:90% 00:35:27.272 --> 00:35:28.264 align:middle line:90% Mm-hmm? 00:35:28.264 --> 00:35:30.248 align:middle line:90% AUDIENCE: So Markov [INAUDIBLE]? 00:35:30.248 --> 00:35:33.470 align:middle line:90% 00:35:33.470 --> 00:35:34.980 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 00:35:34.980 --> 00:35:39.750 align:middle line:84% So the observation was if we try to apply Markov to this-- 00:35:39.750 --> 00:35:42.420 align:middle line:84% maybe I should call it something different. 00:35:42.420 --> 00:35:53.715 align:middle line:84% Let's call it T. So if we try to apply Markov to T, 00:35:53.715 --> 00:35:56.090 align:middle line:84% it's going to be the same thing as when we applied Markov 00:35:56.090 --> 00:35:57.000 align:middle line:90% to R before. 00:35:57.000 --> 00:35:58.920 align:middle line:90% It will give us a bound of 83%. 00:35:58.920 --> 00:36:01.160 align:middle line:84% The problem is that that's wrong. 00:36:01.160 --> 00:36:04.500 align:middle line:84% T does not satisfy the precondition for Markov. 00:36:04.500 --> 00:36:10.785 align:middle line:84% So what we want to do instead is define T prime equals T plus 30. 00:36:10.785 --> 00:36:14.850 align:middle line:90% 00:36:14.850 --> 00:36:18.540 align:middle line:84% And now if we apply Markov to T prime, what do we get? 00:36:18.540 --> 00:36:25.260 align:middle line:84% So the probability that T is greater than 90 00:36:25.260 --> 00:36:34.440 align:middle line:84% is the probability that T prime is at least 120. 00:36:34.440 --> 00:36:39.570 align:middle line:84% And by Markov, this is going to be expectation 00:36:39.570 --> 00:36:45.810 align:middle line:90% of T prime over 120. 00:36:45.810 --> 00:36:48.510 align:middle line:84% And same deal as before, linearity here 00:36:48.510 --> 00:36:55.878 align:middle line:90% gives us 75 plus 30 over 120. 00:36:55.878 --> 00:37:01.410 align:middle line:90% 105 over 120, that's 7/8. 00:37:01.410 --> 00:37:03.585 align:middle line:90% Is that right? 00:37:03.585 --> 00:37:04.085 align:middle line:90% Yeah. 00:37:04.085 --> 00:37:11.030 align:middle line:90% 00:37:11.030 --> 00:37:14.392 align:middle line:84% So in this case, we do get, in some sense, 00:37:14.392 --> 00:37:16.850 align:middle line:84% a weaker bound than we would have gotten by applying Markov 00:37:16.850 --> 00:37:19.130 align:middle line:90% to T directly. 00:37:19.130 --> 00:37:22.650 align:middle line:84% But applying Markov to T directly doesn't actually work. 00:37:22.650 --> 00:37:25.350 align:middle line:84% This looks weaker, but it's a correct bound. 00:37:25.350 --> 00:37:28.910 align:middle line:84% So hopefully, that answers your question more fully. 00:37:28.910 --> 00:37:34.160 align:middle line:90% Does anybody have any questions? 00:37:34.160 --> 00:37:37.040 align:middle line:84% So even though we couldn't apply Markov directly to T, 00:37:37.040 --> 00:37:39.470 align:middle line:84% we can create a new random variable T prime, 00:37:39.470 --> 00:37:42.320 align:middle line:90% to which Markov does apply. 00:37:42.320 --> 00:37:43.295 align:middle line:90% Yeah? 00:37:43.295 --> 00:37:45.600 align:middle line:84% AUDIENCE: Got a question before about 30 00:37:45.600 --> 00:37:49.220 align:middle line:84% being a lower bound to make our statement any stronger 00:37:49.220 --> 00:37:51.210 align:middle line:90% for S [INAUDIBLE]. 00:37:51.210 --> 00:37:55.220 align:middle line:90% 00:37:55.220 --> 00:37:57.470 align:middle line:84% BRYNMOR CHAPMAN: So using the lower bound of 30 making 00:37:57.470 --> 00:38:01.448 align:middle line:90% the statement stronger for T? 00:38:01.448 --> 00:38:02.396 align:middle line:90% AUDIENCE: For S. 00:38:02.396 --> 00:38:08.810 align:middle line:84% BRYNMOR CHAPMAN: Oh, for S. Which one was S. Ah, S here? 00:38:08.810 --> 00:38:15.050 align:middle line:84% So that does not really help us in this case. 00:38:15.050 --> 00:38:22.580 align:middle line:84% Because that corresponds to an upper bound rather than 00:38:22.580 --> 00:38:24.200 align:middle line:90% a lower bound. 00:38:24.200 --> 00:38:28.410 align:middle line:84% And upper bounds don't really appear in the proof of Markov. 00:38:28.410 --> 00:38:30.420 align:middle line:84% We're not really using them in any way. 00:38:30.420 --> 00:38:36.620 align:middle line:84% So imposing a stronger upper bound doesn't really help. 00:38:36.620 --> 00:38:39.950 align:middle line:90% Does that answer your question? 00:38:39.950 --> 00:38:42.800 align:middle line:90% OK, cool. 00:38:42.800 --> 00:38:43.610 align:middle line:90% Yeah? 00:38:43.610 --> 00:38:46.550 align:middle line:84% AUDIENCE: Is that always the case that the lowest lower bound 00:38:46.550 --> 00:38:51.147 align:middle line:84% appears [INAUDIBLE] lowest value here includes 0? 00:38:51.147 --> 00:38:52.730 align:middle line:84% BRYNMOR CHAPMAN: Is it always the case 00:38:52.730 --> 00:38:54.690 align:middle line:84% that the best lower bound appears-- or sorry, 00:38:54.690 --> 00:39:01.010 align:middle line:84% the best upper bound, I guess, appears when the range actually 00:39:01.010 --> 00:39:01.765 align:middle line:90% includes 0? 00:39:01.765 --> 00:39:04.280 align:middle line:90% 00:39:04.280 --> 00:39:14.020 align:middle line:84% Essentially, yes, as far as correct bounds go, anyway. 00:39:14.020 --> 00:39:20.310 align:middle line:90% 00:39:20.310 --> 00:39:26.310 align:middle line:84% So I guess there's a question of morally, 00:39:26.310 --> 00:39:30.410 align:middle line:84% why do we need this non-negativity? 00:39:30.410 --> 00:39:36.230 align:middle line:84% And perhaps a very simple illustrative example 00:39:36.230 --> 00:39:39.320 align:middle line:84% is, suppose we have the random variable R, which 00:39:39.320 --> 00:39:45.090 align:middle line:84% is plus or minus 1, depending on a fair coin flip. 00:39:45.090 --> 00:39:52.100 align:middle line:84% So it's uniform on positive 1 and negative 1. 00:39:52.100 --> 00:39:54.710 align:middle line:90% 00:39:54.710 --> 00:40:00.510 align:middle line:84% So if we try to apply Markov to that directly, 00:40:00.510 --> 00:40:05.130 align:middle line:84% what does that tell us about the probability that R is greater 00:40:05.130 --> 00:40:06.160 align:middle line:90% than 1/2? 