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Any questions? It's almost the weakest convergence in distributions. 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. So remark does not necessarily exist. Sum becomes products of e to the t 1 over square root n xi of x mu. But if it's taken over a long time, it won't be a good choice. That will be our first topic. It's no longer mean or variance. And using that, we can prove this statement. In that case, what you can do is-- you want this to be 0.01. In short, I'll just refer to this condition as iid random variables later. It actually happens for some random variables that you encounter in your life. For example, you have Poisson distribution or exponential distributions. Here, I just use a subscript because I wanted to distinguish f of x and x of y. And the reason we are still using mu and sigma is because of this derivation. So it's centered around the origin, and it's symmetrical on the origin. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. » This is one of over 2,200 courses on OCW. I'll group them. Because normal distribution comes up here. That will give the order of magnitude-- I didn't really calculate here, but it looks like it's close to millions. Because when you want to study it, you don't have to consider each moment separately. Full curriculum of exercises and videos. And most games played in the casinos are designed like this. There may be several reasons, but one reason is that it doesn't take into account the order of magnitude of the price itself. OK. Want to be 99% sure that x minus mu is less than 0.1, or x minus 50 is less than 0.1. The moral is, don't play blackjack. So for example, if you're playing blackjack in a casino, when you're playing against the casino, you have a very small disadvantage. Introduction on basic statistics, probability theory and uncertainty modeling in the context of engineering decision making. And that's mu. So it contains all the statistical information of a random variable. PROFESSOR: Probably right. Thank you. *NOTE: Lecture 4 … » y times. ?]. So assumed that the moment-generating functions exists. Let x1 up to xn be independent random variables with identical distribution. It's known to be e to the t square sigma square over 2. All the more or less advanced probability courses are preceded by this one. Even if they have the same moments, it doesn't necessarily imply that they have the same moment-generating function. It's just some technicality, but at least you can see it really fits in. Now we'll do some estimation. I will denote by x sub y. So here, when I write only x, x should only depend on x, not on theta. The expected amount that the casino will win is $0.52. But if you observed how it works, usually that's not normally distributed. Because I'm giving just discrete increments while these are continuous random variables and so on. Lecture 3: Probability Theory. It gets a unified way. These are just some basic stuff. So mu mean over-- that's one of the most universal random variable distributions, the most important one as well. But in practice, if you use a lot more powerful tool of estimating it, it should only be hundreds or at most thousands. We don't offer credit or certification for using OCW. ), Statistik und Wahrscheinlichkeitsrechnung, Wahrscheinlichkeit und Statistik (M. Schweizer), Wahrscheinlichkeitstheorie und Statistik (Probability Theory and Statistics), Eidgenössische Now we go back to the exponential form. Yeah. From the player's point of view, you only have a very small sample. Because all the derivatives, you know what the functions would be. If mean if mu. 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. So if moment-generating function exists, they pretty much classify your random variables. Probability, Information Theory and Bayesian Inference author: Joaquin Quiñonero Candela , Max Planck Institute for Biological Cybernetics, Max Planck Institute published: July 5, … That's the expectation of 1 over n sum of xi minus mu square. Lecture Description This lecture is a review of the probability theory needed for the course, including random variables, probability distributions, and the Central Limit Theorem. The function from the sample space non-negative reals, but now the integration over the domain. OK. Wiss./HST/Humanmed.) That means if you bet $1 at the beginning of each round, the expected amount you'll win is $0.48. Basic Probability Theory and Statistics. PROFESSOR: OK. Maybe-- yeah. 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) That's one reason, but there are several reasons why that's not a good choice. Lecture 3: Independence. So the tool you'll use there is moment-generating functions, something similar to moment-generating functions. So using this formula, we can find probability distribution function of the log normal distribution using the probabilities distribution of normal. It has the tremendous advantage to make feel the reader the essence of probability theory by using extensively random experiences. There is a correcting factor. The only problem is that because-- poker, you're not playing against the casino. But here, that will not be the case. Introduction to Probability Theory. An example of a continuous random variable is if-- let's say, for example, if x of y is equal to 1 for all y and 0,1, then this is pdf of uniform random variable where the space is 0. So we defined random variables. Instead, we want the percentage change to be normally distributed. AUDIENCE: So usually, independent means all the random variables are independent, like x1 is independent with every others. But really, just try to have it stick in your mind that mu and sigma is no longer mean and variance. It has to be close to millions. Let me just make sure that I didn't mess up in the middle. https://www.patreon.com/ProfessorLeonard Statistics Lecture 4.2: Introduction to Probability Parag Radke. So if there's no tendency-- if the average daily increment is 0, then no matter how far you go, your random variable will be