That's the glitch. Does anyone have experience with the following, and which one would you recommend? So instead, what we want is a relative difference to be normally distributed. It will be more dense here, sparser there, and sparser there. Let's think about our purpose. https://www.patreon.com/ProfessorLeonard Statistics Lecture 4.2: Introduction to Probability Do you want me to add some explanation? Even if they have the same moments, it doesn't necessarily imply that they have the same moment-generating function. So mu mean over-- that's one of the most universal random variable distributions, the most important one as well. When you complete a course, you’ll be eligible to receive a shareable electronic Course Certificate for a small fee. And then you have to figure out what wnt is. So to derive the problem to distribution of this from the normal distribution, we can use the change of variable formula, which says the following-- suppose x and y are random variables such that probability of x minus x-- for all x. What's really interesting here is, no matter what distribution you had in the beginning, if we average it out in this sense, then you converge to the normal distribution. So when we say that several random variables are independent, it just means whatever collection you take, they're all independent. If x and y have a moment-generating function, and they're the same, then they have the same distribution. So a little bit more fun stuff [? Now let's move on to the next topic-- central limit theorem. We don't really know what the distribution is, but we know that they're all the same. That means if you take n to go to infinity, that goes to zero. That will be our first topic. Courses That doesn't imply that the mean is e to the sigma. That doesn't imply that the variance is something like e to the sigma. All the more or less advanced probability courses are preceded by this one. Now, that n can be multiplied to cancel out. Any questions? Then what should it look like? PROFESSOR: So it's designed so that it factors out when it's multiplied. OK. To do that-- let me formally write down what I want to say. The case when mean is 0. PROFESSOR: Yeah. So for all non-zero t, it does not converge for log normal distribution. OK. Most of the material was compiled from a number of text-books, such that A ﬁrst course in probability by Sheldon Ross, An introduction to probability theory and its applications by William Feller, and Weighing the odds by David Williams. OK. » Our k-th moment is defined as expectation of x to the k. And a good way to study all the moments together in one function is a moment-generating function. Oh, sorry. That's equal to the expectation of e to the t over square root n xi minus mu to the n-th power. Product of-- let me split it better. Yeah. The goal of this courseis to prepareincoming PhDstudents in Stanford’s mathematics and statistics departments to do research in probability theory. However, be very careful when you're applying this theorem. It's no longer centered at mu. So this random variable just picks one out of the three numbers with equal probability. Let me just make sure that I didn't mess up in the middle. Other questions? And that gives a different way of writing a moment-generating function. Probability of an event can be computed as probability of a is equal to either sum of all points in a-- this probability mass function-- or integral over a set a depending on what you're using. But from the casino's point of view, they have enough players to play the game so that the law of large numbers just makes them money. Lecture: TTh 8-9:30am, Zoom ... Lecture 20: Central Limit Theorem. It has to be 0.01. That's when your faith in mathematics is being challenged. Because log x as a normal distribution had mean mu. And expectation of y is the integral over omega. Yeah. And if that is the case, what will be the distribution of the random variable? And then because each of the xi's are independent, this sum will split into products. Even if you have a tiny edge, if you can have enough number of trials, if you can trade enough of times using some strategy that you believe is winning over time, then law of large numbers will take it from there and will bring you money profit. And similarly, you can let t2 equals log x and w2 equals mu over sigma. They're not taking chances there. But it says that moment-generating function, if it exists, encodes really all the information about your random variables. Those kind of things are what we want to study. PROFESSOR: It might not converge. Stat 110: Probability. Any mistakes that I made? So take h of x 1 over x. So that disappears. Basic Probability Theory and Statistics. Be careful. I'll make one final remark. But from the casino's point of view, they're taking a very large end there. So that's good. Let's also define x as the average of n random variables. And in each point t, it converges to the value of the moment-generating function of some other random variable x. Yes? Your c theta will be this term and the last term here, because this doesn't depend on x. I will not talk about it in detail. More broadly, the goal of the text Then a big part of it will be review for you. What that means is, this type of statement is not true. So you will parametrize this family in terms