probability theory video lectures
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. But in practice, if you use a lot more powerful tool of estimating it, it should only be hundreds or at most thousands. Let's also define x as the average of n random variables. Some very interesting facts arise from this fact. These tools underlie important advances in many … I assumed it if x-- yeah. Here's the proof. AUDIENCE: When you say the moment-generating function doesn't exist, do you mean that it isn't analytic or it doesn't converge? So we want to study this statistics, whatever that means. However, be very careful when you're applying this theorem. That's an abstract thing. 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. So assumed that the moment-generating functions exists. OK. Topics in Mathematics with Applications in Finance Your mean is 50. They might both not have moment-generating functions. Today, we will review probability theory. And then let's talk a little bit more about more interesting stuff, in my opinion. And that bi-linearity just becomes the sum of. So it becomes 1 over x sigma square root 2 pi 8 to the minus log x minus mu squared. OK. So for example, assume that you have a normal distribution-- one random variable with normal distribution. I'll make one final remark. 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. Those kind of things are what we want to study. Because when you want to study it, you don't have to consider each moment separately. Square. PROFESSOR: OK, so good afternoon. What kind of events are guaranteed to happen with probability, let's say, 99.9%? Your c theta will be this term and the last term here, because this doesn't depend on x. So that's one thing we will use later. Discrete Mathematics and Probability Theory. The question is, what is the distribution of price? This might not converge. So I will mostly focus on-- I'll give you some distributions. And one of the most universal random variable, our distribution is a normal distribution. Probability Theory courses from top universities and industry leaders. So the probability that you deviate from the mean by more than epsilon goes to 0. Yes? OK. It's almost the weakest convergence in distributions. So all logs are natural log. So if moment-generating function exists, they pretty much classify your random variables. And in your homework, one exercise, we'll ask you to compute the mean and variance of the random variable. These are lecture notes written at the University of Zurich during spring 2014 and spring 2015. Toggle navigation. But if it's a hedge fund, or if you're doing high-frequency trading, that's the moral behind it. OK. So the question is, what happens if you replace 1 over n by 1 over square root n? Yeah, log normal distribution. So here, when I write only x, x should only depend on x, not on theta. More broadly, the goal of the text The probability distribution is very similar. Emphasis is given to the aspects of probabilistic model building, hypothesis testing and model verification. As you might already know, two typical theorems of this type will be in this topic. Courses include recorded auto-graded and peer-reviewed assignments, video lectures, and community discussion forums. So this is not a sensible model, not a good model. So a distribution is called to be in an exponential family. But if you observed how it works, usually that's not normally distributed. Fall 2016 CS70 at UC Berkeley. Space. Can somebody tell me the difference between these two for several variables? Afterwards, I will talk about law of large numbers and central limit theorem. Lecture Notes | Probability Theory Manuel Cabral Morais Department of Mathematics Instituto Superior T ecnico Lisbon, September 2009/10 | January 2010/11 So when we say that several random variables are independent, it just means whatever collection you take, they're all independent. Lec : 1; Modules / Lectures. Then the law of large numbers says that this will be very close to the mean. 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. So that does give some estimate, but I should mention that this is a very bad estimate. That means if you bet $1 at the beginning of each round, the expected amount you'll win is $0.48. Modify, remix, and reuse (just remember to cite OCW as the source. That's why moment-generating function won't be interesting to us. 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 now let's look at our purpose. Any questions? Full curriculum of exercises and videos. So we want to see what the distribution of pn will be in this case. And the reason is because-- one reason is because the moment-generating function might not exist. f sum x I will denote. 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. 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. So you want to know the probability that you deviate from your mean by more than 0.1. All the real numbers are between 0 and 1 with equal probability. The same conclusion is true even if you weaken some of the conditions. But when it's clear which random variable we're talking about, I'll just say f. So what is this? And then because each of the xi's are independent, this sum will split into products. 