normally distributed. OK? But pairwise means x1 and x2 are independent, but x1, x2, and x3, they may not be independent. So that disappears. Discrete Mathematics and Probability Theory. So use the Taylor expansion of this. And I want y to be normal distribution or a normal random variable. But when you look at large scale, you know, at least with very high probability, it has to look like this curve. All positive [INAUDIBLE]. Discrete Mathematics and Probability Theory. For all reals. Yes. y. I want x to be-- yes. Basic tools are introduced for assessing probabilities as needed in risk analysis. So 1 over x sigma squared 2 pi e to the minus log x [INAUDIBLE] squared. A probability mass function is a function from the sample space to non-negative reals such that the sum over all points in the domain equals 1. one 10. It says that it's not necessarily the k-th set. What's normally distributed is the percentage of how much it changes daily. What that means is, this type of statement is not true. It's not just some theoretical thing. Play poker. The first part of the notes gives an introduction to probability theory. Just remember-- these are just parameters, some parameters. That doesn't imply that the mean is e to the sigma. It's not clear why this is so useful, at least from the definition. So for example, assume that you have a normal distribution-- one random variable with normal distribution. And that means as long and they have the slightest advantage, they'll be winning money, and a huge amount of money. So for example, one of the distributions you already saw, it does not have moment-generating function. Now, that n can be multiplied to cancel out. No enrollment or registration. So if we're seeing something uniformly about t, that's no longer true. ... Lecture 20: Central Limit Theorem. OK. And that's when you have to believe in yourself. Here's the proof. Download files for later. And say it was $10 here, and $50 here. 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. So plug in that, plug-in your variance, plug in your epsilon. I don't remember exactly what that is, but I think you're right. Product of-- let me split it better. So probabilistic distributions, that will be of interest to us throughout the course. So this is pretty much just e to that term 1 over 2 t square sigma square over n plus little o of 1 over n to the n square. Yes? The log normal distribution does not have any moment-generating function. Knowledge is your reward. If you have an advantage, if your skill-- if you believe that there is skill in poker-- if your skill is better than the other player by, let's say, 5% chance, then you have an edge over that player. And the spirit here is just really the sequence converges if its moment-generating function converges. And I will talk about moment-generating function a little bit. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. Courses Let's think about our purpose. So now let's look at our purpose. And all of these-- normal, log normal, Poisson, and exponential, and a lot more can be grouped into a family of distributions called exponential family. 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. If y is normally distributed, x will be the distribution that we're interested in. That is the percent. And you multiply this epsilon square. Can be rewritten as 1 over x times 1 over sigma squared 2 pi e to the minus log x square over 2 sigma square plus mu log x over sigma square minus m square. So for each round that the players play, they pay some fee to the casino. So that is equal to sigma square. Those kind of things are what we want to study. Thank you. So now we're talking about large-scale behavior. So all logs are natural log. Any questions about this statement, or any corrections? But from the player's point of view, if you're better than the other player, and the amount of edge you have over the other player is larger than the fee that the casino charges to you, then now you can apply law of large numbers to yourself and win. So the stock-- let's say you have a stock price that goes something like that. OK. That's good. And then because they're independent, this product can go out. So suppose there is a random variable x whose mean we do not know, whose mean is unknown. PROFESSOR: Ah. So you can win money. Is it mu? Lecture 2: Conditioning and Bayes' Rule. Dice play a significant role in our understanding of probability and its relation to the universe. So if you take n to go to infinity, that term disappears, and we prove that it converges to that. And that bi-linearity just becomes the sum of. So they are given by its probability distribution-- discrete random variable is given by its probability mass function. PROFESSOR: Yes. Square. Then f of y of the first-- of x of x is equal to y. h of x. So this random variable just picks one out of the three numbers with equal probability. If the random variable is the same mean and same variance as your original random variable, the distribution of this, should it look like the distribution of xi? c theta is equal to 1 over sigma square 2 pi e to the minus mu square. What did I do wrong? OK. For this special case, will it look like xi, or will it not look like xi? 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. Probability Theory and Applications. I'll make one final remark. That means if you take n to go to infinity, that goes to zero. But that's just some technicality. What happens if for the random variable is 1 over square root n times i? So theorem-- let x1 x2 to xn be iid random variables with mean, this time, mu and variance, sigma squared. OK. To do that-- let me formally write down what I want to say. But if it's a hedge fund, or if you're doing high-frequency trading, that's the moral behind it. I don't remember what's there. Locally, it might be good choice. 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. The law of large numbers. 