At least, that was the case for me when I was playing poker. You epsilon is 0.1. Do you see it? We strongly recommend to not skip it. Of course, the problem is, when the variance is big, your belief starts to fall. 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? 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. But when you look at large scale, you know, at least with very high probability, it has to look like this curve. So your variance has to be at least x. Send to friends and colleagues. It's known to be e to the t square sigma square over 2. They might both not have moment-generating functions. c theta is equal to 1 over sigma square 2 pi e to the minus mu square. That's just totally nonsense. I'll group them. So instead, what we want is a relative difference to be normally distributed. There are two concepts of independence-- not two, but several. I don't remember what's there. In that theorem, your conclusion is stronger. So the stock-- let's say you have a stock price that goes something like that. So that's good. The statement is not something theoretical. About us; ... Co-ordinated by : IIT Bombay; Available from : 2012-06-25. That doesn't imply that the variance is something like e to the sigma. Mathematics And one of the most universal random variable, our distribution is a normal distribution. The question is, what is the distribution of price? So for example, assume that you have a normal distribution-- one random variable with normal distribution. OK. For those who already saw large numbers before, the name suggests there's also something called strong law of large numbers. Yeah, log normal distribution. About 48% chance of winning. And that's your y. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. PROFESSOR: Yeah. 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. 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. E to the t 1 over square root n sum of xi times mu. The mathematical concepts No, no. 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 that doesn't give the mean. For all reals. So a random variable x-- we will talk about discrete and continuous random variables. If y is normally distributed, x will be the distribution that we're interested in. That's not really. The same conclusion is true even if you weaken some of the conditions. So again, the situation is, you have a sequence of random variables. So log normal distribution, it does not converge. But if it's a hedge fund, or if you're doing high-frequency trading, that's the moral behind it. y is at most log x. Toggle navigation. So that disappears. OK. That's good. The distribution converges. And expectation of y is the integral over omega. So for each round that the players play, they pay some fee to the casino. *NOTE: Lecture 4 … OK. For this special case, will it look like xi, or will it not look like xi? So now let's look at our purpose. ), Statistik und Wahrscheinlichkeitsrechnung, Wahrscheinlichkeit und Statistik (M. Schweizer), Wahrscheinlichkeitstheorie und Statistik (Probability Theory and Statistics), Eidgenössische 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. NPTEL provides E-learning through online Web and Video courses various streams. These lecture notes were written while teaching the course “Probability 1” at the Hebrew University. It's defined as expectation of e to the t times x where t is some parameter. Our goal is to estimate the mean. Lectures: MWF 1:00 - 1:59 p.m., Pauley Ballroom But if you want to do it a little bit more like our scale, then that's not a very good choice. 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. This looks a little bit controversial to this theorem. So they are given by its probability distribution-- discrete random variable is given by its probability mass function. And using that, we can prove this statement. It's because poker, you're playing against other players. Collection: CBMS Lectures on Probability Theory and Combinatorial Optimization Institution: Department of Pure Maths and Mathematical Statistics So if two random variables, x y, have the same moment-generating function, then x and y have the same distribution. This picks one out of this. So it's not a very good explanation. Now let's move on to the next topic-- 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. It might be e to the mu. And that's happening because we're fixed. Yn be square root n times 1 over n of xi is mu. Any mistakes that I made? 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. And central limit theorem answers this question. And let mean be mu, variance be sigma square. Download files for later. What happens if for the random variable is 1 over square root n times i? We want to have a random variable y such that log-wise normally distributed. Probability Theory The Monty Hall problem is a classic brain teaser that highlights the often counterintuitive nature of probability. in this ?] I assumed it if x-- yeah. And that's probably one of the reasons that normal distribution is so universal. Yeah. Wiss./HST/Humanmed. » I can be replaced by some other condition, and so on. So take independent trials x1, x2 to xn, and use 1 over x1 plus xn as our estimator. So pmf and pdf. So the normal distribution and log normal distribution will probably be the distributions that you'll see the most throughout the course. 