WEBVTT 00:00:05.500 --> 00:00:09.600 Why does going to the airport seem to require extra time compared with coming back from 00:00:09.600 --> 00:00:15.080 the airport even if the traffic is the same in both directions? The answer must somehow 00:00:15.080 --> 00:00:19.900 depend on more than just the average travel time, which we’re assuming is the same and 00:00:19.900 --> 00:00:27.760 often is. In fact, it depends on the distribution of travel times. Probability distributions 00:00:27.769 --> 00:00:32.980 are fully described by listing or graphing every probability. For example, how likely 00:00:32.980 --> 00:00:38.350 is a journey to the airport to be between 10 and 20 minutes? How likely is a 20—30 00:00:38.350 --> 00:00:43.550 minute journey? A 30—40 minute journey? And so on. We’ll answer the airport question 00:00:43.550 --> 00:00:45.590 at the end of the video. 00:00:45.590 --> 00:00:50.379 This video is part of the Probability and Statistics video series. Many natural and 00:00:50.379 --> 00:00:55.960 social phenomena are probabilistic in nature. Engineers, scientists, and policymakers often 00:00:55.960 --> 00:00:59.390 use probability to model and predict system behavior. 00:00:59.390 --> 00:01:04.720 Hi, my name is Sanjoy Mahajan, and I’m a professor of Applied Science and Engineering 00:01:04.720 --> 00:01:10.420 at Olin College. Before watching this video, you should be proficient with integration 00:01:10.420 --> 00:01:14.020 and have some familiarity with probabilities. 00:01:14.020 --> 00:01:16.920 After watching this video, you will be able to: 00:01:16.920 --> 00:01:20.140 Explain what moments of distributions are, and 00:01:20.140 --> 00:01:25.400 Compute moments and understand what they mean 00:01:27.658 --> 00:01:34.658 To illustrate what a probability distribution is, lets consider rolling two fair dice. The 00:01:34.670 --> 00:01:39.549 probability distribution of their sum is this table. For example, the only way to get a 00:01:39.549 --> 00:01:46.329 sum of two is to roll a 1 on each die. And, there are 36 possible rolls for a pair of 00:01:46.329 --> 00:01:53.329 dice. So, getting a sum of two has a probability of 1 over 36. The probability of rolling a 00:01:53.880 --> 00:02:00.880 sum of 3 is 2 over 36. And so on and so forth. You can fill in a table like this yourself. 00:02:01.090 --> 00:02:06.219 But the whole distribution, even for something as simple as two dice, is usually too much 00:02:06.219 --> 00:02:07.860 information. 00:02:07.860 --> 00:02:12.790 We often want to characterize the shape of the distribution using only a few numbers. 00:02:12.790 --> 00:02:19.700 Of course, that throws away information, but throwing away information is the only way 00:02:19.700 --> 00:02:23.200 to fit the complexity of the world into our brains. 00:02:23.200 --> 00:02:29.040 The art comes in keeping the most important information. Finding the moments of a distribution 00:02:29.040 --> 00:02:34.959 can help us reach our goal. Two moments that you are probably already familiar with are 00:02:34.959 --> 00:02:39.690 mean and variance. They are the two most important moments of distributions. 00:02:39.690 --> 00:02:47.150 Let’s define these moments more formally. The mean is the first moment of a distribution. 00:02:47.150 --> 00:02:54.599 It is also called the expected value and is computed as shown. Expected value of x, that’s 00:02:54.599 --> 00:03:00.349 x with angled brackets around it, is equal to this sum. It’s the weighted sum of all 00:03:00.349 --> 00:03:06.069 of the x’s weighted by their probabilities. Let the x sub i be the possible values of 00:03:06.069 --> 00:03:07.569 x. 00:03:07.569 --> 00:03:14.400 For example, for the rolling of two dice, the possible values for x sub i would be 2,3,4 00:03:14.400 --> 00:03:19.220 all the way up through 12. And p sub i would be the corresponding probabilities of rolling 00:03:19.220 --> 00:03:24.540 those sums - so that was 1 over 36, 2 over 36, and so on. 00:03:24.540 --> 00:03:29.840 So, the first moment gives us some idea of what our distribution might look like, but 00:03:29.840 --> 00:03:34.900 not much. Think about it like this, the center of mass in these two images is in the same 00:03:34.900 --> 00:03:39.930 place, but the mass is actually distributed very differently in the two cases. We need 00:03:39.930 --> 00:03:41.409 more information. 