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. We strongly recommend to not skip it. And then the terms after that, because we're only interested in proving that for fixed t, this converges-- so we're only proving pointwise convergence. So for example, the variance does not have to exist. It doesn't always converge. ... with Applications in Finance » Video Lectures » Lecture … But when it's clear which random variable we're talking about, I'll just say f. So what is this? Yes. OK. Then the law of large numbers says that this will be very close to the mean. 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. Yeah? There are much more powerful estimates that can be done here. And then because each of the xi's are independent, this sum will split into products. Let's see how log normal distribution actually falls into the exponential family. PROFESSOR: Yeah. So proof assuming m of xi exists. The linearity of expectation, 1 comes out. The outcome can be anything according to that distribution. So the question is, what happens if you replace 1 over n by 1 over square root n? Home probability Theory and A course on Descriptive Statistics. Oh, sorry. Full curriculum of exercises and videos. That's too abstract. And how the casino makes money at the poker table is by accumulating those fees. So it's not a good choice. That's not really. And that will be represented by the k-th moments of the random variable. That's an abstract thing. Technische Hochschule Zürich, Eidgenössische Technische Hochschule Zürich. Then the distribution of Yn converges to that of normal distribution with mean 0 and variance sigma. That's equal to the expectation of e to the t over square root n xi minus mu to the n-th power. These are lecture notes written at the University of Zurich during spring 2014 and spring 2015. If you're playing at the optimal strategy, you have-- does anybody know the probability? I will not prove this theorem. You may consider t as a fixed number. So-- sorry about that. That will give you some bound on n. If you have more than that number of trials, you can be 99% sure that you don't deviate from your mean by more than epsilon. AUDIENCE: Because it starts with t, and the right-hand side has nothing general. y is at most log x. But there are some other distributions that you'll also see. This lecture is a review of the probability theory needed for the course, including random variables, probability distributions, and the Central Limit Theorem. Lec : 1; Modules / Lectures. So it's not a very good explanation. So let's do that. OK. That's good. Freely browse and use OCW materials at your own pace. But if you have several independent random variables with the exact same distribution, if the number is super large-- let's say 100 million-- and you plot how many random variables fall into each point into a graph, you'll know that it has to look very close to this curve. It can be anywhere. So remember that theorem. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Other questions? This might not converge. But what I'm trying to say here is that normal distribution is not good enough. Today, we will review probability theory. Meets the expectation-- we didn't use independents yet. Toggle navigation. Modify, remix, and reuse (just remember to cite OCW as the source. So that doesn't give the mean. The tools of probability theory, and of the related field of statistical inference, are the keys for being able to analyze and make sense of data. That's why moment-generating function won't be interesting to us. Before that, let's just rewrite that in a different way. recorded lectures on free probability theory, 26 videos, by Roland Speicher, Saarland University, winter term 2018/19 Oh, sorry. So that's the statement we're going to use. Use OCW to guide your own life-long learning, or to teach others. It doesn't get more complicated as you look at the joint density of many variables, and in fact simplifies to the same exponential family. Lecture Notes | Probability Theory Manuel Cabral Morais Department of Mathematics Instituto Superior T ecnico Lisbon, September 2009/10 | January 2010/11 The question is, what is the distribution of price? 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. So let me get this right. Mathematics as a subject is vast and with these online tutorials, we have tried to segregate some major topics into distinct lectures. 18.650 "Statistics for applications" 6.041 "Probabilistic Systems Analysis and Applied Probability" Topics in Mathematics with Applications in Finance. You can model it like this, but it's not a good choice. They were revised in the allF of 2015 and the schedule on the following page re ects that semester. Before going into that, first of all, why is it called moment-generating function? 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. I want to define a log normal distribution y or log over random variable y such that log of y is normally distributed. You have to believe that you have an edge. So for independence, I will talk about independence of several random variables as well. So as you can see from these two theorems, moment-generating function, if it exists, is a really powerful tool that allows you to control the distribution. And if you look in to Wikipedia, you'll see an example of when it happens, of two random variables where this happens. So we want to study this statistics, whatever that means. ), 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. The statement is not something theoretical. Learn statistics and probability for free—everything you'd want to know about descriptive and inferential statistics. When I first saw it, I thought it was really interesting. So we want to see what the distribution of pn will be in this case. 