1 00:00:00,421 --> 00:00:03,457 Hi. In this lecture we're going to talk about 2 00:00:03,457 --> 00:00:05,505 a really simple model of aggregation. 3 00:00:05,505 --> 00:00:07,095 So, here's the thing I want to model 4 00:00:07,095 --> 00:00:09,088 I want to model a situation where I've got a group of people 5 00:00:09,088 --> 00:00:10,855 -- it could be 100, it could be 1000 -- 6 00:00:10,855 --> 00:00:12,560 and each one is independently 7 00:00:12,560 --> 00:00:14,448 going to make a decision to do something. 8 00:00:14,448 --> 00:00:16,545 It could be to, y'know, go to the gym. 9 00:00:16,545 --> 00:00:18,232 It could be to go to the beach. 10 00:00:18,232 --> 00:00:20,009 It could be to go to the grocery store. 11 00:00:20,009 --> 00:00:22,007 What I want to try and understand is that 12 00:00:22,007 --> 00:00:22,764 we've got a whole bunch of people 13 00:00:22,764 --> 00:00:24,639 each one that is making these independent decisions 14 00:00:24,639 --> 00:00:27,667 What's the number of people that shows up? 15 00:00:27,667 --> 00:00:29,844 Now, to characterize that I'm going to use an idea 16 00:00:29,844 --> 00:00:31,824 called the probability distribution. 17 00:00:31,824 --> 00:00:34,713 So, to make this simple, let's suppose that there is 18 00:00:34,713 --> 00:00:37,149 a small group of people, like my family, 19 00:00:37,149 --> 00:00:38,080 which has four people in it. 20 00:00:38,080 --> 00:00:40,374 And I want to know "What's the distribution of number 21 00:00:40,374 --> 00:00:44,104 of four people who go for a walk on a given Saturday?" 22 00:00:44,104 --> 00:00:46,409 Well, if I think about the numbers could be -- 23 00:00:46,409 --> 00:00:48,563 there could be 0 people that go, there could be 1, 24 00:00:48,563 --> 00:00:51,150 there could be 2, there could be 3, or it could be 25 00:00:51,150 --> 00:00:53,763 that all 4 of us decide to go for the walk, right? 26 00:00:53,763 --> 00:00:55,287 The dog would prefer if all four of us went, 27 00:00:55,287 --> 00:00:57,249 but, y'know, there's going be some number that goes. 28 00:00:57,249 --> 00:00:59,263 So, I could take -- I could keep track of data. 29 00:00:59,263 --> 00:01:01,139 I could, y'know, chart this on, like, my wall somewhere. 30 00:01:01,139 --> 00:01:03,418 I doubt we could, right? And you can ask, 31 00:01:03,418 --> 00:01:05,475 "What's the likelihood that nobody went for a walk?" 32 00:01:05,475 --> 00:01:07,880 Maybe that's 10%. 33 00:01:07,880 --> 00:01:10,347 Now, what's the likelihood that 1 person went for a walk? 34 00:01:10,347 --> 00:01:12,742 Well that might be 15%. 35 00:01:12,742 --> 00:01:18,101 What about 2 people? That might be 40%. 36 00:01:18,101 --> 00:01:23,014 And what about 3 people? That might also be 15%. 37 00:01:23,014 --> 00:01:25,784 And then, what's the likelihood that 4 of us went for a walk? 38 00:01:25,784 --> 00:01:27,812 That might be, let's say, 20%. 39 00:01:27,812 --> 00:01:30,066 Now the thing to know about a probability distribution is that 40 00:01:30,066 --> 00:01:31,803 each one of these probabilities is less that one, right? 41 00:01:31,803 --> 00:01:35,411 And if we sum them up we get 25 plus 40 is 65 42 00:01:35,411 --> 00:01:37,959 plus 15 is 80; plus 20 is 100. 43 00:01:37,959 --> 00:01:39,729 So we get a total of 100%. 44 00:01:39,729 --> 00:01:41,208 So a probability distribution tells us is 45 00:01:41,208 --> 00:01:42,980 what are the different things that could happen 46 00:01:42,980 --> 00:01:45,155 -- 0,1,2,3 and 4 -- 47 00:01:45,155 --> 00:01:47,655 and then it tells us the likelihood of each of those things. 48 00:01:47,655 --> 00:01:50,993 OK, so here's sorta the huge result that we're going to 49 00:01:50,993 --> 00:01:52,971 leverage to understand how things add up. 50 00:01:52,971 --> 00:01:55,291 There's a theorem called the Central Limit Theorem. 