of mu with sigma. So I can take it out and square my square. PROFESSOR: Ah. OK. And one of the most universal random variable, our distribution is a normal distribution. This is one of over 2,200 courses on OCW. Our second topic will be we want to study its long-term our large-scale behavior. You'll see some applications later in central limit theorem. So it's not a very good explanation. Michael Steele's series of ten lectures on Probability Theory and Combinatorial Optimization, delivered in Michigan Technological University in 1995. And the spirit here is just really the sequence converges if its moment-generating function converges. Because moment-generating function is defined in terms of the moments. That's just totally nonsense. You should be familiar with this, but I wrote it down just so that we agree on the notation. Week 1. Download files for later. So a random variable x-- we will talk about discrete and continuous random variables. The law of large numbers. It explains the notion of random events and random variables, probability measures, expectation, distributions, characteristic function, independence of random variables, types of convergence and limit theorems. It's because if you take the k-th derivative of this function, then it actually gives the k-th moment of your random variable. OK. I need this. So this is the same as xi. Video lectures; Captions/transcript; Lecture notes; Course Description. I want to define a log normal distribution y or log over random variable y such that log of y is normally distributed. Because when you take many independent events and take the average in this sense, their distribution converges to a normal distribution. The only problem is that because-- poker, you're not playing against the casino. PROFESSOR: Here? MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. PROFESSOR: Probably right. So 1 over x sigma squared 2 pi e to the minus log x [INAUDIBLE] squared. If two random variables have the same moment, we have the same moment-generating function. And what should happen? They might both not have moment-generating functions. i is from [? The edX course focuses on animations, interactive features, readings, and problem-solving, and is complementary to the Stat 110 lecture videos on YouTube, which are available at https://goo.gl/i7njSb The Stat110x animations are available within the course and at https://goo.gl/g7pqTo If you're seeing this message, it means we're having trouble loading external resources on our website. Is this a sensible definition? The first thing you can try is to use normal distribution. E to the t 1 over square root n sum of xi times mu. If they have the same moment-generating function, they have the same distribution. ?]. This lecture is a review of the probability theory needed for the course, including random variables, probability distributions, and the Central Limit Theorem. Any questions about this statement, or any corrections? Before going into that, first of all, why is it called moment-generating function? Thank you. It's a continuous random variable. Learn statistics and probability for free—everything you'd want to know about descriptive and inferential statistics. Because remark, it does not imply that all random variables with identical k-th moments for all k has the same distribution. That's like the Taylor expansion. And that means as long and they have the slightest advantage, they'll be winning money, and a huge amount of money. It actually happens for some random variables that you encounter in your life. Lecture 3: Independence. Here, I just use a subscript because I wanted to distinguish f of x and x of y. It looks like this if it's n 0 1, let's say. But if it's a hedge fund, or if you're doing high-frequency trading, that's the moral behind it. And how the casino makes money at the poker table is by accumulating those fees. Your h of x here will be 1 over x. Afterwards, I will talk about law of large numbers and central limit theorem. PROFESSOR: I didn't get it. Equal to the product from 1 to n expectation e to the t times square root n. OK. Now they're identically distributed, so you just have to take the n-th power of that. But in practice, if you use a lot more powerful tool of estimating it, it should only be hundreds or at most thousands. Now we'll do some estimation. I'll just write it down. Log x is centered at mu, but when it takes exponential, it becomes skewed. But one good thing is, they exhibit some good statistical behavior, the things-- when you group them into-- all distributions in the exponential family have some nice statistical properties, which makes it good. You can use either definition. The reason that the rule of law of large numbers doesn't apply, at least in this sense, to poker-- can anybody explain why? There are two concepts of independence-- not two, but several. y is at most log x. Independent Identically-distributed random variables. Lecture-01-Basic principles of counting; Lecture-02-Sample space , events, axioms of probability; Lecture-03-Conditional probability, Independence of events. It can be anywhere. What we get is expectation of 1 plus that t over square root n xi minus mu plus 1 over 2 factorial, that squared, t over square root n, xi minus mu squared plus 1 over 3 factorial, that cubed plus so on. So whenever