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 So log normal distribution, it does not converge. f sum x I will denote. So you want to know the probability that you deviate from your mean by more than 0.1. You can use either definition. Massachusetts Institute of Technology. Your mean is 50. That just can be compute. About us; Courses; Contact us; Courses; Mathematics; NOC:Introduction to Probability Theory and Stochastic Processes (Video) Syllabus; Co-ordinated by : IIT Delhi; Available from : 2018-05-02. Description: This lecture is a review of the probability theory needed for the course, including random variables, probability distributions, and the Central Limit Theorem. So this distribution, how it looks like-- I'm sure you saw this bell curve before. It might be e to the mu. This will be less than or equal to the variance of x. Expectation-- probability first. Lectures: MWF 1:00 - 1:59 p.m., Pauley Ballroom 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. In light of this theorem, it should be the case that the distribution of this sequence gets closer and closer to the distribution of this random variable x. These lecture notes were written while teaching the course “Probability 1” at the Hebrew University. Both of the boards don't slide. If two random variables have the same moment, we have the same moment-generating function. So what's an example of this? CS 70 at UC Berkeley. The proof is quite easy. You epsilon is 0.1. Yeah. E to the t 1 over square root n sum of xi times mu. So you will see something about this. And expectation, our mean is expectation of x is equal to the sum over all x, x times that. If x and y have a moment-generating function, and they're the same, then they have the same distribution. Because I believed that I had an edge, but when there is really swing, it looks like your expectation is negative. If you take some graduate probability course, you'll see that there's several possible ways to define convergence. NPTEL provides E-learning through online Web and Video courses various streams. For fixed t, we have to prove it. Their moment-generating function exists. 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. And that's probably one of the reasons that normal distribution is so universal. It factors out well. If you look at a very small scale, it might be OK, because the base price doesn't change that much. So it looks like the mean doesn't matter, because the variance takes over in a very short scale. Courses include recorded auto-graded and peer-reviewed assignments, video lectures, and community discussion forums. Toggle navigation. x4 and x2, x1 is independent with x2, x1 is independent with 3x, x2 is with x3. There are two main things that we're interested in. About us; ... Co-ordinated by : IIT Bombay; Available from : 2012-06-25. » One thing I should mention is, in this case, if each discriminant is normally distributed, then the price at day n will still be a normal random variable distributed like that. That's when your faith in mathematics is being challenged. What does the distribution of price? And also, the condition I gave here is a very strong condition. Then the probability that x for all. Another thing that we will use later, it's a statement very similar to that, but it says something about a sequence of random variables. The problem is typically stated as follows: Suppose you're a contestant on a game show and asked to select one of three doors for your prize. Pointwise convergence implies pointwise convergence. They're not taking chances there. PROFESSOR: I didn't get it. And this part is well known. And the reason it happened was because this had mean mu and variance sigma square over n. We've exploited the fact that variance vanishes to get this. I can be replaced by some other condition, and so on. It might be mu. Wiss./HST/Humanmed. We evaluated that theorem. And if you take an example as poker, it looks like-- OK, I'm not going to play poker. So it's not a good choice. And you at least get the spirit of what's happening. And then you have to figure out what wnt is. The mathematical concepts I need this. Yn be square root n times 1 over n of xi is mu. 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. And to make it formal, to make that information formal, what we can conclude is, for all x, the probability xi is less than or equal to x tends to the probability that at x. OK. And that's one thing you have to be careful. i is from [? When we look at this long-term behavior or large scale of behavior, what can we say? So weak law of large numbers says that if you have IID random variables, 1 over n times sum over x i's converges to mu, the mean in some weak sense. So what they do instead is they take rake. So instead, what we want is a relative difference to be normally distributed. That's just totally nonsense. Lec : 1; Modules / Lectures. So if two random variables, x y, have the same moment-generating function, then x and y have the same distribution. And let v-- or Yn. Normal distribution doesn't make sense, but we can say the price at day n minus the price at day n minus 1 is normal distribution. These are the lecture notes for a year long, PhD level course in Probability Theory that I taught at Stanford University in 2004, 2006 and 2009. Do you see it? Learn more », © 2001–2018 There's no signup, and no start or end dates. International Relations and Security Network, D-BSSE: Lunch Meetings Molecular Systems Engineering, Empirical Process Theory and Applications, Limit Shape Phenomenon in Integrable Models in Statistical Mechanics, Mass und Integral (Measure and Integration), Selected Topics in Life Insurance Mathematics, Statistik I (für Biol./Pharm. Yes? More broadly, the goal of the text You can just think of it as these random variables converge to that random variable. As you might already know, two typical theorems of this type will be in this topic. And in each point t, it converges to the value of the moment-generating function of some other random variable x. But altogether, it's not independent. Then what should it look like? Probability Theory courses from top universities and industry leaders. I hope it doesn't happen to you. Here-- so you can put log of x here. And then you're summing n terms of sigma square. And because normal distribution have very small tails, the tail distributions is really small, we will get really close really fast.

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