00:40:06.160 --> 00:40:27.270 align:middle line:90% 00:40:27.270 --> 00:40:30.960 align:middle line:84% Well, if we try to apply Markov directly, the expectation of R 00:40:30.960 --> 00:40:31.520 align:middle line:90% is 0. 00:40:31.520 --> 00:40:34.110 align:middle line:90% 00:40:34.110 --> 00:40:38.250 align:middle line:84% So no matter what we put here, Markov 00:40:38.250 --> 00:40:45.180 align:middle line:84% would try to tell us that the probability is at most, 0. 00:40:45.180 --> 00:40:48.190 align:middle line:90% That is clearly rubbish. 00:40:48.190 --> 00:40:50.640 align:middle line:84% Because R can be greater than 1/2. 00:40:50.640 --> 00:40:52.770 align:middle line:84% R takes value 1 with probability 1/2. 00:40:52.770 --> 00:40:56.368 align:middle line:90% 00:40:56.368 --> 00:40:59.060 align:middle line:90% So basically, in our proof-- 00:40:59.060 --> 00:41:00.455 align:middle line:90% where was it? 00:41:00.455 --> 00:41:00.955 align:middle line:90% Yeah. 00:41:00.955 --> 00:41:04.550 align:middle line:90% 00:41:04.550 --> 00:41:08.160 align:middle line:84% So if we get rid of the non-negativity constraint, 00:41:08.160 --> 00:41:11.580 align:middle line:90% then this no longer works. 00:41:11.580 --> 00:41:17.330 align:middle line:84% We can't discard this term, because that 0, 00:41:17.330 --> 00:41:20.310 align:middle line:84% if we don't know that the random variable is non-negative, 00:41:20.310 --> 00:41:23.670 align:middle line:84% we could, in principle, have a negative value there, 00:41:23.670 --> 00:41:26.335 align:middle line:90% which we can't get rid of. 00:41:26.335 --> 00:41:35.090 align:middle line:90% 00:41:35.090 --> 00:41:39.175 align:middle line:84% So another question, is Markov tight? 00:41:39.175 --> 00:41:57.030 align:middle line:90% 00:41:57.030 --> 00:42:02.070 align:middle line:84% So, in other words, can we make a stronger statement 00:42:02.070 --> 00:42:05.550 align:middle line:84% than what Markov tells us out of the 10, 00:42:05.550 --> 00:42:08.790 align:middle line:84% just with the information that we have the mean 00:42:08.790 --> 00:42:10.940 align:middle line:84% and, we have a non-negative random variable? 00:42:10.940 --> 00:42:13.663 align:middle line:90% 00:42:13.663 --> 00:42:15.455 align:middle line:84% Is there anything stronger that we can say? 00:42:15.455 --> 00:42:18.270 align:middle line:90% 00:42:18.270 --> 00:42:19.025 align:middle line:90% Who thinks yes? 00:42:19.025 --> 00:42:21.840 align:middle line:90% 00:42:21.840 --> 00:42:23.080 align:middle line:90% A couple of tentative hands. 00:42:23.080 --> 00:42:26.010 align:middle line:90% Who thinks no? 00:42:26.010 --> 00:42:28.090 align:middle line:84% A couple of less tentative hands. 00:42:28.090 --> 00:42:28.790 align:middle line:90% Who has no idea? 00:42:28.790 --> 00:42:32.070 align:middle line:90% 00:42:32.070 --> 00:42:36.782 align:middle line:84% Well, let's think about some simple examples then. 00:42:36.782 --> 00:42:39.480 align:middle line:84% So let's go back to the cell phone check problem 00:42:39.480 --> 00:42:43.170 align:middle line:90% from, what was it, last week? 00:42:43.170 --> 00:42:45.330 align:middle line:84% Don't remember which of the lectures last week. 00:42:45.330 --> 00:42:46.300 align:middle line:90% Do you say Thursday? 00:42:46.300 --> 00:42:47.192 align:middle line:90% Both? 00:42:47.192 --> 00:42:50.260 align:middle line:90% Both, OK. 00:42:50.260 --> 00:42:55.260 align:middle line:84% So if we consider the Lazy Susan version, 00:42:55.260 --> 00:43:03.995 align:middle line:84% and we let R be the number of people who get their phone-- 00:43:03.995 --> 00:43:09.750 align:middle line:90% 00:43:09.750 --> 00:43:21.000 align:middle line:84% so this is the Lazy Susan How could we 00:43:21.000 --> 00:43:24.873 align:middle line:84% use Markov to bound the probability that everybody 00:43:24.873 --> 00:43:25.790 align:middle line:90% gets their phone back? 00:43:25.790 --> 00:43:33.030 align:middle line:90% 00:43:33.030 --> 00:43:36.470 align:middle line:84% So if everybody gets their phone back, that's R equals n. 00:43:36.470 --> 00:43:48.090 align:middle line:90% 00:43:48.090 --> 00:43:50.470 align:middle line:84% What does Markov tell us about this probability? 00:43:50.470 --> 00:44:04.140 align:middle line:90% 00:44:04.140 --> 00:44:04.830 align:middle line:90% Yeah? 00:44:04.830 --> 00:44:06.083 align:middle line:90% AUDIENCE: At most, 1 over n. 00:44:06.083 --> 00:44:08.000 align:middle line:84% BRYNMOR CHAPMAN: Yeah, it's at most, 1 over n. 00:44:08.000 --> 00:44:12.600 align:middle line:90% 00:44:12.600 --> 00:44:13.600 align:middle line:90% Why is that? 00:44:13.600 --> 00:44:17.730 align:middle line:84% Well, the expectation of R is 1, as we saw last week. 00:44:17.730 --> 00:44:21.390 align:middle line:90% The bound we're looking at is n. 00:44:21.390 --> 00:44:24.180 align:middle line:84% So there's at most, a 1 in n probability 00:44:24.180 --> 00:44:26.745 align:middle line:84% that everybody gets their phone back. 00:44:26.745 --> 00:44:28.620 align:middle line:84% What is the actual probability that everybody 00:44:28.620 --> 00:44:29.537 align:middle line:90% gets their phone back? 00:44:29.537 --> 00:44:35.370 align:middle line:90% 00:44:35.370 --> 00:44:36.090 align:middle line:90% Yeah? 00:44:36.090 --> 00:44:37.043 align:middle line:90% AUDIENCE: 1 over n. 00:44:37.043 --> 00:44:37.960 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:44:37.960 --> 00:44:40.620 align:middle line:84% The actual probability is 1 over n. 00:44:40.620 --> 00:44:43.140 align:middle line:90% So this is tight. 00:44:43.140 --> 00:44:45.480 align:middle line:84% We can't say anything stronger than Markov 00:44:45.480 --> 00:44:47.230 align:middle line:90% without additional information. 00:44:47.230 --> 00:44:50.490 align:middle line:90% 00:44:50.490 --> 00:44:51.770 align:middle line:90% Does that make sense? 