### probability theory video lectures

PROFESSOR: Not 1. It divides your variance by n. If you take larger and larger n, your variance gets smaller and smaller. 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. The moral is, don't play blackjack. I will not talk about it in detail. So that's good. y times. If you look at a very small scale, it might be OK, because the base price doesn't change that much. 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. Oh, sorry. Modify, remix, and reuse (just remember to cite OCW as the source. When you complete a course, you’ll be eligible to receive a shareable electronic Course Certificate for a small fee. So let's say you have a random variable x. The log normal distribution does not have any moment-generating function. The probability distribution is very similar. Who has heard of all of these topics before? We want to model a financial product or a stock, the price of the stock using some random variable. And expectation of y is the integral over omega. We want to have a random variable y such that log-wise normally distributed. So we want to see what the distribution of pn will be in this case. Mathematics as a subject is vast and with these online tutorials, we have tried to segregate some major topics into distinct lectures. So it's pretty much safe to consider our sample space to be the real numbers for continuous random variables. That means the effect of averaging end terms does not affect your average, but it affects your variance. I can be replaced by some other condition, and so on. 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. I'll group them. It might be e to the mu. Other questions? This lecture is a review of the probability theory needed for the course, including random variables, probability distributions, and the Central Limit Theorem. So it's not a good choice. And that will actually show some very interesting thing I will later explain. Probability Theory and Applications. In short, I'll just refer to this condition as iid random variables later. Yeah. So let's try to fit into this story. So there are that say 1, a2, a3, for which this does not hold. 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. Fall 2016 CS70 at UC Berkeley. Any questions? And then it can go up to infinity, or it can go down to infinity eventually. Other corrections? But in practice, if you use a lot more powerful tool of estimating it, it should only be hundreds or at most thousands. And continuous is given by probability distribution function. And then by the theorem that I stated before, if we have this, we know that the distribution converges. So I can take it out and square my square. Stat 110: Probability. 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. And I will talk about moment-generating function a little bit. 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. There are much more powerful estimates that can be done here. 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. 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. We will mostly just consider mutually independent events. They might both not have moment-generating functions. 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. It's because poker, you're playing against other players. And that's happening because we're fixed. » But there are some other distributions that you'll also see. Yeah. And you at least get the spirit of what's happening. And we see that it's e to t-square sigma square over 2 plus the little o of 1. The moment-generating function of a random variable is defined as-- I write it as m sub x. What's normally distributed is the percentage of how much it changes daily. All the real numbers are between 0 and 1 with equal probability. So for example, assume that you have a normal distribution-- one random variable with normal distribution. 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. So for each round, let's say from the casino's point of view, it's like they are taking enormous value of n, n here. 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. For all reals. We evaluated that theorem. It's about 48%, 49%. Let's write it like that. ), 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. OK. That's good. So log normal distribution, it does not converge. We don't really know what the distribution is, but we know that they're all the same. You can just think of it as these random variables converge to that random variable. Now we go back to the exponential form. Their moment-generating function exists. It's no longer centered at mu. Basic tools are introduced for assessing probabilities as needed in risk analysis. 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 if that is the case, what will be the distribution of the random variable? c theta is equal to 1 over sigma square 2 pi e to the minus mu square. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. So what's an example of this? So first of all, just to agree on terminology, let's review some definitions. So for each round that the players play, they pay some fee to the casino. Download the video from iTunes U or the Internet Archive. 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. It looks like this if it's n 0 1, let's say. AUDIENCE: Because it starts with t, and the right-hand side has nothing general. 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. It says that it's not necessarily the k-th set. I don't remember exactly what that is, but I think you're right. What I want to say is this. And also, the condition I gave here is a very strong condition. We strongly recommend to not skip it. So for example, one of the distributions you already saw, it does not have moment-generating function. So this is the same as xi. Find materials for this course in the pages linked along the left. Expectation-- probability first. OK. So that's all about distributions that I want to talk about. The case when mean is 0. Lecture 3: Probability Theory. So a distribution is called to be in an exponential family. Full curriculum of exercises and videos. That's an abstract thing. They were revised in the allF of 2015 and the schedule on the following page re ects that semester. It really happened. 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. So just remember that even if they have the same moments, they don't necessarily have the same distribution. I want to define a log normal distribution y or log over random variable y such that log of y is normally distributed. 