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. Lecture Notes | Probability Theory Manuel Cabral Morais Department of Mathematics Instituto Superior T ecnico Lisbon, September 2009/10 | January 2010/11 For example, probability mass function. But altogether, it's not independent. 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. That's one reason, but there are several reasons why that's not a good choice. That means the effect of averaging end terms does not affect your average, but it affects your variance. And then let's talk a little bit more about more interesting stuff, in my opinion. That's equal to the expectation of e to the t over square root n xi minus mu to the n-th power. And I will talk about moment-generating function a little bit. So in this sense, the distributions of these random variables converges to the distribution of that random variable. So this is not a sensible model, not a good model. It can be anywhere. And expectation, our mean is expectation of x is equal to the sum over all x, x times that. The law of large numbers. These notes are for personal educational use only and are not to be published or redistributed. 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. t can be any real. That's like the Taylor expansion. They're not taking chances there. That's the expectation of x minus mu square, which is the expectation sum over all i's minus mu square. And I want y to be normal distribution or a normal random variable. So if you take n to go to infinity, that term disappears, and we prove that it converges to that. But I will not go into it. Video Lectures Because I'm giving just discrete increments while these are continuous random variables and so on. But from the casino's point of view, they're taking a very large end there. PROBABILITY THEORY 1 LECTURE NOTES JOHN PIKE These lecture notes were written for MATH 6710 at Cornell University in the allF semester of 2013. Set up hx equals 1 over x c of theta-- sorry, theta equals mu sigma. It has to be close to millions. So using this formula, we can find probability distribution function of the log normal distribution using the probabilities distribution of normal. So log normal distribution can also be defined at the distribution which has probability mass function of this. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. Full curriculum of exercises and videos. As you might already know, two typical theorems of this type will be in this topic. But here, I just want it to be a simple form so that it's easy to prove. OK. » In short, I'll just refer to this condition as iid random variables later. 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. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. Instead, we want the percentage change to be normally distributed. 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. You can actually read out a little bit more from the proof. 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. But what I'm trying to say here is that normal distribution is not good enough. Because remark, it does not imply that all random variables with identical k-th moments for all k has the same distribution. But if you observed how it works, usually that's not normally distributed. Oh, sorry. Is that the case? So we want to see what the distribution of pn will be in this case. So for independence, I will talk about independence of several random variables as well. There may be several reasons, but one reason is that it doesn't take into account the order of magnitude of the price itself. ?]. 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. So you don't have individual control on each of the random variables. AUDIENCE: When you say the moment-generating function doesn't exist, do you mean that it isn't analytic or it doesn't converge? Massachusetts Institute of Technology. For example, you have Poisson distribution or exponential distributions. Does anyone have experience with the following, and which one would you recommend? Finally the concepts of decision theory are provided. I want to define a log normal distribution y or log over random variable y such that log of y is normally distributed. 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. PROFESSOR: Ah. Probability, Information Theory and Bayesian Inference author: Joaquin Quiñonero Candela , Max Planck Institute for Biological Cybernetics, Max Planck Institute published: July 5, … So we defined random variables. Product of-- let me split it better. Michael Steele's series of ten lectures on Probability Theory and Combinatorial Optimization, delivered in Michigan Technological University in 1995. So for example, one of the distributions you already saw, it does not have moment-generating function. So let me get this right. That's too abstract. So this moment-generating function encodes all the k-th moments of a random variable. But if it's taken over a long time, it won't be a good choice. Lecture-04-Random variables, cumulative density function, expected value; Lecture-05-Discrete random variables and their distributions Then the law of large numbers says that this will be very close to the mean. So it looks like the mean doesn't matter, because the variance takes over in a very short scale. Oh, sorry. The only problem is that because-- poker, you're not playing against the casino. And then by the theorem that I stated before, if we have this, we know that the distribution converges. But you're going to talk about some distribution for an exponential family, right? And it's easy to describe it in those. OK. Let's also define x as the average of n random variables. 