00:44:51.770 --> 00:44:54.753 align:middle line:90% 00:44:54.753 --> 00:44:56.920 align:middle line:84% So if we know something else about our distribution, 00:44:56.920 --> 00:44:58.753 align:middle line:84% then we might be able to say something more. 00:44:58.753 --> 00:45:01.230 align:middle line:84% But if all we know is the expectation 00:45:01.230 --> 00:45:04.430 align:middle line:84% and the non-negativity, then Markov 00:45:04.430 --> 00:45:05.555 align:middle line:90% is the best that we can do. 00:45:05.555 --> 00:45:12.550 align:middle line:90% 00:45:12.550 --> 00:45:15.990 align:middle line:84% On the other hand, suppose we're permuting all of the phones 00:45:15.990 --> 00:45:17.855 align:middle line:84% instead of putting them on the Lazy Susan. 00:45:17.855 --> 00:45:35.520 align:middle line:90% 00:45:35.520 --> 00:45:37.047 align:middle line:84% So in this case, what does Markov 00:45:37.047 --> 00:45:38.880 align:middle line:84% tell us about the probability that everybody 00:45:38.880 --> 00:45:41.395 align:middle line:90% gets their phone back? 00:45:41.395 --> 00:45:45.120 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:45:45.120 --> 00:45:47.700 align:middle line:84% BRYNMOR CHAPMAN: Well, it's still 1 over n. 00:45:47.700 --> 00:45:50.580 align:middle line:84% We've still got a non-negative random variable 00:45:50.580 --> 00:45:51.640 align:middle line:90% with the same mean. 00:45:51.640 --> 00:45:54.088 align:middle line:84% We're still looking at the same threshold. 00:45:54.088 --> 00:45:55.505 align:middle line:84% So Markov gives us the same bound. 00:45:55.505 --> 00:46:06.695 align:middle line:90% 00:46:06.695 --> 00:46:08.688 align:middle line:84% What is the actual probability that everybody 00:46:08.688 --> 00:46:09.605 align:middle line:90% gets their phone back? 00:46:09.605 --> 00:46:27.660 align:middle line:90% 00:46:27.660 --> 00:46:31.360 align:middle line:84% Well, that's exactly when you have the identity permutation. 00:46:31.360 --> 00:46:33.490 align:middle line:84% You've got n factorial possible permutations. 00:46:33.490 --> 00:46:36.150 align:middle line:84% One of them gives everybody's phone back. 00:46:36.150 --> 00:46:41.770 align:middle line:84% So what we're looking for is actually 1 over n factorial. 00:46:41.770 --> 00:46:45.575 align:middle line:84% So Markov is very much not tight in this example. 00:46:45.575 --> 00:47:01.120 align:middle line:90% 00:47:01.120 --> 00:47:03.060 align:middle line:84% So the next question is, what can we 00:47:03.060 --> 00:47:06.420 align:middle line:84% say if we do have more information 00:47:06.420 --> 00:47:11.615 align:middle line:90% about a random variable? 00:47:11.615 --> 00:47:18.030 align:middle line:90% 00:47:18.030 --> 00:47:21.450 align:middle line:84% So, suppose we know the variance. 00:47:21.450 --> 00:47:25.620 align:middle line:84% So the variance of those two distributions is very different. 00:47:25.620 --> 00:47:28.530 align:middle line:84% In the case where we put the phones on the Lazy Susan, 00:47:28.530 --> 00:47:30.412 align:middle line:90% we've got a very high variance. 00:47:30.412 --> 00:47:32.370 align:middle line:84% In the case where we have a random permutation, 00:47:32.370 --> 00:47:34.830 align:middle line:90% it's much lower. 00:47:34.830 --> 00:47:36.930 align:middle line:84% So maybe the variance can tell us 00:47:36.930 --> 00:47:40.990 align:middle line:90% more about these probabilities. 00:47:40.990 --> 00:47:50.350 align:middle line:90% 00:47:50.350 --> 00:47:54.236 align:middle line:84% So this is a different tail bound, due to Chebyshev. 00:47:54.236 --> 00:47:58.200 align:middle line:90% 00:47:58.200 --> 00:48:08.400 align:middle line:84% So probability that R minus the expectation of R, 00:48:08.400 --> 00:48:13.110 align:middle line:84% an absolute value is at least x, is 00:48:13.110 --> 00:48:14.640 align:middle line:84% less than or equal to the variance 00:48:14.640 --> 00:48:19.640 align:middle line:90% of x divided by x squared. 00:48:19.640 --> 00:48:29.586 align:middle line:90% 00:48:29.586 --> 00:48:33.240 align:middle line:84% So note that we no longer require non-negativity here. 00:48:33.240 --> 00:48:37.830 align:middle line:84% We just require information about the variance 00:48:37.830 --> 00:48:41.280 align:middle line:84% And, well, we have to compute it relative to the mean. 00:48:41.280 --> 00:48:46.120 align:middle line:90% 00:48:46.120 --> 00:48:49.040 align:middle line:84% But R can be any random variable. 00:48:49.040 --> 00:48:52.580 align:middle line:84% We're no longer constrained to looking at non-negative ones. 00:48:52.580 --> 00:48:53.660 align:middle line:90% It can be anything. 00:48:53.660 --> 00:48:55.202 align:middle line:84% We just have to compute the variance. 00:48:55.202 --> 00:48:58.180 align:middle line:90% 00:48:58.180 --> 00:49:00.560 align:middle line:84% How might we prove Chebyshev's inequality? 00:49:00.560 --> 00:49:12.040 align:middle line:90% 00:49:12.040 --> 00:49:12.840 align:middle line:90% Any ideas? 00:49:12.840 --> 00:49:20.200 align:middle line:90% 00:49:20.200 --> 00:49:22.420 align:middle line:90% Is that a hand or a stretch? 00:49:22.420 --> 00:49:23.717 align:middle line:90% Yeah? 00:49:23.717 --> 00:49:25.180 align:middle line:90% AUDIENCE: [INAUDIBLE] Markov? 00:49:25.180 --> 00:49:26.330 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 00:49:26.330 --> 00:49:29.890 align:middle line:84% So we're going to prove Chebyshev using Markov. 00:49:29.890 --> 00:49:40.900 align:middle line:84% So in order to use Markov, what do we need? 00:49:40.900 --> 00:49:43.120 align:middle line:84% Well, we need a non-negative random variable. 00:49:43.120 --> 00:49:47.500 align:middle line:84% So similar to what we were doing earlier 00:49:47.500 --> 00:49:49.810 align:middle line:84% with shifting our random variable, 00:49:49.810 --> 00:49:52.990 align:middle line:84% we're going to define a new random variable that we can then 00:49:52.990 --> 00:49:54.610 align:middle line:90% apply Markov to. 00:49:54.610 --> 00:49:57.690 align:middle line:84% And that should give us this result here. 00:49:57.690 --> 00:50:05.176 align:middle line:90% 00:50:05.176 --> 00:50:08.500 align:middle line:84% Does that make sense to everybody? 00:50:08.500 --> 00:50:09.340 align:middle line:90% Yeah? 00:50:09.340 --> 00:50:13.458 align:middle line:84% AUDIENCE: For a [INAUDIBLE] is variance of x supposed to be R? 00:50:13.458 --> 00:50:15.250 align:middle line:84% BRYNMOR CHAPMAN: Oh, variance of R-- sorry. 