Hi, Welcome back. In this set of lectures, we're talking about randomness and random locks. What I want in these very short lectures is just unpack a little bit of what I mean by randomness. Because randomness means different things to different people, and oftentimes when you see randomness in a model It has different conceptual underpinnings. It means different things to different people. So let's talk bit about what's meant by the concept of randomness. Now to start, we've got to start by talking about a probability distribution. Remember we talked about a probability saying here's our set of possible outcomes. Along here and here's the probability. [sound]. [sound]. They occur, if they are equally likely, like they just were. In the example we saw of the path dependent process, then you're gonna get a distribution that looks like this. But more commonly, we see distributions that are normally distributed and we get this nice bell curve. Right, remember the bell curve when we talked about the [inaudible] theorem. So when we talk about randomness, we're gonna care about what the distribution of that randomness is and also where it comes from. And that randomness can have lots of shapes, alright. It can be uniform, it can be a bell curve, and it can even be a long tail as we saw on that distribution on networks graph. So, lots of different distributions out there Lots of different forms of randomness. Now when we write randomness in models, what we often do is the following concede. We say, instead of just having some value X, there's also a little error term, epsilon. So, there's an X term, which is the value we're concerned with and there's some randomness or error that gets in our way. We want to talk about where that error can come from, were the randomness can come from. In models, and so many models, whether you look at economics, sociology, physics. Biology, engineering, You're always going to see these epsilon terms. They're everywhere. Where they come from, what's the reason for it? Well, one reason is just noise or measurement error. So if you think about measuring the luminosity of a star, let's say. If you look through this telescope and you get some measurement. Well, there's different levels of ambient light, there's different humidity. There's maybe a little bit of dust on the glass. All those things are gonna contribute to creating small deviations in actually getting the two measurements. So instead of getting, if the true luminosity's L, you might get L+epsilon, or L-epsilon. You're gonna be off by a tiny bit. So noise is one reason Just, variation up there in the environment. Second is. Error. So suppose you run an ice cream store and you want people to, you know, dip ice cream scoops so that they weigh exactly four ounces. So each ice cream cone has exactly four ounces of ice cream on it. Well, different people are gonna be dipping, the ice cream's hard and soft. People are gonna make mistakes. So instead of selling exactly four ounces of ice cream in each cone, some people are gonna get big cones, some people are gonna get small cones, and that's just how it's gonna be, 'cause people make mistakes. So error is another reason you see epsilon. Remember we talked about six sigma early on. Well, six sigma is about reducing the size of that error. But even if you [laugh] [inaudible] as much as you can there's still going to be some and that's another cause of randomness. Third cause of randomness is uncertainty. So suppose you undertake a huge project, like building Big Ben, for instance. Well, what you have is you have. An estimate of how much it's gonna cost to build that project. But that's just an estimate. What's actually gonna play out in reality is gonna be different because some materials may cost more than you expected, some may cost less, there may be problems along the way as the project unfolds. And so as a result, instead of it costing what you expect it to be, there's gonna be some uncertainty because the world may change as you're moving along in the process. So, therefore, uncertainty i s another reason why we see randomness in models. Fourth reason is complexity. [sound] Remember how we talked in our models how systems can go to equilibria. They can be periodic. They can be random. Or they can be complex. The thing is, remember we talked about these processes. In which lots of little interacting things. Well those can produce all sorts of interesting patterns that are complex, that you can't really predict, we can't know. So if the world is really complex, we may not know what's gonna happen. So we say, okay, well x is our best bet, and because I don't really know what's happening, I'm going to throw in an error term just because that'll correct for maybe what I get wrong. So I know I'm not going to be exactly right, so there's going to be some error. So complexity and uncertainty can actually get modeled in the same way, it's just a plus epsilon term. Now there's going to be problems with doing that. We'll see those a little bit later in the course, but for the moment. It's not a bad idea to think about just, if you're not sure what's going to happen, whether you're uncertain or the process is complex, by throwing in some sort of error term. And finally, epsilon can be that error term that randomness can occur just through capriciousness. People, you know, people are hard to predict. So if I'm writing a model of people, I don't wanna say, I know what these people are gonna do. Instead, I might say, well, you know, they're probably gonna do this, but who knows. You know, they're people. They're crazy. They might do anything. So we put in a little bit of an error term, All sorts of reasons why things may not go as we expect. There can be noise, there can be error, there can be capriciousness, there can be uncertainty, there can be complexity in the underlying process. So when we think about these models, these random models that we're gonna study, there's all sorts of things that can come into play to make the outcome not be what we expect, but to include little error term. And that error is go nna introduce things like luck, that make it really interesting when you think about why is someone successful, and why is someone not successful? Alright, thank you.