51 00:01:55,291 --> 00:01:57,409 And what the Central Limit Theorem tells us is that 52 00:01:57,409 --> 00:02:01,199 if I add up all the whole bunch of individual, independent events 53 00:02:01,199 --> 00:02:03,213 So what does 'independent' mean? 54 00:02:03,213 --> 00:02:05,401 It means my decision to go to the beach 55 00:02:05,401 --> 00:02:07,389 is independent of your decision to go to the beach, 56 00:02:07,389 --> 00:02:09,518 which is independent of your cousin Mary's 57 00:02:09,518 --> 00:02:10,750 decision to go to the beach. 58 00:02:10,750 --> 00:02:13,533 So, by independent, I mean not influenced. 59 00:02:13,533 --> 00:02:15,371 So, I don't care whether you're going to the beach or not. 60 00:02:15,371 --> 00:02:17,002 I'm going to make my decision on my own, 61 00:02:17,002 --> 00:02:19,326 completely independent of what you decide to do 62 00:02:19,326 --> 00:02:20,972 ... or your cousin Mary. 63 00:02:20,972 --> 00:02:23,268 So, what the Central Limit Theorem tells us is that 64 00:02:23,268 --> 00:02:24,983 if a whole bunch of people make a whole bunch 65 00:02:24,983 --> 00:02:27,394 of independent decisions, the distribution that we get 66 00:02:27,394 --> 00:02:29,382 has this nice bell-shaped curve. 67 00:02:29,382 --> 00:02:31,893 And this bell-shaped curve means that like 68 00:02:31,893 --> 00:02:34,591 the most likely outcome is the one right in the middle. 69 00:02:34,591 --> 00:02:36,755 So, there's a lot of structure to what happens. 70 00:02:36,755 --> 00:02:38,342 And that means that we can predict a lof ot things. 71 00:02:38,342 --> 00:02:40,077 We can tell a lot about what's going on in the world 72 00:02:40,077 --> 00:02:41,793 And that's what we're going to learn about in this lecture. 73 00:02:41,793 --> 00:02:43,015 It's going to be a lot of fun. 74 00:02:43,015 --> 00:02:45,256 To get an understanding of where these distributions 75 00:02:45,256 --> 00:02:48,236 come from, let's start really simple. 76 00:02:48,236 --> 00:02:50,114 Suppose I flip a coin twice. 77 00:02:50,114 --> 00:02:53,257 And I want to know "What are the odds of getting a head?" 78 00:02:53,257 --> 00:02:55,760 What's the probability distribution over heads. 79 00:02:55,760 --> 00:02:57,570 Well, what could I get? 80 00:02:57,570 --> 00:02:59,981 I could get tails-tails, and that would be 0 heads. 81 00:02:59,981 --> 00:03:03,742 I could get tails-heads, or heads-tails 82 00:03:03,742 --> 00:03:07,215 both of these would be 1 head. 83 00:03:07,215 --> 00:03:10,033 Or I could get heads-heads. 84 00:03:10,033 --> 00:03:12,079 And that would be 2 heads. 85 00:03:12,079 --> 00:03:13,650 So, what's the probability of each of these? 86 00:03:13,650 --> 00:03:15,709 The probability of getting tails-tails is just 1/4. 87 00:03:15,709 --> 00:03:18,562 The probability of getting 1 head is 1/2. 88 00:03:18,562 --> 00:03:21,194 And the probability of getting 2 heads is 1/4. 89 00:03:21,194 --> 00:03:22,755 So, I'm going to get a probability distribution, 90 00:03:22,755 --> 00:03:25,009 if I do it out like this 0, 1, 2 91 00:03:25,009 --> 00:03:28,662 There's a 1/4 chance of that and a 1/2 chance of that 92 00:03:28,662 --> 00:03:29,697 and a 1/4 chance of that 93 00:03:29,697 --> 00:03:31,601 You notice, it sorta looks like a little bell curve. 94 00:03:31,601 --> 00:03:34,866 OK. Let's suppose I flip it 4 times. 95 00:03:34,866 --> 00:03:36,446 Well, it gets harder. 96 00:03:36,446 --> 00:03:38,863 I could think, OK, what are the odds of getting no heads? 97 00:03:38,863 --> 00:03:43,085 I could get tails-tails-tails-tails 98 00:03:43,085 --> 00:03:45,496 well, how do I figure out the probability of that? 99 00:03:45,496 --> 00:03:49,539 Well, 1/2 time 1/2 times 1/2 times 1/2 -- 4 one halves -- 100 00:03:49,539 --> 00:03:52,759 that's 2 times 2 times 2 times 2 ... so that's 1/16. 