And because normal distribution have very small tails, the tail distributions is really small, we will get really close really fast. So the stock-- let's say you have a stock price that goes something like that. And that's probably one of the reasons that normal distribution is so universal. If mean if mu. The moment-generating function of a random variable is defined as-- I write it as m sub x. NPTEL provides E-learning through online Web and Video courses various streams. Some very interesting facts arise from this fact. So we want to somehow show that the moment-generating function of this Yn converges to that. Even if they have the same moments, it doesn't necessarily imply that they have the same moment-generating function. Dice and the Theory of Probability. So it looks like the mean doesn't matter, because the variance takes over in a very short scale. And that's happening because we're fixed. Then our probability mass function is fx 1 equals fx minus 1 1/3, just like that. Courses Here, I just use a subscript because I wanted to distinguish f of x and x of y. 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. 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. Lecture-01-Basic principles of counting; Lecture-02-Sample space , events, axioms of probability; Lecture-03-Conditional probability, Independence of events. Statistics Lecture 4.2: Introduction to Probability So for independence, I will talk about independence of several random variables as well. 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. If it doesn't look like xi, can we say anything interesting about the distribution of this? » There is a correcting factor. Emphasis is given to the aspects of probabilistic model building, hypothesis testing and model verification. Video Lectures Course Home Syllabus ... Lecture 1: Probability Models and Axioms. Today, we will review probability theory. So there are that say 1, a2, a3, for which this does not hold. The law of large numbers. So let's start with our first topic-- the moment-generating function. So as n goes to infinity-- if n is really, really large, all these terms will be smaller order of magnitude than n, 1 over n. Something like that happens. But you're going to talk about some distribution for an exponential family, right? Do you see it? If you look at a very small scale, it might be OK, because the base price doesn't change that much. There are two main things that we're interested in. 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. You have to believe that you have an edge. Our second topic will be we want to study its long-term our large-scale behavior. If that's the case, x is e to the mu will be the mean. Discrete Mathematics and Probability Theory. You can just think of it as these random variables converge to that random variable. So when we say that several random variables are independent, it just means whatever collection you take, they're all independent. I will not prove this theorem. 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. So if you model in terms of ratio, our if you model it in an absolute way, it doesn't matter that much. Be careful. Lecture 3: Probability Theory. The case when mean is 0. recorded lectures on free probability theory, 26 videos, by Roland Speicher, Saarland University, winter term 2018/19 And central limit theorem answers this question. Any questions about this statement, or any corrections? No, no. So it's just a technical issue. In that case, what you can do is-- you want this to be 0.01. Probability Theory courses from top universities and industry leaders. For all reals. But if you want to do it a little bit more like our scale, then that's not a very good choice. Lec : 1; Modules / Lectures. I hope it doesn't happen to you. This picks one out of this. It has the tremendous advantage to make feel the reader the essence of probability theory by using extensively random experiences. So take h of x 1 over x. So-- sorry about that. So that doesn't give the mean. Stat 110: Probability. So now we're talking about large-scale behavior. Anyway, that's proof of there's numbers. And you multiply this epsilon square. So you will see something about this. In short, I'll just refer to this condition as iid random variables later. It might be mu. So remember that theorem. OK. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. So this moment-generating function encodes all the k-th moments of a random variable. So we want to study this statistics, whatever that means. And the reason is because-- one reason is because the moment-generating function might not exist. So assumed that the moment-generating functions exists. So instead, what we want is a relative difference to be normally distributed. So probabilistic distributions, that will be of interest to us throughout the course. So it's centered around the origin, and it's symmetrical on the origin. That just can be compute. These are just some basic stuff. I will prove it when the moment-generating function exists. Modify, remix, and reuse (just remember to cite OCW as the source. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at Selected Topics in Probability FS 2020; Statistik I (für Biol./Pharm. Video Lectures It will be more dense here, sparser there, and sparser there. 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. If X takes 1 with probability 1/3 minus 1 of probability 1/3 and 0 with probability 1/3. But I will not go into it. So proof assuming m of xi exists. 1 over n is inside the square. Product of-- let me split it better. So that's good. Independent Identically-distributed random variables. I don't remember what's there. And that will actually show some very interesting thing I will later explain. And we see that it's e to t-square sigma square over 2 plus the little o of 1. So this random variable just picks one out of the three numbers with equal probability. PROFESSOR: Ah. So it's not a good choice. In that theorem, your conclusion is stronger. But it's designed so that the variance is so big that this expectation is hidden, the mean is hidden. t can be any real. I assumed it if x-- yeah. That's equal to the expectation of e to the t over square root n xi minus mu to the n-th power. OK. 