00:03:41.409 --> 00:03:46.099 The second moment can help us. The second moment is very similar in structure to the 00:03:46.099 --> 00:03:51.819 first moment. We write it the same way with angled brackets, but now we’re talking about 00:03:51.819 --> 00:03:58.379 the expected value of x squared. So it’s still a sum and it’s still weighted by the 00:03:58.379 --> 00:04:04.340 probabilities p sub i, but now we square each possible x value. For the dice example that 00:04:04.340 --> 00:04:10.920 was the values from two through twelve. This is also called the mean square. First you 00:04:10.920 --> 00:04:16.829 square the x values, then you take the mean, weighting each x sub i by its probability, 00:04:16.829 --> 00:04:17.779 p sub i. 00:04:17.779 --> 00:04:24.780 In general, the nth moment is defined as follows. 00:04:27.590 --> 00:04:32.479 So how does the second moment help us get a better picture of our distribution? Because 00:04:32.479 --> 00:04:38.300 it can help us calculate something called the variance. The variance measures how spread 00:04:38.300 --> 00:04:44.229 out the distribution is around the mean. To calculate the variance, you first subtract 00:04:44.229 --> 00:04:49.710 the mean from each x sub i – this is like finding the distance of each x sub i from 00:04:49.710 --> 00:04:56.710 the mean - and then you square the result and multiply by p sub i. 00:04:59.620 --> 00:05:04.930 What are the dimensions of the variance? The square of the dimensions of x. For example 00:05:04.930 --> 00:05:10.660 if the dimension is a length, then the variance is a length squared. But we often want a measure 00:05:10.660 --> 00:05:16.490 of dispersion like the variance, but one that has the same dimensions as x itself. That 00:05:16.490 --> 00:05:22.320 measure is the standard deviation, sigma. Sigma is defined as the square root of the 00:05:22.320 --> 00:05:27.520 variance. So if the variable x has dimensions of length, then the variance will have dimensions 00:05:27.520 --> 00:05:32.470 of length squared, but the standard deviation, sigma, will have dimensions of length so it’s 00:05:32.470 --> 00:05:35.000 comparable to x directly. 00:05:35.000 --> 00:05:40.350 This expression for the variance looks like a pain to compute, but it has an alternative 00:05:40.350 --> 00:05:45.320 expression that is much simpler. And you get to show that as one of the exercises after 00:05:45.320 --> 00:05:51.490 the video. The alternative expression, the much simpler one, is that the variance is 00:05:51.490 --> 00:05:57.159 equal to the second moment, our old friend, minus the square of the first moment, or the 00:05:57.159 --> 00:05:58.240 mean. 00:05:58.240 --> 00:06:05.240 Pause the video here to convince yourself that this difference is always non-negative. 00:06:09.729 --> 00:06:15.050 This alternative expression for the variance, this much more useful one, is also the parallel 00:06:15.050 --> 00:06:19.990 axis theorem in mechanics, which says that the moment of inertia of an object about the 00:06:19.990 --> 00:06:25.160 center of mass is equal to the moment of inertia about an axis shifted by h from the center 00:06:25.160 --> 00:06:29.860 of mass, a parallel shift, minus mh squared. 00:06:29.860 --> 00:06:36.350 So how does this analogy work? This, the dispersion around the mean, which is here at the center 00:06:36.350 --> 00:06:42.610 of mass, is like the variance. This is like the second moment if we make h equal to the 00:06:42.610 --> 00:06:50.389 mean. So this is the dispersion around zero or its second moment. So this is like x squared, 00:06:50.389 --> 00:06:56.580 the expected value. The mass is the sum total of all the weights here for each of xi which 00:06:56.580 --> 00:07:03.580 all add up to one. So this is just like one in this problem. And then the h squared, well 00:07:03.639 --> 00:07:06.840 h is the mean, so this is x squared. 00:07:06.840 --> 00:07:12.910 So you can see the exact same structure repeated with h, the shift of axis as the mean, and 00:07:12.910 --> 00:07:19.500 m the mass, as the sum of all probabilities which is one. So this formula for the variance 00:07:19.500 --> 00:07:24.080 is also the parallel axis theorem. 