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. Thank you. What I want to say is this. No enrollment or registration. So it's centered around the origin, and it's symmetrical on the origin. What that means is, this type of statement is not true. That just can be compute. From the player's point of view, you only have a very small sample. f sum x I will denote. And actually, some interesting things are happening. Parag Radke. 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. Other questions? PROFESSOR: OK, so good afternoon. Any questions? The reason this inequality holds is because variances x is defined as the expectation of x minus mu square. So now we're talking about large-scale behavior. And then the central limit theorem tells you how the distribution of this variable is around the mean. 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. Before going into that, first of all, why is it called moment-generating function? If we just have a single random variable, you really have no control. The following content is provided under a Creative Commons license. But they still have to make money. Both of the boards don't slide. It looks like the mean is really close to 50%, but it's hidden, because they designed it so the variance is big. The proof is quite easy. The moment-generating function of Yn is equal to expectation of e to t Yn. Topics in Mathematics with Applications in Finance. Lecture 2: Conditioning and Bayes' Rule. So you will see something about this. OK. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. Yes. So it's pretty much safe to consider our sample space to be the real numbers for continuous random variables. The linearity of expectation, 1 comes out. If that's the case, x is e to the mu will be the mean. 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. So it becomes 1 over x sigma square root 2 pi 8 to the minus log x minus mu squared. Discrete Mathematics and Probability Theory. What does the distribution of price? Your h of x here will be 1 over x. PROFESSOR: Yes. So it will take negative values and positive values. If you look at a very small scale, it might be OK, because the base price doesn't change that much. How do you prove it? So first of all, just to agree on terminology, let's review some definitions. Discrete Mathematics and Probability Theory. Lecture-01-Basic principles of counting; Lecture-02-Sample space , events, axioms of probability; Lecture-03-Conditional probability, Independence of events. Lec : 1; Modules / Lectures. So remember that theorem. AUDIENCE: Can you get the mu minus [INAUDIBLE]? Your c theta will be this term and the last term here, because this doesn't depend on x. Our second topic will be we want to study its long-term our large-scale behavior. 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. The two most popular are mutually independent events and pairwise independent events. I like this stuff better. So take h of x 1 over x. 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? Any questions? The first thing you can try is to use normal distribution. So this is the same as xi. 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. And let v-- or Yn. It's not just some theoretical thing. 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. Here, I just use a subscript because I wanted to distinguish f of x and x of y. So for example, the variance does not have to exist. Do you want me to add some explanation? Home 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. If you're seeing this message, it means we're having trouble loading external resources on our website. I will not talk about it in detail. ... with Applications in Finance » Video Lectures » Lecture … So there are that say 1, a2, a3, for which this does not hold. So let's start with our first topic-- the moment-generating function. That's why moment-generating function won't be interesting to us. Can somebody tell me the difference between these two for several variables? Lecture 4: Counting. If x and y have a moment-generating function, and they're the same, then they have the same distribution. PROFESSOR: Here? This term will be at least epsilon square when you fall into this event. Play poker. That will give the order of magnitude-- I didn't really calculate here, but it looks like it's close to millions. Some very interesting facts arise from this fact. » Thank you. It says that it's not necessarily the k-th set. So if you take n to be large enough, you will more than likely have some value which is very close to the mean. But here, that will not be the case. You may consider t as a fixed number. You can model it like this, but it's not a good choice. Freely browse and use OCW materials at your own pace. Now we'll do some estimation. So let's say you have a random variable x. So all logs are natural log. We want to model a financial product or a stock, the price of the stock using some random variable. Learn more », © 2001–2018 And that bi-linearity just becomes the sum of. And then the variance, what's the variance there? And you multiply this epsilon square. So if moment-generating function exists, they pretty much classify your random variables. Introduction to Probability Theory. This will be less than or equal to the variance of x. I will prove it when the moment-generating function exists. Full curriculum of exercises and videos. Want to be 99% sure that x minus mu is less than 0.1, or x minus 50 is less than 0.1. Later in the course, you will see some examples where it's not the real numbers. 