101 00:03:52,759 --> 00:03:55,480 What are the odds of getting one head? 102 00:03:55,480 --> 00:04:00,176 Well, I could get the head first, and then 3 tails... 103 00:04:00,176 --> 00:04:01,650 I could get it second, 104 00:04:01,650 --> 00:04:04,564 I could get it third, 105 00:04:04,564 --> 00:04:07,026 and it could come last. 106 00:04:07,026 --> 00:04:10,759 So, there's four places that it could show up. 107 00:04:10,759 --> 00:04:12,699 So that means there's a 4/16 chance. 108 00:04:12,699 --> 00:04:15,339 Well, I could do all sorts of math again for 109 00:04:15,339 --> 00:04:17,152 what are the odds of getting two heads? 110 00:04:17,152 --> 00:04:19,314 And I'd actually get 6/16. 111 00:04:19,314 --> 00:04:22,557 And 3 heads, well, that's the same as getting 1 head 112 00:04:22,557 --> 00:04:24,631 really, right, because tails and heads are interchangable. 113 00:04:24,631 --> 00:04:27,705 So what I'd get, I'd get this again -- 114 00:04:27,705 --> 00:04:31,234 If I drew this distribution out, I'd get a peak at 2 heads, right? 115 00:04:31,234 --> 00:04:33,110 I'm going to get a nice bell curve, right? 116 00:04:33,110 --> 00:04:34,434 So, I'm going to get this thing where 117 00:04:34,434 --> 00:04:36,372 there's very little chance of getting no heads 118 00:04:36,372 --> 00:04:38,386 not that much chance of getting 4 heads 119 00:04:38,386 --> 00:04:40,816 but the most likely thing is getting 2 heads. 120 00:04:40,816 --> 00:04:42,753 So I can count all this stuff and it's fun... 121 00:04:42,753 --> 00:04:48,949 big data, lots of data, and we want to try 122 00:04:48,949 --> 00:04:52,415 But, here is the problem: 123 00:04:52,415 --> 00:04:53,729 Remember we have talk about it. [inaudible] 124 00:04:53,729 --> 00:04:54,562 and understand it. 125 00:04:54,562 --> 00:04:55,934 Often we have more than 2 or 4, we have 'n' 126 00:04:55,934 --> 00:04:57,228 and that is a huge number. 127 00:04:57,228 --> 00:04:58,656 So if we're talking about New York City that can be 10 million people. 128 00:04:58,656 --> 00:05:00,508 If we're talking about Ann Harbour, where I live, that's still like a hundred thousand people. 129 00:05:00,508 --> 00:05:04,148 So I don't want to be sitting there writing tails, tails, tails, tails, tails a hundred thousand times. 130 00:05:04,148 --> 00:05:07,082 I want to have a model that will help me explain it. 131 00:05:07,082 --> 00:05:15,019 So what you can do is if you have n things, the mean, the expected number should be 132 00:05:15,019 --> 00:05:17,582 N over 2, right, should be half of n. 133 00:05:17,582 --> 00:05:21,774 But what we'd like to do is understand sort of what that distribution looks like. 134 00:05:21,774 --> 00:05:26,372 Well, what we know from statistics is that distribution is actually gonna be a nice bell-curve 135 00:05:26,372 --> 00:05:31,752 and the mean, right in the middle of this thing, is gonna be N/2 and this just gonna 136 00:05:31,752 --> 00:05:34,355 sort of flow out nice and symmetrically from each side 137 00:05:34,355 --> 00:05:38,381 Now there's a fancy equation, a formula that tells you what this line looks like. 138 00:05:38,381 --> 00:05:40,969 We're not going to get into that but if you take the Statistics class 139 00:05:40,969 --> 00:05:43,693 which I'd encourage you do - it's a lot of fun - you could learn exactly 140 00:05:43,693 --> 00:05:45,876 what this formula is and how it works, OK? 141 00:05:45,876 --> 00:05:49,358 We just wanna use it as a model for understanding how things aggregate. 142 00:05:49,358 --> 00:05:52,533 So, we're gonna take some leaps ahead in statistics. 143 00:05:52,533 --> 00:05:56,188 Here's the trick though, we gotta be a little bit careful. 