00:07:26.900 --> 00:07:32.100 Let’s use the definitions of the moments, and also of the related quantity, the variance, 00:07:32.110 --> 00:07:34.460 and practice on a few distributions. 00:07:34.460 --> 00:07:39.639 A simple discrete distribution is a single coin flip. Instead of thinking of the coin 00:07:39.639 --> 00:07:43.889 flip as resulting in heads or tails, let’s think about the coin as turning up a zero 00:07:43.889 --> 00:07:47.970 or one. Let p be the probability of a one. 00:07:47.970 --> 00:07:53.560 So the mean is the weighted sum of the xi’s, weighted by the probabilities. So the mean 00:07:53.560 --> 00:08:02.340 x is the sum pi xi which is equal to one minus p times zero plus p times one which is equal 00:08:02.349 --> 00:08:03.620 to p. 00:08:03.620 --> 00:08:11.040 What about the second moment? X squared, it’s equal to the weighted sum of the xi’s squared 00:08:11.050 --> 00:08:18.970 so the weights are the same and we can square each value here, the xi’s, but since they’re 00:08:18.970 --> 00:08:24.919 all zero or one, squaring doesn’t change them. So the second moment and the third moment 00:08:24.919 --> 00:08:32.759 and every higher moment are all p. Pause the video here and compute the variance and sketch 00:08:32.760 --> 00:08:35.919 it as a function of p. 00:08:40.690 --> 00:08:44.750 The variance from our old convenient form of the formula is… variance of x is the 00:08:44.750 --> 00:08:49.680 mean squared, mean square minus the squared mean and all the moments themselves were just 00:08:49.680 --> 00:08:56.580 p. So that’s p minus p squared which is equal to p times 1 minus p. 00:08:56.580 --> 00:09:03.339 What does that look like? We sketch it. P on this axis, variance on that axis and the 00:09:03.339 --> 00:09:09.310 curve starts at zero (something I can’t understand) and goes back to zero. 00:09:09.310 --> 00:09:15.450 This is a p equals 1 and that’s p equals zero. Does that make sense? 00:09:15.450 --> 00:09:22.080 Yeah, it does… from the meaning of variance as dispersion around the mean. So take the 00:09:22.080 --> 00:09:27.430 first extreme case of p equals zero. In other words, the coin has no chance of producing 00:09:27.430 --> 00:09:33.730 a one, always produces a zero every time. There the mean is zero and there is no dispersion 00:09:33.730 --> 00:09:40.060 because it always produces zero. The same applies when p equals one here at this extreme. 00:09:40.060 --> 00:09:45.560 The coin always produces a one with no dispersion. There is no variation, there is no variance 00:09:45.560 --> 00:09:52.420 and it’s plausible that the variance should be a maximum right in between… here at p 00:09:52.420 --> 00:09:59.100 equals one half which it is on this curve. So everything looks good. Our calculation 00:09:59.100 --> 00:10:03.540 seems reasonable and checks out in the extreme cases. 00:10:03.540 --> 00:10:07.900 Before we go back to the airport problem, let’s extend the idea of moments to continuous 00:10:07.900 --> 00:10:09.620 distributions. 00:10:09.630 --> 00:10:14.459 Here, instead of a list of probabilities for each possible x, we have a probability density 00:10:14.459 --> 00:10:21.010 p as a function of x, where x is now a continuous variable. That’s the continuous version 00:10:21.010 --> 00:10:27.880 for the nth moment was a sum of xi to the nth weighted by the probabilities. Here, the 00:10:27.880 --> 00:10:34.540 nth moment, x sub n, in equal to instead of a sum, an integral. Weighted again, as always, 00:10:34.540 --> 00:10:42.340 by the probability times x sub n, as before and with a dx because p of x times dx is the 00:10:42.340 --> 00:10:48.389 probability and you add them all up over all possible values of x. That’s the formula 00:10:48.399 --> 00:10:50.769 for a continuous distribution, for the moments of a continuous distribution. 00:10:50.769 --> 00:10:57.170 Let’s practice on the simplest continuous distribution, the uniform distribution. X 00:10:57.170 --> 00:11:03.420 is equally likely to be any real number between zero and one. That’s the distribution and 00:11:03.420 --> 00:11:07.450 we can compute the first and second moments and the variance. 