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. Who has heard of all of these topics before? If that's the case, x is e to the mu will be the mean. How do you prove it? It can be anywhere. Is that the case? All the more or less advanced probability courses are preceded by this one. That's where the name comes from. So for each round that the players play, they pay some fee to the casino. Do you see it? OK. And one more basic concept I'd like to review is two random variables x1 x2 are independent if probability that x1 is in A and x2 is in B equals the product of the probabilities for all events A and B. OK. All agreed? So this moment-generating function encodes all the k-th moments of a random variable. It has to be 0.01. 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. Because log x as a normal distribution had mean mu. Topics in Mathematics with Applications in Finance. I will not talk about it in detail. ... C onditional Probability Conditional Probability is a measure of the probability of an event given that (by assumption, presumption, assertion or evidence) another event has already occurred. Collection: CBMS Lectures on Probability Theory and Combinatorial Optimization Institution: Department of Pure Maths and Mathematical Statistics And that's one thing you have to be careful. The reason that the rule of law of large numbers doesn't apply, at least in this sense, to poker-- can anybody explain why? Full curriculum of exercises and videos. It gets a unified way. Before proving it, example of this theorem in practice can be seen in the Casino. Anyway, that's proof of there's numbers. So it's not a good choice. The moment-generating function of a random variable is defined as-- I write it as m sub x. The goal of this courseis to prepareincoming PhDstudents in Stanford’s mathematics and statistics departments to do research in probability theory. So we want to somehow show that the moment-generating function of this Yn converges to that. I don't remember what's there. It has the tremendous advantage to make feel the reader the essence of probability theory by using extensively random experiences. Our goal is to estimate the mean. recorded lectures on free probability theory, 26 videos, by Roland Speicher, Saarland University, winter term 2018/19 So it's not a good choice. Introduction to Probability Theory. So now we're talking about large-scale behavior. All positive [INAUDIBLE]. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. 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. So the stock-- let's say you have a stock price that goes something like that. OK. That's good. So this random variable just picks one out of the three numbers with equal probability. It's defined as expectation of e to the t times x where t is some parameter. In our video lectures, we are going to talk about finite mathematics, differential equations, statistical probability, Laplace transforms, Fourier series and more. Lecture 4: Counting. So the normal distribution and log normal distribution will probably be the distributions that you'll see the most throughout the course. It's known to be e to the t square sigma square over 2. 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. AUDIENCE: Can you get the mu minus [INAUDIBLE]? So remark does not necessarily exist. This looks a little bit controversial to this theorem. 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. You're not losing anything. Lecture: TTh 8-9:30am, Zoom So for independence, I will talk about independence of several random variables as well. And if that is the case, what will be the distribution of the random variable? The reason I'm making this choice of 1 over square root n is because if you make this choice, now the average has mean mu and variance sigma square just as in xi's. If we just have a single random variable, you really have no control. That doesn't imply that the variance is something like e to the sigma. Don't play against casino. The statement is not something theoretical. So you can win money. For all reals. If X takes 1 with probability 1/3 minus 1 of probability 1/3 and 0 with probability 1/3. Description: This lecture is a review of the probability theory needed for the course, including random variables, probability distributions, and the Central Limit Theorem. But for now, just consider it as real Numbers. PROFESSOR: Ah. I don't remember exactly what that is, but I think you're right. That's like the Taylor expansion. But here, that will not be the case. Then the distribution of Yn converges to that of normal distribution with mean 0 and variance sigma. That's when your faith in mathematics is being challenged. Other questions? 18.650 "Statistics for applications" 6.041 "Probabilistic Systems Analysis and Applied Probability" And let v-- or Yn. So for example, one of the distributions you already saw, it does not have moment-generating function. The function from the sample space non-negative reals, but now the integration over the domain. PROFESSOR: It might not converge. And I want y to be normal distribution or a normal random variable. For fixed t, we have to prove it. But they still have to make money. And then you have to figure out what wnt is. But still, we can use normal distribution to come up with a pretty good model. There is a hole in this argument. And most games played in the casinos are designed like this. The distribution converges. OK. That's good. 