144 00:05:56,188 --> 00:06:00,436 Flipping a coin is always equally likely, it's either a head or a tail, each one is 50/50 145 00:06:00,436 --> 00:06:02,937 But if I'm worried about people going to the beach 146 00:06:02,937 --> 00:06:07,315 right, or people going to the supermarket, or people showing up for their flight 147 00:06:07,315 --> 00:06:10,037 that's not a 50/50 proposition, right. 148 00:06:10,037 --> 00:06:12,521 So maybe 90% of people may show up for their flight 149 00:06:12,521 --> 00:06:15,767 and maybe only 10% of people of 15% go out to the beach. 150 00:06:15,767 --> 00:06:19,101 So I'd like to change that 1/2 into something else 151 00:06:19,101 --> 00:06:21,874 Well, I can introduce something called the binomial distribution 152 00:06:21,874 --> 00:06:26,745 where instead of having 1/2, that gives some probability p of doing the thing. 153 00:06:26,745 --> 00:06:29,847 So let's suppose going to the beach happens 15% of the time 154 00:06:29,847 --> 00:06:39,373 Well then, if I had a 1000 people, and p = 15%, then p times N is a 150 155 00:06:39,373 --> 00:06:42,156 so I expect to have a 150 show up. 156 00:06:42,156 --> 00:06:45,130 So that makes sense but then I can ask well, what's the distribution now 157 00:06:45,130 --> 00:06:49,778 I mean 150 is the average but I could have 200, I could have 74 158 00:06:49,778 --> 00:06:53,435 Well again, what the central limit theorem tells us is that we're 159 00:06:53,435 --> 00:06:56,133 gonna get a nice bell-curve, right here you got this nice shape here 160 00:06:56,133 --> 00:06:59,635 [inaudible] with the mean here, which is p times N 161 00:06:59,635 --> 00:07:01,442 well this will be provided if N is big enough, right 162 00:07:01,442 --> 00:07:06,575 but if I can get a pretty large N, you're gonna get this nice bell-curve and the mean is gonna be right at p times N 163 00:07:06,575 --> 00:07:14,134 Okay, there's more of that. Here's where it gets a little bit complicated but also interesting. 164 00:07:14,134 --> 00:07:20,013 There's something called the standard deviation and this is this thing called sigma 165 00:07:20,013 --> 00:07:24,789 which is [inaudible] called the standard deviation. Now, when I draw the normal curve, 166 00:07:24,789 --> 00:07:28,745 there's gonna be a mean, that's this point right here at the center 167 00:07:28,745 --> 00:07:34,255 And then there's gonna be a standard deviation which basically tells us how far spread out 168 00:07:34,255 --> 00:07:35,830 that curve is 169 00:07:35,830 --> 00:07:39,862 And what I mean by that is how far spread out the different outcomes are 170 00:07:39,862 --> 00:07:44,109 So it turns out there's this nice structure to any normal distribution. If you tell me the mean, 171 00:07:44,109 --> 00:07:46,340 and then you tell me the standard deviation, 172 00:07:46,340 --> 00:07:49,997 it's always gonna be the case that 68% 173 00:07:49,997 --> 00:07:56,832 of all outcomes will be between -1 and +1 standard deviation 174 00:07:56,832 --> 00:07:59,413 So, if it's got a big standard deviation 175 00:07:59,413 --> 00:08:00,753 That means that that range could be really wide 176 00:08:00,753 --> 00:08:04,024 If it's got a small standard deviation that means that range should be really tight but 177 00:08:04,024 --> 00:08:06,263 if you tell me the mean and tell me the standard deviation 178 00:08:06,263 --> 00:08:12,136 it's always gonna be the case that 68% of the time I'm between -1 and +1 standard deviation 179 00:08:12,136 --> 00:08:19,187 Now, in fact since that's true for one, it's also gonna be true for 2, true for 3 and true for 4, right 180 00:08:19,187 --> 00:08:24,635 So there's gonna be a 95% chance I'm within 2 standard deviations 181 00:08:24,635 --> 00:08:29,234 So wait, why do we care about this, why do we care about this stuff 182 00:08:29,234 --> 00:08:33,763 Here's why. Now I got this model that says if I add up a bunch of independent events, 183 00:08:33,763 --> 00:08:37,809 here's what the mean is, right. 