00:50:15.250 --> 00:50:17.920 align:middle line:90% Thank you. 00:50:17.920 --> 00:50:20.363 align:middle line:84% Yes, variance of x does not make any sense. 00:50:20.363 --> 00:50:30.322 align:middle line:90% 00:50:30.322 --> 00:50:32.380 align:middle line:84% So what might be a useful random variable 00:50:32.380 --> 00:50:33.585 align:middle line:90% that we can apply Markov to? 00:50:33.585 --> 00:50:46.736 align:middle line:90% 00:50:46.736 --> 00:50:48.403 align:middle line:84% Well, maybe here's the way we can cheat. 00:50:48.403 --> 00:50:52.480 align:middle line:90% 00:50:52.480 --> 00:50:55.575 align:middle line:84% In Markov's inequality, what's the bound that we get? 00:50:55.575 --> 00:50:59.050 align:middle line:90% 00:50:59.050 --> 00:51:00.580 align:middle line:84% The bound should be the expectation 00:51:00.580 --> 00:51:01.455 align:middle line:90% of a random variable. 00:51:01.455 --> 00:51:04.020 align:middle line:90% 00:51:04.020 --> 00:51:07.390 align:middle line:84% So can we define a random variable whose expectation 00:51:07.390 --> 00:51:08.835 align:middle line:90% is the variance of R? 00:51:08.835 --> 00:51:11.950 align:middle line:90% 00:51:11.950 --> 00:51:13.108 align:middle line:90% Yeah? 00:51:13.108 --> 00:51:15.100 align:middle line:90% AUDIENCE: [INAUDIBLE]. 00:51:15.100 --> 00:51:17.110 align:middle line:90% BRYNMOR CHAPMAN: Yeah, exactly. 00:51:17.110 --> 00:51:24.330 align:middle line:84% So R prime equals R minus its expectation squared. 00:51:24.330 --> 00:51:28.340 align:middle line:90% 00:51:28.340 --> 00:51:30.650 align:middle line:90% So this is non-negative. 00:51:30.650 --> 00:51:32.115 align:middle line:90% It's a square of some value. 00:51:32.115 --> 00:51:35.140 align:middle line:90% 00:51:35.140 --> 00:51:41.740 align:middle line:84% Its expectation by definition is the variance of R. 00:51:41.740 --> 00:51:48.880 align:middle line:84% And also, if we squint, we can fiddle around 00:51:48.880 --> 00:51:52.270 align:middle line:84% with this to actually threshold R 00:51:52.270 --> 00:51:56.380 align:middle line:84% prime instead of this absolute value thing that we've got here. 00:51:56.380 --> 00:51:58.990 align:middle line:90% How do we do that? 00:51:58.990 --> 00:52:01.550 align:middle line:84% Well, we just square the stuff in the absolute value. 00:52:01.550 --> 00:52:05.390 align:middle line:84% It has the same square as if we didn't have the absolute value. 00:52:05.390 --> 00:52:13.420 align:middle line:84% So the probability that R minus its expectation greater than 00:52:13.420 --> 00:52:17.650 align:middle line:84% or equal to x equals the probability 00:52:17.650 --> 00:52:23.260 align:middle line:84% that R prime is greater than or equal to x squared-- 00:52:23.260 --> 00:52:27.320 align:middle line:84% because those are exactly the same event. 00:52:27.320 --> 00:52:31.480 align:middle line:84% They're the same event, so they have the same probability. 00:52:31.480 --> 00:52:35.110 align:middle line:84% And now this looks like something we can apply Markov, 00:52:35.110 --> 00:52:38.980 align:middle line:84% less than or equal to the expectation of R 00:52:38.980 --> 00:52:43.815 align:middle line:90% prime divided by x squared. 00:52:43.815 --> 00:52:52.660 align:middle line:90% 00:52:52.660 --> 00:52:55.550 align:middle line:84% Does that make sense to everybody? 00:52:55.550 --> 00:53:10.540 align:middle line:90% 00:53:10.540 --> 00:53:12.133 align:middle line:84% So let's take a look at an example. 00:53:12.133 --> 00:53:19.000 align:middle line:90% 00:53:19.000 --> 00:53:22.120 align:middle line:84% So let's look at those test scores again. 00:53:22.120 --> 00:53:25.290 align:middle line:90% Oops, sorry, my bad. 00:53:25.290 --> 00:53:38.680 align:middle line:90% 00:53:38.680 --> 00:53:41.713 align:middle line:84% So R, same deal, we've got test scores. 00:53:41.713 --> 00:53:45.550 align:middle line:90% 00:53:45.550 --> 00:53:49.990 align:middle line:84% Expectation is-- what did we have? 00:53:49.990 --> 00:53:51.675 align:middle line:90% 70, 75? 00:53:51.675 --> 00:53:52.175 align:middle line:90% 75. 00:53:52.175 --> 00:53:55.450 align:middle line:90% 00:53:55.450 --> 00:53:57.083 align:middle line:90% And say that the variance. 00:53:57.083 --> 00:54:00.220 align:middle line:90% 00:54:00.220 --> 00:54:01.380 align:middle line:90% Is 25. 00:54:01.380 --> 00:54:06.106 align:middle line:90% 00:54:06.106 --> 00:54:10.510 align:middle line:84% So now what can we say about the probability 00:54:10.510 --> 00:54:14.263 align:middle line:90% that R is at most, 65? 00:54:14.263 --> 00:54:22.030 align:middle line:90% 00:54:22.030 --> 00:54:22.860 align:middle line:90% Any ideas? 00:54:22.860 --> 00:54:26.935 align:middle line:90% 00:54:26.935 --> 00:54:28.620 align:middle line:84% How could we apply Chebyshev to this? 00:54:28.620 --> 00:54:51.868 align:middle line:90% 00:54:51.868 --> 00:54:54.285 align:middle line:84% That doesn't quite look like the theorem statement, right? 00:54:54.285 --> 00:54:59.540 align:middle line:90% 00:54:59.540 --> 00:55:06.050 align:middle line:84% But we can observe that this is at most, the probability 00:55:06.050 --> 00:55:13.803 align:middle line:84% that R is at most, 65, or R is greater than or equal to 85. 00:55:13.803 --> 00:55:20.460 align:middle line:90% 00:55:20.460 --> 00:55:23.960 align:middle line:84% If we take the union of two events, 00:55:23.960 --> 00:55:27.200 align:middle line:84% that union has at least the probability of one 00:55:27.200 --> 00:55:30.140 align:middle line:90% of those two constituent events. 00:55:30.140 --> 00:55:32.270 align:middle line:84% So this is only increasing the probability, 00:55:32.270 --> 00:55:39.550 align:middle line:84% if we add in this extra half of the event. 00:55:39.550 --> 00:55:49.200 align:middle line:90% 00:55:49.200 --> 00:55:51.420 align:middle line:84% And what do we know about this probability? 00:55:51.420 --> 00:55:55.040 align:middle line:90% 00:55:55.040 --> 00:55:57.205 align:middle line:84% How could we rewrite that in a nicer way? 00:55:57.205 --> 00:56:04.190 align:middle line:90% 00:56:04.190 --> 00:56:05.345 align:middle line:90% Yeah? 00:56:05.345 --> 00:56:06.720 align:middle line:84% AUDIENCE: Probability of actually 00:56:06.720 --> 00:56:10.430 align:middle line:84% having [INAUDIBLE] expectation is greater than or equal to 10. 