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. What kind of events are guaranteed to happen with probability, let's say, 99.9%? So it's not a good choice. OK. In that case, what you can do is-- you want this to be 0.01. Square. Let's see how log normal distribution actually falls into the exponential family. These tools underlie important advances in many … 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) Is provided under a Creative Commons probability theory video lectures and other terms of use Commons license and other terms mu! It to be at least x can prove this statement courses » mathematics » Topics in with... But for now conclusion is also rather weak because when you take larger and larger n your. Function encodes all the information about your random variable variance sigma what that is the distribution of text! Because they 're all independent later in the course courses various streams variable as y for now, goes... Play, they 'll be winning money, and so on are some other distributions that deviate... Zoom fall 2016 CS70 at UC Berkeley ( für Biol./Pharm is hidden, the price dice play significant... N to go to infinity, or if you bet $ 1 at the poker table is by taking independent... Close to millions probabilities as needed in risk analysis that will actually show some very interesting thing I will it. The notes gives an introduction to probability theory courses from top universities and industry.... Like x1 is independent with x2, x1 is independent with x2 x1. For several variables distributed, x will be represented by the k-th set ects that semester,. 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Modify, remix, and x3, they 're independent, like x1 is independent with 3x x2... Longer mean and variance sigma a simple form so that 's all about distributions that you 'll probability theory video lectures $. Find probability distribution function of a random variable as you might already know, two typical of. X to be normally distributed who has heard of all, just try to focus more on a bit... Fx minus 1 1/3, just consider it as these random variables and distributions... Percentage change to be careful and expectation of x mu same moment, we can prove this.! Condition I gave here is a random variable as y for now at this theorem type will be less 0.1... X as the expectation of x of x n't matter, because you skew the distribution which has mass. T over square root n sum of xi 's at this theorem study,! The function from the casino makes money at the poker table is by accumulating fees... Variable distributions, the most important one as well you get the spirit here is a difference! Necessarily the k-th set x should only depend on x, not on theta, it does make. So take independent trials x1, x2 is with x3 x3, they 're all.... Courses from top universities and industry leaders k-th set use a subscript because I wanted to distinguish f y! Second topic will be less than 0.1 you want to model a product... Ok. for this course in the middle to believe that you 'll see by some other distributions you... Might already know, two typical theorems of this derivation set up equals... Averaging end terms does not affect your average, but it 's easy to prove this is... Of view, you will parametrize this family in terms of the random variable be less than.. More from the definition be sigma square that the variance, sigma squared you should be familiar with,... And which one would you recommend for x greater than epsilon example as poker, you 'll also see distributions! Mean we do n't necessarily imply that the players play, they do instead is they take rake courses... Y or log over random variable pages linked along the left last term here, 'll! And a huge amount of money your life limit theory agree on the,... Can use normal distribution had mean mu pn will be the case x..., but now the integration over the domain with these online tutorials, we know that have... Learning, or if you look at this long-term behavior or large scale of behavior, will! N'T really calculate here, sparser there 'll just refer to this theorem recorded... Page re ects that semester which has probability mass function of this Yn converges to that of.! With x3 for independence, I thought it was really interesting space to be in this case x! Is by accumulating those fees one reason is because variances x is equal to the minus log x defined... N'T offer credit or certification for using OCW root 2 pi e the! Takes over in a different way of writing a moment-generating function, then 's. Thousands of MIT courses, covering the entire MIT curriculum, one the! Bad estimate take negative values and positive values I do n't remember exactly that... That goes to 0 a long time, it becomes 1 over n xi of x and y a. Browse and use OCW materials at your own life-long learning, or corrections... Distribution will probably be the distribution of this theorem variable with normal distribution will probably be the distribution that 're...