you have identical independent distributions, when you take their average, if you take a large enough number of samples, they will be very close to the mean, which makes sense. It's not clear why this is so useful, at least from the definition. And I want y to be normal distribution or a normal random variable. This looks a little bit controversial to this theorem. You can model it like this, but it's not a good choice. Sl.No Chapter Name MP4 Download; 1: Advanced Probability Theory (Lec 01) Download: 2: Advanced Probability Theory (Lec 02) Download: 3: Advanced Probability Theory (Lec 03) So if moment-generating function exists, they pretty much classify your random variables. So that's the belief you should have. So that's where that becomes very useful. The question is, what is the distribution of price? So all logs are natural log. So first of all, just to agree on terminology, let's review some definitions. Their moment-generating function exists. » Emphasis is given to the aspects of probabilistic model building, hypothesis testing and model verification. A distribution belongs to exponential family if there exists a theta, a vector that parametrizes the distribution such that the probability density function for this choice of parameter theta can be written as h of x times c of theta times the exponent of sum from i equal 1 to k. Yes. And the reason we are still using mu and sigma is because of this derivation. That's an abstract thing. We will mostly just consider mutually independent events. Flash and JavaScript are required for this feature. So assumed that the moment-generating functions exists. Yeah. So use the Taylor expansion of this. So now we're talking about large-scale behavior. The distribution converges. There's only one thing you have to notice-- that the probability that x minus mu is greater than epsilon. Yes. And I will talk about moment-generating function a little bit. So if you take n to be large enough, you will more than likely have some value which is very close to the mean. AUDIENCE: Can you get the mu minus [INAUDIBLE]? Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. And so with this exponential family, if you have random variables from the same exponential family, products of this density function factor out into a very simple form. And let mean be mu, variance be sigma square. And this is known to be sigma square over n. So probability that x minus mu is greater than epsilon is at most sigma square over ne squared. And central limit theorem answers this question. So the normal distribution and log normal distribution will probably be the distributions that you'll see the most throughout the course. Which seems like it doesn't make sense if you look at this theorem. Our continuous random variable has normal distribution, is said to have normal distribution if n mu sigma if the probability distribution function is given as 1 over sigma square root 2 pi e to the minus x minus mu squared. *NOTE: Lecture 4 … Two random variables, which have identical moments-- so all k-th moments are the same for two variables-- even if that's the case, they don't necessarily have to have the same distribution. PROFESSOR: Yes. This picks one out of this. If it doesn't look like xi, can we say anything interesting about the distribution of this? It's because poker, you're playing against other players. I want x to be the log normal distribution. So we don't know what the real value is, but we know that the distribution of the value that we will obtain here is something like that around the mean. So if you model in terms of ratio, our if you model it in an absolute way, it doesn't matter that much. Lectures: MWF 1:00 - 1:59 p.m., Pauley Ballroom Then by using this change of variable formula, probability density function of x is equal to probability density function of y at log x times the differentiation of log x of 1 over x. ), Learn more at Get Started with MIT OpenCourseWare, MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. Try to recall that theorem where if you know that the moment-generating function of Yn's converges to the moment-generating function of the normal, then we have the statement. So what they do instead is they take rake. So let's do that. There are two main things that we're interested in. So you want to know the probability that you deviate from your mean by more than 0.1. This course presents the basics of probability theory and the theory of stochastic processes in discrete time. These are just some basic stuff. You can let w1 of x be log x square t1-- no, t1 of x be log x square, w1 of theta be minus 1 over 2 sigma square. Of course, only if it exists. f sum x I will denote. In short, I'll just refer to this condition as iid random variables later. Here's the proof. You have to be careful. Let's write it like that. What this means-- I'll write it down again-- it means for all x, probability that Yn is less than or equal to x converges the probability that normal distribution is less than or equal to x. Send to friends and colleagues. Do you see it? And this part is well known. Toggle navigation. I assumed it if x-- yeah. And we see that it's e to t-square sigma square over 2 plus the little o of 1. So for example, one of the distributions you already saw, it does not have moment-generating function.