00:56:10.430 --> 00:56:11.810 align:middle line:90% BRYNMOR CHAPMAN: Exactly. 00:56:11.810 --> 00:56:15.020 align:middle line:84% So this is just saying R deviates from its mean 00:56:15.020 --> 00:56:16.060 align:middle line:90% by at least 10. 00:56:16.060 --> 00:56:30.365 align:middle line:90% 00:56:30.365 --> 00:56:34.370 align:middle line:84% So now we've got it in the form that Chebyshev likes. 00:56:34.370 --> 00:56:36.830 align:middle line:90% So we can apply Chebyshev. 00:56:36.830 --> 00:56:42.453 align:middle line:84% This is at most, variance of R divided by-- 00:56:42.453 --> 00:56:45.470 align:middle line:90% 00:56:45.470 --> 00:56:47.190 align:middle line:90% R threshold is 10? 00:56:47.190 --> 00:56:48.115 align:middle line:90% Yeah, 10 squared. 00:56:48.115 --> 00:57:00.980 align:middle line:90% 00:57:00.980 --> 00:57:05.005 align:middle line:90% 25 over 100 equals 1/4. 00:57:05.005 --> 00:57:11.330 align:middle line:90% 00:57:11.330 --> 00:57:17.990 align:middle line:84% Now, we've got a much stronger bound this time 00:57:17.990 --> 00:57:19.170 align:middle line:90% than we did earlier. 00:57:19.170 --> 00:57:19.795 align:middle line:90% Yeah, question? 00:57:19.795 --> 00:57:21.420 align:middle line:84% AUDIENCE: Can you explain one more time 00:57:21.420 --> 00:57:22.970 align:middle line:84% how you got the probability of R is 00:57:22.970 --> 00:57:27.290 align:middle line:84% that [INAUDIBLE] 65 or greater than 85 to the next probability? 00:57:27.290 --> 00:57:31.150 align:middle line:84% BRYNMOR CHAPMAN: Ah, OK, so explain this equality? 00:57:31.150 --> 00:57:33.524 align:middle line:90% AUDIENCE: Yes. 00:57:33.524 --> 00:57:35.780 align:middle line:84% BRYNMOR CHAPMAN: So R less than or equal to 65 00:57:35.780 --> 00:57:38.840 align:middle line:84% or R greater than or equal to 85, 00:57:38.840 --> 00:57:44.060 align:middle line:84% we could rewrite this as R minus-- 00:57:44.060 --> 00:57:53.270 align:middle line:84% well, R minus 75 is less than or equal to negative 10. 00:57:53.270 --> 00:57:57.480 align:middle line:84% And this one is R minus 75 greater than 00:57:57.480 --> 00:57:58.480 align:middle line:90% or equal to positive 10. 00:57:58.480 --> 00:58:01.430 align:middle line:90% 00:58:01.430 --> 00:58:06.530 align:middle line:84% So if you absolute value both of these, they're both greater than 00:58:06.530 --> 00:58:07.550 align:middle line:90% or equal to 10. 00:58:07.550 --> 00:58:10.316 align:middle line:90% Does that make sense? 00:58:10.316 --> 00:58:10.816 align:middle line:90% OK. 00:58:10.816 --> 00:58:13.938 align:middle line:90% 00:58:13.938 --> 00:58:17.275 align:middle line:84% And other questions about this example? 00:58:17.275 --> 00:58:26.600 align:middle line:90% 00:58:26.600 --> 00:58:31.950 align:middle line:84% So because we have more information than we had here, 00:58:31.950 --> 00:58:33.175 align:middle line:90% we can get a stronger bound. 00:58:33.175 --> 00:58:35.840 align:middle line:90% 00:58:35.840 --> 00:58:37.510 align:middle line:84% Knowing the variance helps us out. 00:58:37.510 --> 00:58:46.940 align:middle line:90% 00:58:46.940 --> 00:58:50.570 align:middle line:84% So equivalently, we could rephrase 00:58:50.570 --> 00:58:56.840 align:middle line:84% this in terms of standard deviations. 00:58:56.840 --> 00:58:59.990 align:middle line:84% Where would be a good place to write this. 00:58:59.990 --> 00:59:01.970 align:middle line:90% How about here? 00:59:01.970 --> 00:59:15.380 align:middle line:84% So the probability of being at least two standard deviations 00:59:15.380 --> 00:59:21.560 align:middle line:90% from the mean is at most, 1/4. 00:59:21.560 --> 00:59:42.376 align:middle line:90% 00:59:42.376 --> 00:59:45.360 align:middle line:84% Does anybody have any objections to that last statement? 00:59:45.360 --> 01:00:10.010 align:middle line:90% 01:00:10.010 --> 01:00:12.340 align:middle line:84% So if you've taken a stats class, 01:00:12.340 --> 01:00:14.590 align:middle line:84% you probably heard something very different from that. 01:00:14.590 --> 01:00:19.700 align:middle line:90% 01:00:19.700 --> 01:00:20.630 align:middle line:90% No? 01:00:20.630 --> 01:00:24.140 align:middle line:90% Nobody's taking stats? 01:00:24.140 --> 01:00:24.810 align:middle line:90% That's good. 01:00:24.810 --> 01:00:26.540 align:middle line:90% High school stats is terrible. 01:00:26.540 --> 01:00:27.920 align:middle line:90% Oh, that's on camera. 01:00:27.920 --> 01:00:28.710 align:middle line:90% Never mind. 01:00:28.710 --> 01:00:30.020 align:middle line:90% Stats are great. 01:00:30.020 --> 01:00:32.380 align:middle line:84% it's good that you've saved it for university instead 01:00:32.380 --> 01:00:33.505 align:middle line:90% of doing it in high school. 01:00:33.505 --> 01:00:37.424 align:middle line:90% 01:00:37.424 --> 01:00:42.750 align:middle line:84% So if you're looking at a normal distribution, 01:00:42.750 --> 01:00:45.600 align:middle line:84% you actually have a much better bound than 1/4. 01:00:45.600 --> 01:00:47.680 align:middle line:84% It's something like a few percent. 01:00:47.680 --> 01:00:50.531 align:middle line:90% 01:00:50.531 --> 01:00:53.240 align:middle line:90% So why is that. 01:00:53.240 --> 01:00:53.950 align:middle line:90% Any ideas? 01:00:53.950 --> 01:01:00.410 align:middle line:90% 01:01:00.410 --> 01:01:02.270 align:middle line:84% Well, for a normal distribution, we 01:01:02.270 --> 01:01:09.544 align:middle line:84% know a lot more about the random variable than just its variance. 01:01:09.544 --> 01:01:12.320 align:middle line:84% So if we know even more about a random variable 01:01:12.320 --> 01:01:14.000 align:middle line:84% than its variance, it turns out we can 01:01:14.000 --> 01:01:18.110 align:middle line:90% make even stronger statements. 01:01:18.110 --> 01:01:21.438 align:middle line:84% Just like adding in the variance was an upgrade from Markov, 01:01:21.438 --> 01:01:23.105 align:middle line:84% let's see how we can upgrade this again. 