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. 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. Any questions? Lectures: MWF 1:00 - 1:59 p.m., Pauley Ballroom So you will parametrize this family in terms of mu with sigma. It's no longer centered at mu. ), Statistik und Wahrscheinlichkeitsrechnung, Wahrscheinlichkeit und Statistik (M. Schweizer), Wahrscheinlichkeitstheorie und Statistik (Probability Theory and Statistics), Eidgenössische Video Lectures y times. At least, that was the case for me when I was playing poker. I will prove it when the moment-generating function exists. But those are not the mean and variance anymore, because you skew the distribution. This term will be at least epsilon square when you fall into this event. Now we'll do some estimation. Mathematics 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. Other questions? There may be several reasons, but one reason is that it doesn't take into account the order of magnitude of the price itself. But I will not go into it. This is one of over 2,200 courses on OCW. Yes? 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. The moral is, don't play blackjack. I need this. 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 also, the condition I gave here is a very strong condition. In this case, what will be the distribution of the price? It's for all integers. Want to be 99% sure that x minus mu is less than 0.1, or x minus 50 is less than 0.1. Then what should it look like? And similarly, you can let t2 equals log x and w2 equals mu over sigma. Massachusetts Institute of Technology. That's not really. Let me just make sure that I didn't mess up in the middle. That's the expectation of 1 over n sum of xi minus mu square. Now, that n can be multiplied to cancel out. It will be law of large numbers and central limit theory. Parag Radke. We don't really know what the distribution is, but we know that they're all the same. A few more stuff. These lecture notes were written while teaching the course “Probability 1” at the Hebrew University. 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. Video lectures; Captions/transcript; Lecture notes; Course Description. And because normal distribution have very small tails, the tail distributions is really small, we will get really close really fast. in this ?] Which seems like it doesn't make sense if you look at this theorem. Does anyone have experience with the following, and which one would you recommend? AUDIENCE: The notion of independent random variables, you went over how the-- well, the probability density functions of collections of random variables if they're mutually independent is the product of the probability densities of the individual variables. Lecture-01-Basic principles of counting; Lecture-02-Sample space , events, axioms of probability; Lecture-03-Conditional probability, Independence of events. So if you just take this model, what's going to happen over a long period of time is it's going to hit this square root of n, negative square root of n line infinitely often. And using that, we can prove this statement. And then you're summing n terms of sigma square. Michael Steele's series of ten lectures on Probability Theory and Combinatorial Optimization, delivered in Michigan Technological University in 1995. Then that's equal to 1 to the n-th power. You can use either definition. Dice play a significant role in our understanding of probability and its relation to the universe. And I'll try to focus more on a little bit more of the advanced stuff. But there are some other distributions that you'll also see. Let's write it like that. That's too abstract. Download files for later. So a random variable x-- we will talk about discrete and continuous random variables. » So this is the same as xi. So they are given by its probability distribution-- discrete random variable is given by its probability mass function. 1 over n is inside the square. Because normal distribution comes up here. They're not taking chances there. So that's all about distributions that I want to talk about. So one very important thing to remember is log normal distribution are referred to in terms of the parameters mu and sigma, because that's the mu and sigma up here and here coming from the normal distribution. There is a correcting factor. And that will be represented by the k-th moments of the random variable. And one way to do that is by taking many independent trials of this random variable. And let mean be mu, variance be sigma square. So the convergence is stronger than this type of convergence. Oh, sorry. OK. For those who already saw large numbers before, the name suggests there's also something called strong law of large numbers. And then the central limit theorem tells you how the distribution of this variable is around the mean. What does the distribution of price? But it's designed so that the variance is so big that this expectation is hidden, the mean is hidden. Now let's move on to the next topic-- central limit theorem. t can be any real. Introduction on basic statistics, probability theory and uncertainty modeling in the context of engineering decision making.
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