184 00:08:37,809 --> 00:08:39,962 Now, in a second, I'm going to show you the formula for the standard deviation 185 00:08:39,962 --> 00:08:42,604 so then it'll tell you what sigma is here 186 00:08:42,604 --> 00:08:44,935 Well, if you know the mean and you know sigma, 187 00:08:44,935 --> 00:08:50,210 then I can give you range and I can tell you, you know, that 95% of the time 188 00:08:50,210 --> 00:08:54,176 I'm gonna be between -2 sigma and +2 sigma 189 00:08:54,176 --> 00:08:57,060 So if I said the mean number of people that showed up is a hundred 190 00:08:57,060 --> 00:09:02,812 and that's the mean, right, and the standard deviation is only 2 191 00:09:02,812 --> 00:09:09,012 Well then you'd know 95% of the time, you're gonna be between 96 and 104 192 00:09:09,012 --> 00:09:12,786 So you'd know, okay, I should prepare for pretty much exactly 100 people 193 00:09:12,786 --> 00:09:16,098 If I told you the standard deviation was 15, 194 00:09:16,098 --> 00:09:19,575 then you'd know it can be anywhere between 70 to130 195 00:09:19,575 --> 00:09:22,313 So that's what we want to try and use this model to explain [inaudible] 196 00:09:22,313 --> 00:09:26,727 how wide a range of outcomes we're likely to see in any particular setting 197 00:09:26,727 --> 00:09:31,030 So let's go back to our simple binomial distribution where the probability was 1/2 198 00:09:31,030 --> 00:09:33,733 The mean, remember, is just N over 2 199 00:09:33,733 --> 00:09:38,720 the standard deviation is the square root of N over 2 200 00:09:38,720 --> 00:09:41,132 Well, you can do a little bit of math and show that 201 00:09:41,132 --> 00:09:43,496 So, let's suppose I have N = 100 202 00:09:43,496 --> 00:09:47,593 So if N = 100, that tells me the mean is gonna be 50 so if I flip a coin a hundred times, 203 00:09:47,593 --> 00:09:51,387 guess what, the average is 50, no surprise 204 00:09:51,387 --> 00:09:54,202 But, standard deviation is the square root of N/2 205 00:09:54,202 --> 00:09:56,047 What's the square root of 100, that's 10 206 00:09:56,047 --> 00:09:59,032 So this is 10 over 2 so that gives 5 207 00:09:59,032 --> 00:10:02,359 So what that tells me is if I think in binomial distribution 208 00:10:02,359 --> 00:10:03,758 right, if I draw this thing out 209 00:10:03,758 --> 00:10:07,783 I've got a mean of 50, and then I've got a standard deviation of 5 210 00:10:07,783 --> 00:10:12,827 so that means between 55 and 45 -- 68% of all outcomes 211 00:10:12,827 --> 00:10:16,385 So, if you want, you can do this at home -- it'll take a while -- flip a coin a hundred times 212 00:10:16,385 --> 00:10:19,949 Count how many heads you get. Flip it again, count how many heads again 213 00:10:19,949 --> 00:10:26,069 Do that a whole bunch of times, you'll find that 68% of the time, you get between 45 heads and 55 heads 214 00:10:26,069 --> 00:10:30,945 So what this model gives us is it gives a sense of how strange of outcomes we'll get 215 00:10:30,945 --> 00:10:37,277 So, we know that most of the time, 68% of the time, we'll be between 45 and 55, right 216 00:10:37,277 --> 00:10:46,657 So, our mean is 50, 1 standard deviation is 55 and 45, that means 2 standard deviations is 60 and 40 217 00:10:46,657 --> 00:10:53,316 What that tells us is 95% of the time, you're gonna be between 40 and 60 heads 218 00:10:53,316 --> 00:11:00,059 And, 99% of the time, you're gonna be between 35 and 65 219 00:11:00,059 --> 00:11:07,449 So basically it's, you're almost never gonna throw fewer than 35 heads, and never throw more than 65 heads 220 00:11:07,449 --> 00:11:10,299 And so this is what sort of power that Central Limit Theorem is, right 221 00:11:10,299 --> 00:11:14,671 It gives us a sense of not only the average, but also what the spread will be 222 00:11:14,671 --> 00:11:18,774 Okay, remember this is a simple case. This is the P = 1/2 case 223 00:11:18,774 --> 00:11:23,656 And what we'd like is we want it for the more general case where the probability of something happening can be anything 224 00:11:23,656 --> 00:11:24,989 Right, this is this p over N thing 225 00:11:24,989 --> 00:11:33,866 What turns out here, we're okay because the standard deviation is just p times 1 - p times N then square root the whole thing 226 00:11:33,866 --> 00:11:40,828 So the case where p = 1/2 right, then we have the square root of 1/2 times 1/2 times N 227 00:11:40,828 --> 00:11:47,334 But notice I've got a 1/2 squared here inside so we can just pull that outside so it's just 1/2 the square root of N 228 00:11:47,334 --> 00:11:50,195 so that's where that square root of N over 2 came from 229 00:11:50,195 --> 00:11:53,959 So now, for the binomial distribution, I've got this clean formula as well 230 00:11:53,959 --> 00:12:00,476 And we can use that to model and understand stuff that's a little bit more interesting than just flipping a coin 231 00:12:00,476 --> 00:12:03,647 Let's a real example, let's have some fun 232 00:12:03,647 --> 00:12:07,486 So, how, most of us have probably been bumped off a plane before 233 00:12:07,486 --> 00:12:11,573 You show up at the airport and there's like too many people showed up for the plane 234 00:12:11,573 --> 00:12:15,438 And you think why did they do this, but the reason that they sometimes have to [inaudible] is they oversell 235 00:12:15,438 --> 00:12:19,755 And the reason they oversell tickets is because not everybody shows up 236 00:12:19,755 --> 00:12:26,450 So if you're running an airline and you've got 400 seats, and you know people show up, you know, 90% of the time 237 00:12:26,450 --> 00:12:32,659 You want to sell more than those 400 seats, right, so that your plane is pretty much full 238 00:12:32,659 --> 00:12:37,754 So let's do an example. Let's suppose, make it simple, that our plane [inaudible] got 380 seats 239 00:12:37,754 --> 00:12:41,724 So let's suppose we got a Boeing 747 with 380 seats 240 00:12:41,724 --> 00:12:45,047 Let's suppose that 90% of the time, people show up 241 00:12:45,047 --> 00:12:47,969 So we've gathered, we run an airline, we've gathered lots of data 242 00:12:47,969 --> 00:12:52,549 We pretty much know 90% of the time, people show up and that it's independent 243 00:12:52,549 --> 00:12:56,045 So one person's decision to show up doesn't [inaudible] have anything to do with anybody else's 244 00:12:56,045 --> 00:13:00,661 Now, that might not be true, right. Because if it's snowy, if I'm late, you're likely to be late 245 00:13:00,661 --> 00:13:06,557 But let's just suppose that these things are independent. And let's suppose that we sell 400 tickets 246 00:13:06,557 --> 00:13:10,652 Now we're trying to get some understanding why, what is that mean 247 00:13:10,652 --> 00:13:15,086 What's the likelihood that if we sell 400, that we're going to have more than 380 people show up 248 00:13:15,086 --> 00:13:21,020 Here's where the model can help us. It'll be able to tell us what the mean is, it will also tell us what the standard deviation is 249 00:13:21,020 --> 00:13:31,060 So the mean, right if I sell 400 tickets, and on average 90% of people show up, that means I should sell on average 360 tickets 250 00:13:31,060 --> 00:13:39,092 That's less than 380 seats but it should be fine, but what I care about is more than 380 people show up 251 00:13:39,092 --> 00:13:40,773 'cause they're gonna be like, I paid for this to go to Florida, I want to go to Florida, I don't want to be bumped 252 00:13:40,773 --> 00:13:42,394 So more than 380 show up, guess what, they're gonna be mad, right 253 00:13:42,394 --> 00:13:51,255 So the 360 doesn't tell us enough, we want to know something