01:01:23.105 --> 01:01:35.140 align:middle line:90% 01:01:35.140 --> 01:01:36.120 align:middle line:90% AUDIENCE: [INAUDIBLE]? 01:01:36.120 --> 01:01:37.330 align:middle line:84% BRYNMOR CHAPMAN: Yeah, what's up? 01:01:37.330 --> 01:01:38.640 align:middle line:84% AUDIENCE: What's a normal distribution? 01:01:38.640 --> 01:01:40.980 align:middle line:84% BRYNMOR CHAPMAN: Oh, what's a normal distribution? 01:01:40.980 --> 01:01:44.770 align:middle line:84% Think of a normal distribution as numb er of heads, 01:01:44.770 --> 01:01:46.920 align:middle line:84% if you flip a whole bunch of coins. 01:01:46.920 --> 01:01:51.150 align:middle line:84% That's not quite it, but roughly speaking-- 01:01:51.150 --> 01:01:51.890 align:middle line:90% good catch. 01:01:51.890 --> 01:02:15.365 align:middle line:90% 01:02:15.365 --> 01:02:19.200 align:middle line:84% So as I just said, if you flip a whole bunch of coins, 01:02:19.200 --> 01:02:23.705 align:middle line:84% let your random variable R be the number of heads. 01:02:23.705 --> 01:02:27.540 align:middle line:84% You know a lot more about this random variable 01:02:27.540 --> 01:02:29.870 align:middle line:90% than just its variance. 01:02:29.870 --> 01:02:32.610 align:middle line:84% You can decompose your random variable 01:02:32.610 --> 01:02:37.570 align:middle line:84% into a sum of a whole bunch of indicators. 01:02:37.570 --> 01:02:40.210 align:middle line:84% And all of those indicators are mutually independent. 01:02:40.210 --> 01:02:45.726 align:middle line:90% 01:02:45.726 --> 01:02:49.620 align:middle line:84% In order to use Chebyshev, you don't need mutual independence. 01:02:49.620 --> 01:02:51.100 align:middle line:84% If you had something of that form, 01:02:51.100 --> 01:02:52.680 align:middle line:84% you just need pairwise independence. 01:02:52.680 --> 01:02:54.055 align:middle line:84% You need to compute the variance. 01:02:54.055 --> 01:02:56.475 align:middle line:90% 01:02:56.475 --> 01:02:59.183 align:middle line:84% But if, on top of that, you have mutual independence, 01:02:59.183 --> 01:03:00.725 align:middle line:84% it turns out you can do a lot better. 01:03:00.725 --> 01:03:03.620 align:middle line:90% 01:03:03.620 --> 01:03:32.070 align:middle line:84% So suppose T1, T2, T3, up to Tn are mutually independent 01:03:32.070 --> 01:03:33.930 align:middle line:90% random variables. 01:03:33.930 --> 01:03:38.645 align:middle line:84% And, well, let's say, in addition, they are bounded. 01:03:38.645 --> 01:03:54.162 align:middle line:90% 01:03:54.162 --> 01:03:56.580 align:middle line:84% And the random variable we actually care about 01:03:56.580 --> 01:03:58.580 align:middle line:90% is T, which is their sum. 01:03:58.580 --> 01:04:17.790 align:middle line:90% 01:04:17.790 --> 01:04:20.340 align:middle line:84% Oh, and say we've also got some threshold 01:04:20.340 --> 01:04:24.240 align:middle line:90% C, which is at least 1. 01:04:24.240 --> 01:04:30.986 align:middle line:84% The probability that T is at least C times its expectation-- 01:04:30.986 --> 01:04:36.970 align:middle line:90% 01:04:36.970 --> 01:04:39.480 align:middle line:90% I'll close that as well-- 01:04:39.480 --> 01:04:44.850 align:middle line:84% less than or equal to e to the negative C 01:04:44.850 --> 01:05:04.505 align:middle line:84% log of C minus C plus 1 times the expectation of T. 01:05:04.505 --> 01:05:12.330 align:middle line:84% So a little bit of a mouthful, but what 01:05:12.330 --> 01:05:13.445 align:middle line:90% exactly is going on here? 01:05:13.445 --> 01:05:23.220 align:middle line:90% 01:05:23.220 --> 01:05:29.940 align:middle line:84% So basically, Markov was saying that the probability 01:05:29.940 --> 01:05:34.320 align:middle line:84% that you deviate from the mean by a large amount, 01:05:34.320 --> 01:05:43.200 align:middle line:84% it's at most linear in how far you're deviating. 01:05:43.200 --> 01:05:46.350 align:middle line:84% Chebyshev said it's at most, quadratic. 01:05:46.350 --> 01:05:48.480 align:middle line:84% This is saying it's at most, exponential, 01:05:48.480 --> 01:05:51.510 align:middle line:90% or inverse exponential. 01:05:51.510 --> 01:05:54.258 align:middle line:84% So you have a much stronger condition that gives you 01:05:54.258 --> 01:05:55.175 align:middle line:90% a much stronger bound. 01:05:55.175 --> 01:05:59.480 align:middle line:90% 01:05:59.480 --> 01:06:00.905 align:middle line:90% And how could we prove it? 01:06:00.905 --> 01:06:08.550 align:middle line:90% 01:06:08.550 --> 01:06:09.485 align:middle line:90% Not like this. 01:06:09.485 --> 01:06:54.720 align:middle line:90% 01:06:54.720 --> 01:06:55.625 align:middle line:90% Any ideas? 01:06:55.625 --> 01:07:12.420 align:middle line:90% 01:07:12.420 --> 01:07:14.410 align:middle line:90% Well, maybe a proof sketch-- 01:07:14.410 --> 01:07:16.005 align:middle line:84% we won't go through the entire proof. 01:07:16.005 --> 01:07:25.810 align:middle line:90% 01:07:25.810 --> 01:07:28.780 align:middle line:84% So what tools do we have already? 01:07:28.780 --> 01:07:30.120 align:middle line:90% What might we be able to do? 01:07:30.120 --> 01:08:03.100 align:middle line:90% 01:08:03.100 --> 01:08:07.150 align:middle line:84% Well, what happens if we apply Markov? 01:08:07.150 --> 01:08:08.415 align:middle line:90% How can we apply Markov? 01:08:08.415 --> 01:08:14.230 align:middle line:90% 01:08:14.230 --> 01:08:18.405 align:middle line:84% In order to use Markov, we need a non-negative random variable. 01:08:18.405 --> 01:08:24.069 align:middle line:90% 01:08:24.069 --> 01:08:27.250 align:middle line:84% So in order to prove Chebyshev, we 01:08:27.250 --> 01:08:31.300 align:middle line:84% took a square that made something non-negative. 01:08:31.300 --> 01:08:33.614 align:middle line:84% What might be a reasonable thing to try in this case? 01:08:33.614 --> 01:08:49.870 align:middle line:90% 01:08:49.870 --> 01:08:50.500 align:middle line:90% Yeah? 01:08:50.500 --> 01:08:51.760 align:middle line:90% AUDIENCE: Add 1. 01:08:51.760 --> 01:08:54.250 align:middle line:90% BRYNMOR CHAPMAN: Add 1. 