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. Lecture: TTh 8-9:30am, Zoom Sum becomes products of e to the t 1 over square root n xi of x mu. The expected amount that the casino will win is $0.52. Learn statistics and probability for free—everything you'd want to know about descriptive and inferential statistics. Now let's move on to the next topic-- central limit theorem. So just remember that even if they have the same moments, they don't necessarily have the same distribution. So I will mostly focus on-- I'll give you some distributions. I want to define a log normal distribution y or log over random variable y such that log of y is normally distributed. In our video lectures, we are going to talk about finite mathematics, differential equations, statistical probability, Laplace transforms, Fourier series and more. The first thing you can try is to use normal distribution. *NOTE: Lecture 4 … It gets a unified way. When I write some function of theta, it should only depend on theta, not on x. So plug in that, plug-in your variance, plug in your epsilon. 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. 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. You'll see some applications later in central limit theorem. So that is equal to sigma square. We don't really know what the distribution is, but we know that they're all the same. OK. There's no signup, and no start or end dates. 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. Week 1. I will denote by x sub y. If you're seeing this message, it means we're having trouble loading external resources on our website. Want to be 99% sure that x minus mu is less than 0.1, or x minus 50 is less than 0.1. That's when your faith in mathematics is being challenged. I need this. ), 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. Yeah? You may consider t as a fixed number. So it's not a good choice. Does anyone have experience with the following, and which one would you recommend? 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 it becomes 1 over x sigma square root 2 pi 8 to the minus log x minus mu squared. And that gives a different way of writing a moment-generating function. It's a continuous random variable. But when it's clear which random variable we're talking about, I'll just say f. So what is this? These tools underlie important advances in many … Thank you very much. The log normal distribution does not have any moment-generating function. I don't remember exactly what that is, but I think you're right. You epsilon is 0.1. These are some other distributions that you'll see. OK. For this special case, will it look like xi, or will it not look like xi? MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Because when you want to study it, you don't have to consider each moment separately. So for example, one of the distributions you already saw, it does not have moment-generating function. All positive [INAUDIBLE]. But when you look at large scale, you know, at least with very high probability, it has to look like this curve. These lecture notes were written while teaching the course “Probability 1” at the Hebrew University. And one of the most universal random variable, our distribution is a normal distribution. So all logs are natural log. We will mostly just consider mutually independent events. So it's pretty much safe to consider our sample space to be the real numbers for continuous random variables. Of course, only if it exists. Yeah. Any questions? It looks like this if it's n 0 1, let's say. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. Can somebody tell me the difference between these two for several variables? Michael Steele's series of ten lectures on Probability Theory and Combinatorial Optimization, delivered in Michigan Technological University in 1995. It's just some technicality, but at least you can see it really fits in. Because I'm giving just discrete increments while these are continuous random variables and so on. 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. 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. You can actually read out a little bit more from the proof. So if x1, x2 up to xn is a sequence of random variables such that the moment-generating function exists, and it goes to infinity. The linearity of expectation, 1 comes out. What happens if for the random variable is 1 over square root n times i? And actually, some interesting things are happening. And then you have to figure out what wnt is. Lec : 1; Modules / Lectures. And say it was $10 here, and $50 here. When you complete a course, you’ll be eligible to receive a shareable electronic Course Certificate for a small fee. So use the Taylor expansion of this. Courses include recorded auto-graded and peer-reviewed assignments, video lectures, and community discussion forums. Afterwards, I will talk about law of large numbers and central limit theorem. So log normal distribution, it does not converge. Because of that, we may write the moment-generating function as a sum from k equals 0 to infinity, t to the k, k factorial, times a k-th moment. So pmf and pdf. If they have the same moment-generating function, they have the same distribution. So let's see-- for the example of three random variables, it might be the case that each pair are independent. Lecture-04-Random variables, cumulative density function, expected value; Lecture-05-Discrete random variables and their distributions Space. But if it's taken over a long time, it won't be a good choice. Because remark, it does not imply that all random variables with identical k-th moments for all k has the same distribution. That doesn't imply that the mean is e to the sigma. There's only one thing you have to notice-- that the probability that x minus mu is greater than epsilon. I can be replaced by some other condition, and so on. Other corrections? Let's say you want to be 99% sure. OK. So that disappears. Freely browse and use OCW materials at your own pace. So the normal distribution and log normal distribution will probably be the distributions that you'll see the most throughout the course. Play poker. The following content is provided under a Creative Commons license. one 10. f sum x I will denote. 