about the distribution 254 00:13:51,255 --> 00:13:54,627 Okay, well look, we've got a formula, right, remember 255 00:13:54,627 --> 00:14:02,298 So N was 400, and p was .9 so p times N is 360, that's our mean 256 00:14:02,298 --> 00:14:06,729 Now, the standard deviation we can solve for pretty easily. That's just the square root 257 00:14:06,729 --> 00:14:13,058 of p, which is .9 times 1-p, which is .1, times N which is 400 258 00:14:13,058 --> 00:14:16,209 So if we multiply right out that's .9 times .1 times 400 259 00:14:16,209 --> 00:14:23,717 .1 times 400 is 40, times .9 is 36, that gives the squared of 36, which is 6 260 00:14:23,717 --> 00:14:33,180 So 6, is our standard deviation. Now I get a bell-curve with a mean of 360 and a standard deviation of 6 261 00:14:33,180 --> 00:14:38,287 Well, that's useful, that can help us 'cause let's go back and let's look 262 00:14:38,287 --> 00:14:49,435 That means our mean's 360, our standard deviation is 6, so that means 68% of the time, we're gonna between 354 and 366. That's great. 263 00:14:49,435 --> 00:14:56,525 It means that 95% of the time, we'll between 348 and 372, also great 264 00:14:56,525 --> 00:15:03,356 It means 99.75% of the time, we'll be between 378 and 342. 265 00:15:03,356 --> 00:15:12,997 Well, how many seats do we have, we have 380 seats, so this means that 99.75 -- actually more than that, right 266 00:15:12,997 --> 00:15:16,390 More than 99.75% of the time we won't overbook. 267 00:15:16,390 --> 00:15:19,903 So here's the Central Limit Theorem, let's, let's say it formally. 268 00:15:19,903 --> 00:15:23,475 Central Limit Theorem [inaudible] is the following. We got a whole bunch of random variables 269 00:15:23,475 --> 00:15:28,237 so those could be decisions to show up to a flight or not so in most case the random variables are just 1s and 0s 270 00:15:28,237 --> 00:15:34,005 Or they could be, you know, the weight of your bag. Each person's weight of their bag is [inaudible] independent variable. 271 00:15:34,005 --> 00:15:43,194 As long as those things are independent, so that means it, each person's decision doesn't depend on somebody else's or how much stuff I jam in my bag doesn't affect how much stuff you jam in your bag 272 00:15:43,194 --> 00:15:49,888 And that those things have finite variance -- what does that mean -- that means that they're bounded 273 00:15:49,888 --> 00:15:56,068 So we know we can't have super huge values, like so my bag couldn't weigh billions and billions of pounds 274 00:15:56,068 --> 00:16:01,609 So long as there's sort of you know, the possible range of [inaudible] that each one can take is bounded in some way 275 00:16:01,609 --> 00:16:06,209 Or doesn't with some high probability take huge, huge values then when you add those things up 276 00:16:06,209 --> 00:16:12,677 When you sum them up, you're gonna get a normal distribution which means a bell-curve, which means we can predict stuff 277 00:16:12,677 --> 00:16:16,255 We can use that model [inaudible] make sense of how the world works 278 00:16:16,255 --> 00:16:22,506 Now, let's step back for just a second and think about like, why this is so cool 279 00:16:22,506 --> 00:16:25,838 Suppose it weren't true, here's a little thought experiment 280 00:16:25,838 --> 00:16:28,394 Suppose it were the case that when I added up a bunch of independent events 281 00:16:28,394 --> 00:16:35,767 Most of the time, I get something nice, then there were some spiky probability of some huge event 282 00:16:35,767 --> 00:16:39,928 over here. What would this mean. Well, this would mean like sometimes 283 00:16:39,928 --> 00:16:44,266 you go to the grocery store, and there would be like, 1000 people there 284 00:16:44,266 --> 00:16:48,626 or sometimes you'd be like I'm just gonna run to the bathroom and there'd be 300 people in line, right 285 