01:08:54.250 --> 01:08:59.429 align:middle line:84% OK, so look at T plus 1 instead of T? 01:08:59.429 --> 01:09:04.840 align:middle line:90% 01:09:04.840 --> 01:09:10.210 align:middle line:84% So that's actually not going to help us. 01:09:10.210 --> 01:09:12.430 align:middle line:84% T is already the sum of a bunch of 01:09:12.430 --> 01:09:14.540 align:middle line:90% non-negative random variables. 01:09:14.540 --> 01:09:16.216 align:middle line:90% So it's already non-negative. 01:09:16.216 --> 01:09:19.359 align:middle line:90% 01:09:19.359 --> 01:09:23.529 align:middle line:84% We're not looking to turn a non-negative-- or sorry, 01:09:23.529 --> 01:09:25.060 align:middle line:84% turn an arbitrary random variable 01:09:25.060 --> 01:09:26.649 align:middle line:90% into a non-negative one. 01:09:26.649 --> 01:09:31.060 align:middle line:84% We are trying to find a more useful non-negative random 01:09:31.060 --> 01:09:33.040 align:middle line:90% variable. 01:09:33.040 --> 01:09:35.859 align:middle line:90% Yeah? 01:09:35.859 --> 01:09:37.007 align:middle line:90% Yeah? 01:09:37.007 --> 01:09:38.165 align:middle line:90% STUDENT: Log of both sides? 01:09:38.165 --> 01:09:40.040 align:middle line:84% BRYNMOR CHAPMAN: Take the log of both sides-- 01:09:40.040 --> 01:09:40.540 align:middle line:90% close. 01:09:40.540 --> 01:10:01.960 align:middle line:90% 01:10:01.960 --> 01:10:03.446 align:middle line:90% What happens if we exponentiate? 01:10:03.446 --> 01:10:06.910 align:middle line:90% 01:10:06.910 --> 01:10:14.360 align:middle line:84% So take T prime equals C to the T. 01:10:14.360 --> 01:10:47.020 align:middle line:90% 01:10:47.020 --> 01:10:48.405 align:middle line:90% What do we know about T prime? 01:10:48.405 --> 01:11:29.740 align:middle line:90% 01:11:29.740 --> 01:11:33.010 align:middle line:84% Well, we're taking something that's at least 1, right? 01:11:33.010 --> 01:11:35.720 align:middle line:84% We're raising it to some positive power-- 01:11:35.720 --> 01:11:37.260 align:middle line:90% well, some non-negative power. 01:11:37.260 --> 01:11:47.844 align:middle line:90% 01:11:47.844 --> 01:11:49.980 align:middle line:84% So T prime is a non-negative random variable. 01:11:49.980 --> 01:11:56.170 align:middle line:90% 01:11:56.170 --> 01:11:57.675 align:middle line:84% Do we know anything more about it? 01:11:57.675 --> 01:12:11.830 align:middle line:90% 01:12:11.830 --> 01:12:12.330 align:middle line:90% Yeah? 01:12:12.330 --> 01:12:13.660 align:middle line:90% AUDIENCE: It's at least 1. 01:12:13.660 --> 01:12:16.750 align:middle line:84% BRYNMOR CHAPMAN: It's at least 1, yeah. 01:12:16.750 --> 01:12:18.750 align:middle line:84% So maybe we try applying Markov to this. 01:12:18.750 --> 01:12:25.010 align:middle line:90% 01:12:25.010 --> 01:12:26.590 align:middle line:84% So now we've got a random variable 01:12:26.590 --> 01:12:30.490 align:middle line:84% that takes values in the range from 0 to C to the n, C 01:12:30.490 --> 01:12:31.305 align:middle line:90% to the n minus 1. 01:12:31.305 --> 01:12:35.690 align:middle line:90% 01:12:35.690 --> 01:12:39.910 align:middle line:84% Now, how could we rewrite this condition, 01:12:39.910 --> 01:12:45.940 align:middle line:84% T greater than or equal to C times expectation of T 01:12:45.940 --> 01:12:47.055 align:middle line:90% in terms of T prime? 01:12:47.055 --> 01:13:12.640 align:middle line:90% 01:13:12.640 --> 01:13:19.270 align:middle line:84% Well, because C is at least 1, exponentiation is monotone. 01:13:19.270 --> 01:13:23.620 align:middle line:84% So this is the same as C to the T greater than or equal to C 01:13:23.620 --> 01:13:54.680 align:middle line:84% to the C times expectation of T. And we don't have too much time. 01:13:54.680 --> 01:13:58.850 align:middle line:84% So I think I will refer you to the book, if you are interested. 01:13:58.850 --> 01:14:07.460 align:middle line:84% But the main idea is this step here. 01:14:07.460 --> 01:14:09.470 align:middle line:84% You take the random variable you care about, 01:14:09.470 --> 01:14:12.260 align:middle line:90% and you exponentiate it. 01:14:12.260 --> 01:14:17.430 align:middle line:84% And then if you apply Markov to that, 01:14:17.430 --> 01:14:19.850 align:middle line:84% it actually gives you a much stronger bound. 01:14:19.850 --> 01:14:23.540 align:middle line:84% If you're using the mutual independence of all 01:14:23.540 --> 01:14:25.950 align:middle line:84% of the component random variables, 01:14:25.950 --> 01:14:28.430 align:middle line:84% this will give you a much stronger bound than even 01:14:28.430 --> 01:14:29.480 align:middle line:90% Chernoff. 01:14:29.480 --> 01:14:33.800 align:middle line:84% So I promised you last week that we 01:14:33.800 --> 01:14:36.570 align:middle line:90% would revisit the coin tosses. 01:14:36.570 --> 01:14:37.735 align:middle line:90% So let's do that briefly. 01:14:37.735 --> 01:14:40.834 align:middle line:90% 01:14:40.834 --> 01:14:53.890 align:middle line:84% So suppose R is the number of heads if you flip n coins. 01:14:53.890 --> 01:15:05.690 align:middle line:90% 01:15:05.690 --> 01:15:10.040 align:middle line:84% So if you recall, we were looking at the probability 01:15:10.040 --> 01:15:17.800 align:middle line:84% that R is equal to n over 4 before. 01:15:17.800 --> 01:15:20.810 align:middle line:90% 01:15:20.810 --> 01:15:27.650 align:middle line:84% Or maybe equivalently, 3n over 4 might be a better way 01:15:27.650 --> 01:15:30.416 align:middle line:90% to analogize. 01:15:30.416 --> 01:15:33.296 align:middle line:90% 01:15:33.296 --> 01:15:37.520 align:middle line:84% And I promised you that we would revisit this probability, 01:15:37.520 --> 01:15:39.775 align:middle line:84% R greater than or equal to 3n over 4. 01:15:39.775 --> 01:15:53.440 align:middle line:90% 01:15:53.440 --> 01:15:58.670 align:middle line:84% Now, what happens if we apply each of our three tail bounds 01:15:58.670 --> 01:15:59.380 align:middle line:90% to R? 01:15:59.380 --> 01:16:11.717 align:middle line:90% 01:16:11.717 --> 01:16:13.300 align:middle line:84% Well, first, what happens with Markov? 01:16:13.300 --> 01:16:21.590 align:middle line:90% 01:16:21.590 --> 01:16:22.790 align:middle line:90% So let's call this p. 01:16:22.790 --> 01:16:29.526 align:middle line:90% 01:16:29.526 --> 01:16:30.775 align:middle line:90% p is at most, what? 01:16:30.775 --> 01:16:54.080 align:middle line:90% 01:16:54.080 --> 01:16:58.640 align:middle line:84% Well, the probability that R is at least some threshold, 01:16:58.640 --> 01:17:06.000 align:middle line:84% it's going to be the expectation of R divided by this threshold. 