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. Yes? And then the central limit theorem tells you how the distribution of this variable is around the mean. So for all non-zero t, it does not converge for log normal distribution. 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 should be log ln. But those are not the mean and variance anymore, because you skew the distribution. It really happened. That doesn't imply that the variance is something like e to the sigma. So that's one thing we will use later. Because it will also take negative values, for example. We don't offer credit or certification for using OCW. I'll make one final remark. x4 and x2, x1 is independent with x2, x1 is independent with 3x, x2 is with x3. There is a hole in this argument. What does the distribution of price? Let me just make sure that I didn't mess up in the middle. AUDIENCE: Because it starts with t, and the right-hand side has nothing general. in this ?] For all the events when you have x minus mu at least epsilon, you're multiplying factor x square will be at least epsilon square. Full curriculum of exercises and videos. Probability Theory and Applications. OK. This looks a little bit controversial to this theorem. 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. ), Statistik und Wahrscheinlichkeitsrechnung, Wahrscheinlichkeit und Statistik (M. Schweizer), Wahrscheinlichkeitstheorie und Statistik (Probability Theory and Statistics), Eidgenössische So first of all, just to agree on terminology, let's review some definitions. Now, that n can be multiplied to cancel out. Your mean is 50. So suppose there is a random variable x whose mean we do not know, whose mean is unknown. Knowledge is your reward. Because pointwise, this conclusion is also rather weak. Lecture 3: Independence. And in your homework, one exercise, we'll ask you to compute the mean and variance of the random variable. But here, I just want it to be a simple form so that it's easy to prove. So now let's look at our purpose. OK. Then the law of large numbers says that this will be very close to the mean. And how the casino makes money at the poker table is by accumulating those fees. But in practice, if you use a lot more powerful tool of estimating it, it should only be hundreds or at most thousands. Yn be square root n times 1 over n of xi is mu. The moral is, don't play blackjack. 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. So you can win money. 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. This course presents the basics of probability theory and the theory of stochastic processes in discrete time. Wiss./HST/Humanmed.) Before going into that, first of all, why is it called moment-generating function? Technische Hochschule Zürich, Eidgenössische Technische Hochschule Zürich. 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. 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. Yeah, log normal distribution. xi minus mu square, when you take expectation, that will be sigma square. That's the expectation of x minus mu square, which is the expectation sum over all i's minus mu square. Because normal distribution comes up here. But if it's a hedge fund, or if you're doing high-frequency trading, that's the moral behind it. Description: This lecture is a review of the probability theory needed for the course, including random variables, probability distributions, and the Central Limit Theorem. That means our goal is to prove that the moment-generating function of these Yn's converge to the moment-generating function of the normal for all t pointwise convergence. Send to friends and colleagues. It's for all integers. So we defined random variables. That's the glitch. The probability distribution is very similar. That's the definition of log over distribution. 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. 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. It can be anywhere. So your variance has to be at least x. So this is the same as xi. Lecture Notes | Probability Theory Manuel Cabral Morais Department of Mathematics Instituto Superior T ecnico Lisbon, September 2009/10 | January 2010/11 So for example, the variance does not have to exist. But for now, just consider it as real Numbers. I will not talk about it in detail. It's almost the weakest convergence in distributions. And this part is well known. Let's think about our purpose. For fixed t, we have to prove it. 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. Set up hx equals 1 over x c of theta-- sorry, theta equals mu sigma. Any questions? For example, you have Poisson distribution or exponential distributions. Then the probability that x is at most x equals the probability that y is at most-- sigma. So we want to see what the distribution of pn will be in this case. 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. And that will be represented by the k-th moments of the random variable. There are two concepts of independence-- not two, but several. And then you're summing n terms of sigma square. Log x equals mu might be the center. Home And that's one thing you have to be careful. The mathematical concepts
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