00:16:48,626 --> 00:16:53,527 A lot of the predictability of the world, a lot of the predictability of these sort of daily comings and goings 286 00:16:53,527 --> 00:16:57,841 stem from the fact that this can't happen and that we get these nice bell-curves 287 00:16:57,841 --> 00:17:02,390 Because if individual people, individual firms, individual groups of people 288 00:17:02,390 --> 00:17:05,409 make decisions that don't depend on what other people decide 289 00:17:05,409 --> 00:17:08,229 [inaudible] independent decisions, then what you're gonna get is 290 00:17:08,229 --> 00:17:11,419 you're gonna get sort of nice, regular stuff, according to a bell-curve 291 00:17:11,419 --> 00:17:13,845 Yeah, sure there'll be traffic jams, sure there'll be a lot of people at the mall 292 00:17:13,845 --> 00:17:17,518 There will be days where you get a lot coming on, and there'll be days where nothing much is going on 293 00:17:17,518 --> 00:17:22,627 But most of the time, you're gonna get things in that little region which is gonna be predictable and understandable 294 00:17:22,627 --> 00:17:26,350 Now, is everything normally distributed? No, it's not. 295 00:17:26,350 --> 00:17:29,873 What about stock returns? If you look at stock returns, you'll actually see that 296 00:17:29,873 --> 00:17:35,783 there's far too many days where really nothing happens, and there's far too many days where there's huge gains 297 00:17:35,783 --> 00:17:37,826 and far too many days where there's huge losses 298 00:17:37,826 --> 00:17:41,836 And what's going on there is this is that the actions are no longer independent 299 00:17:41,836 --> 00:17:46,387 For example, prices are going up, a lot of people may buy and that's going to cause prices to go up even further 300 00:17:46,387 --> 00:17:50,956 And if prices start to fall, people may sell and that can cause prices to fall even further 301 00:17:50,956 --> 00:17:55,168 So when events fail to become independent, fail to satisfy independence's assumption, 302 00:17:55,168 --> 00:18:00,400 then we can get more big events than we'd expected and more small events that we'd expected 303 00:18:00,400 --> 00:18:03,661 So let's put a [inaudible] on this, what have we got 304 00:18:03,661 --> 00:18:07,229 If we use the Central Limit Theorem as a model, and we use this model to explain 305 00:18:07,229 --> 00:18:10,531 how if we add up a bunch of independent events, then what we get is 306 00:18:10,531 --> 00:18:12,861 we get a nice normal distribution, right 307 00:18:12,861 --> 00:18:15,891 And we can understand the mean, we can understand the standard deviation 308 00:18:15,891 --> 00:18:19,901 We can use that to predict how likely things are to occur, right 309 00:18:19,901 --> 00:18:24,023 We also learn that like, it's that independence that gives us that normality, right 310 00:18:24,023 --> 00:18:27,236 Without independence, we could get really big events, really small events 311 00:18:27,236 --> 00:18:29,273 We can get all sorts of strange stuff happening 312 00:18:29,273 --> 00:18:32,810 So where we're gonna go next, I'm gonna get take, there's a brief lecture on something called 313 00:18:32,810 --> 00:18:36,278 the Six Sigma that pushes this idea sort of, the predictability of the system a little bit 314 00:18:36,278 --> 00:18:39,348 further than we had before. But then after that, we're gonna start 315 00:18:39,348 --> 00:18:41,717 y'know, we're [inaudible] 316 00:18:41,717 --> 00:18:44,359 having systems where there's interdependent actions 317 00:18:44,359 --> 00:18:48,046 and we have those interdependent actions, we're no longer going to get these sort of nice bell-curves 318 00:18:48,046 --> 00:18:50,133 We're going to get all sorts of really interesting, strange stuff 319 00:18:50,133 --> 00:18:52,300 It's going to be a lot of fun. Alright, thank you.