01:17:06.000 --> 01:17:09.260 align:middle line:90% 01:17:09.260 --> 01:17:11.470 align:middle line:84% What's the expectation of R? n over 2. 01:17:11.470 --> 01:17:19.832 align:middle line:90% 01:17:19.832 --> 01:17:21.280 align:middle line:90% So this is going to be 2/3. 01:17:21.280 --> 01:17:26.390 align:middle line:90% 01:17:26.390 --> 01:17:27.570 align:middle line:90% What about Chebyshev? 01:17:27.570 --> 01:17:43.550 align:middle line:90% 01:17:43.550 --> 01:17:48.890 align:middle line:84% So, similar to what we did with the first Chebyshev example, 01:17:48.890 --> 01:17:52.130 align:middle line:84% p is going to be at most, the probability 01:17:52.130 --> 01:17:57.890 align:middle line:84% that R differs from its expectation in either direction 01:17:57.890 --> 01:17:58.825 align:middle line:90% by another 4. 01:17:58.825 --> 01:18:02.060 align:middle line:90% 01:18:02.060 --> 01:18:05.540 align:middle line:90% So what does that give us? 01:18:05.540 --> 01:18:10.970 align:middle line:84% Probability R minus, let's say, n over 01:18:10.970 --> 01:18:18.500 align:middle line:84% 2 greater than or equal to n over 01:18:18.500 --> 01:18:31.180 align:middle line:84% 4 is at most var of r divided by n over 4 squared. 01:18:31.180 --> 01:18:40.370 align:middle line:90% 01:18:40.370 --> 01:18:42.130 align:middle line:90% So what is the variance of R? 01:18:42.130 --> 01:18:55.868 align:middle line:90% 01:18:55.868 --> 01:18:56.785 align:middle line:90% Does anybody remember? 01:18:56.785 --> 01:19:01.820 align:middle line:90% 01:19:01.820 --> 01:19:04.680 align:middle line:84% We've got a sum of a bunch of indicators. 01:19:04.680 --> 01:19:07.250 align:middle line:84% They're all mutually independent. 01:19:07.250 --> 01:19:10.070 align:middle line:90% So the variance is linear. 01:19:10.070 --> 01:19:12.060 align:middle line:84% What's the variance of a single indicator? 01:19:12.060 --> 01:19:19.250 align:middle line:90% 01:19:19.250 --> 01:19:21.950 align:middle line:90% Nobody? 01:19:21.950 --> 01:19:23.352 align:middle line:90% Yeah? 01:19:23.352 --> 01:19:24.650 align:middle line:90% AUDIENCE: It's n over 4. 01:19:24.650 --> 01:19:26.120 align:middle line:84% BRYNMOR CHAPMAN: Yeah, n over 4-- 01:19:26.120 --> 01:19:28.015 align:middle line:84% well, for R, not for a single one. 01:19:28.015 --> 01:19:31.930 align:middle line:90% 01:19:31.930 --> 01:19:38.060 align:middle line:84% So n over 4 divided by n squared over 16 equals 4 over n. 01:19:38.060 --> 01:19:41.000 align:middle line:90% 01:19:41.000 --> 01:19:45.997 align:middle line:84% So, much better, much better than the 2/3 we just got. 01:19:45.997 --> 01:19:47.330 align:middle line:90% What happens if we use Chernoff? 01:19:47.330 --> 01:20:13.840 align:middle line:90% 01:20:13.840 --> 01:20:15.600 align:middle line:90% Any ideas? 01:20:15.600 --> 01:20:16.380 align:middle line:90% Yeah? 01:20:16.380 --> 01:20:18.780 align:middle line:90% AUDIENCE: Plug in 3/2 for C. 01:20:18.780 --> 01:20:23.725 align:middle line:84% BRYNMOR CHAPMAN: Plug in 3/2 for C. So C equals 3/2. 01:20:23.725 --> 01:20:24.225 align:middle line:90% OK. 01:20:24.225 --> 01:20:33.120 align:middle line:90% 01:20:33.120 --> 01:20:35.240 align:middle line:84% Why don't I put Chernoff back up here. 01:20:35.240 --> 01:20:49.206 align:middle line:90% 01:20:49.206 --> 01:20:50.980 align:middle line:90% and what does that give us? 01:20:50.980 --> 01:21:03.720 align:middle line:84% Probability that R is at least 3/2 times expectation of R is 01:21:03.720 --> 01:21:14.950 align:middle line:84% less than or equal to e to the minus C log C minus C plus 1. 01:21:14.950 --> 01:21:19.620 align:middle line:84% So this is a constant times the expectation 01:21:19.620 --> 01:21:27.660 align:middle line:84% of R. So the expectation of R is n over 2. 01:21:27.660 --> 01:21:29.260 align:middle line:90% This is a constant. 01:21:29.260 --> 01:21:31.380 align:middle line:84% So we've got something that's exponential, 01:21:31.380 --> 01:21:33.020 align:middle line:90% or inverse exponential, in n-- 01:21:33.020 --> 01:21:37.170 align:middle line:90% 01:21:37.170 --> 01:21:45.425 align:middle line:84% oops-- e to the minus order of n. 01:21:45.425 --> 01:21:52.173 align:middle line:90% 01:21:52.173 --> 01:21:59.130 align:middle line:84% So, Markov gave us a really bad bound; Chebyshev, linear-- 01:21:59.130 --> 01:22:00.750 align:middle line:90% pretty reasonable. 01:22:00.750 --> 01:22:03.760 align:middle line:84% Chernoff gives us an exponential. 01:22:03.760 --> 01:22:05.620 align:middle line:84% So that's all we have time for today. 01:22:05.620 --> 01:22:07.530 align:middle line:84% Sorry, I went a little bit over time. 01:22:07.530 --> 01:22:10.470 align:middle line:84% It's been a pleasure teaching you all this semester. 01:22:10.470 --> 01:22:14.190 align:middle line:84% And let's see, we have the rest of our instructors 01:22:14.190 --> 01:22:15.610 align:middle line:90% and at least one TA. 01:22:15.610 --> 01:22:17.640 align:middle line:90% Hello, Lily. 01:22:17.640 --> 01:22:21.420 align:middle line:84% If anything went wrong, it was probably my fault. Anything 01:22:21.420 --> 01:22:24.840 align:middle line:84% that went right, you have your TAs to thank for it, 01:22:24.840 --> 01:22:25.890 align:middle line:90% including Lily. 01:22:25.890 --> 01:22:28.750 align:middle line:90% Lily's amazing. 01:22:28.750 --> 01:22:33.870 align:middle line:84% Perhaps I will see some of you next term in 6.1210. 01:22:33.870 --> 01:22:34.990 align:middle line:90% Good luck on the final. 01:22:34.990 --> 01:22:36.910 align:middle line:84% Make sure you take the right one. 01:22:36.910 --> 01:22:39.160 align:middle line:84% ERIK DEMAINE: Thanks for being students in this class. 01:22:39.160 --> 01:22:40.077 align:middle line:90% It's been lots of fun. 01:22:40.077 --> 01:22:41.560 align:middle line:90% BRYNMOR CHAPMAN: Yeah. 01:22:41.560 --> 01:22:44.610 align:middle line:90% [APPLAUSE] 01:22:44.610 --> 01:22:50.000 align:middle line:90%