00:11:07.450 --> 00:11:12.700 Pause the video here, use the definition of moments for a continuous distribution and 00:11:12.720 --> 00:11:19.720 compute the mean, first moment, the second moment, and from those two, the variance. 00:11:27.930 --> 00:11:33.240 What you should have found is … for the mean, it’s the integral of one because p 00:11:33.240 --> 00:11:40.980 of x is one, times x between zero and one dx, which is x squared over two evaluated 00:11:40.990 --> 00:11:47.649 between zero and one, which equal one half… which makes sense. The mean here, the average 00:11:47.649 --> 00:11:52.680 value is just one-half right in the middle of the distribution of the possible values 00:11:52.680 --> 00:11:54.279 of x. 00:11:54.279 --> 00:12:02.159 What about the mean square? For that, you should have found almost the same calculation, one 00:12:02.160 --> 00:12:09.519 times x squared dx, which equals x cubed over 3 between zero and one equals one-third. And 00:12:09.519 --> 00:12:15.200 thus, the variance is equal to one-third, that’s the mean square minus the squared 00:12:15.200 --> 00:12:22.720 mean, which is… one twelfth. And that number is familiar. That’s the same 1/12 that shows 00:12:22.730 --> 00:12:28.500 up in the moment of inertia of a ruler of length l and mass m. Its moment of inertia 00:12:28.500 --> 00:12:34.600 is 1/12 ml squared which illustrates again the connection between moments of inertia 00:12:34.600 --> 00:12:36.570 and moments of distributions. 00:12:36.570 --> 00:12:42.589 Let’s apply our knowledge to understand quantitatively, or in a formal way, what happens 00:12:42.589 --> 00:12:47.600 with airport travel – why does it seem so much longer on the way there, than on the 00:12:47.600 --> 00:12:48.370 way back? 00:12:48.370 --> 00:12:53.839 Here is the ideal travel experience to the airport, the distribution of travel times 00:12:53.839 --> 00:12:59.079 t. Here's the probability of each particular travel time, p 00:12:59.079 --> 00:13:06.219 of t. In the ideal world, the travel time would be very predictable. Let’s say it 00:13:06.230 --> 00:13:11.570 would be almost always twenty minutes. In that case, you would allow twenty minutes 00:13:11.570 --> 00:13:16.329 to get to the airport and you would allow twenty minutes on the way back. Going there 00:13:16.329 --> 00:13:18.399 and coming back would seem the same. 00:13:18.399 --> 00:13:24.070 But, here’s what travel to the airport actually looks like. Let’s say the mean is still 00:13:24.070 --> 00:13:30.139 the same, but the reality is that there’s lots of dispersion. And so the curve actually 00:13:30.139 --> 00:13:36.360 looks like that. Sometimes the travel time will be 30 minutes, sometimes 40, sometimes 00:13:36.360 --> 00:13:38.630 10. 00:13:38.630 --> 00:13:44.079 So now, what do you have to do?... this is reality. Well, on the way home, it’s no 00:13:44.079 --> 00:13:48.680 problem. On average, you get home in twenty minutes. You leave whenever you get out of 00:13:48.680 --> 00:13:54.269 the baggage claim. And while it’s true that the trip to the airport follows the same distribution, 00:13:54.269 --> 00:13:59.639 the risk to you of not making it to the airport on time is much greater. If you just allow 00:13:59.639 --> 00:14:03.800 twenty minutes, yeah, sometimes you’ll get lucky, but every once in a while it will take 00:14:03.800 --> 00:14:06.410 you twenty-five or thirty minutes. 00:14:06.410 --> 00:14:09.740 So what you have to do is allow more time on the way there so that you don’t miss 00:14:09.740 --> 00:14:14.540 your flight - maybe thirty minutes, maybe even forty minutes. It all depends on the 00:14:14.540 --> 00:14:19.730 dispersion, or standard deviation, of the distribution. On the way to the airport, you 00:14:19.730 --> 00:14:25.290 are much more aware of the distribution, if you will, than you are on the way back. 00:14:29.400 --> 00:14:34.320 In this video, we saw how to calculate the moments of a distribution and how these moments can 00:14:34.320 --> 00:14:39.000 help us quickly summarize the distribution. Like life... 00:14:39.000 --> 00:14:46.000 when something is complicated, simplify it, grasp it, and understand it by appreciating its moments!