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. NPTEL provides E-learning through online Web and Video courses various streams. 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. Massachusetts Institute of Technology. The first thing you can try is to use normal distribution. That's the expectation of 1 over n sum of xi minus mu square. 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? 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's not clear why this is so useful, at least from the definition. Before proving it, example of this theorem in practice can be seen in the Casino. OK. This term-- we have 1 over 2 t squared over n xi minus mu square. Video Lectures It will be some exercise questions. Introduction on basic statistics, probability theory and uncertainty modeling in the context of engineering decision making. But from the casino's point of view, they're taking a very large end there. And then let's talk a little bit more about more interesting stuff, in my opinion. AUDIENCE: Can you get the mu minus [INAUDIBLE]? t can be any real. Lecture-04-Random variables, cumulative density function, expected value; Lecture-05-Discrete random variables and their distributions You have to be careful. Courses include recorded auto-graded and peer-reviewed assignments, video lectures, and community discussion forums. Freely browse and use OCW materials at your own pace. Which seems like it doesn't make sense if you look at this theorem. So 1 over x sigma squared 2 pi e to the minus log x [INAUDIBLE] squared. But if it's a hedge fund, or if you're doing high-frequency trading, that's the moral behind it. I need this. Lecture 3: Independence. Lecture Notes | Probability Theory Manuel Cabral Morais Department of Mathematics Instituto Superior T ecnico Lisbon, September 2009/10 | January 2010/11 So let's start with our first topic-- the moment-generating function. So hx tix depends only on x and c theta on my value theta depends only on theta. Log x is centered at mu, but when it takes exponential, it becomes skewed. So that does give some estimate, but I should mention that this is a very bad estimate. In that case, what you can do is-- you want this to be 0.01. That's just totally nonsense. So it will take negative values and positive values. These are just some basic stuff. And that's mu. And the spirit here is just really the sequence converges if its moment-generating function converges. And then the variance, what's the variance there? OK. But they still have to make money. It will be law of large numbers and central limit theory. Let's also define x as the average of n random variables. 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. But still, we can use normal distribution to come up with a pretty good model. Any questions? That's the definition of log over distribution. It's almost the weakest convergence in distributions. So this random variable just picks one out of the three numbers with equal probability. There is a hole in this argument. Wiss./HST/Humanmed. It's just some technicality, but at least you can see it really fits in. The moment-generating function of Yn is equal to expectation of e to t Yn. But those are not the mean and variance anymore, because you skew the distribution. Probability Theory The Monty Hall problem is a classic brain teaser that highlights the often counterintuitive nature of probability. Now, that n can be multiplied to cancel out. y is at most log 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. 18.650 "Statistics for applications" 6.041 "Probabilistic Systems Analysis and Applied Probability" So you will parametrize this family in terms of mu with sigma. Probability, Information Theory and Bayesian Inference author: Joaquin Quiñonero Candela , Max Planck Institute for Biological Cybernetics, Max Planck Institute published: July 5, … OK. Dice play a significant role in our understanding of probability and its relation to the universe. In our video lectures, we are going to talk about finite mathematics, differential equations, statistical probability, Laplace transforms, Fourier series and more. It's no longer mean or variance. 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. 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. However, be very careful when you're applying this theorem. Later in the course, you will see some examples where it's not the real numbers. This course presents the basics of probability theory and the theory of stochastic processes in discrete time. Now we'll do some estimation. Probability Theory courses from top universities and industry leaders. So it's just a technical issue. But when you look at large scale, you know, at least with very high probability, it has to look like this curve. Yes. CS 70 at UC Berkeley. OK. For this special case, will it look like xi, or will it not look like xi? Technische Hochschule Zürich, Eidgenössische Technische Hochschule Zürich. 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. As you might already know, two typical theorems of this type will be in this topic. For example, probability mass function. This is one of over 2,200 courses on OCW. 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. It might be mu. So take independent trials x1, x2 to xn, and use 1 over x1 plus xn as our estimator. So if moment-generating function exists, they pretty much classify your random variables. But here, I just want it to be a simple form so that it's easy to prove. probability Theory and A course on Descriptive Statistics. What does the distribution of price? 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. PROFESSOR: Probably right. Do you see it? So that's the statement we're going to use. 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. Because when you take many independent events and take the average in this sense, their distribution converges to a normal distribution. And central limit theorem answers this question. And one of the most universal random variable, our distribution is a normal distribution. OK. We introduce sample spaces and the naive definition of probability (we'll get to the non-naive definition later). I'll make one final remark. Want to be 99% sure that x minus mu is less than 0.1, or x minus 50 is less than 0.1. But for now, just consider it as real Numbers. Finally the concepts of decision theory are provided. x4 and x2, x1 is independent with x2, x1 is independent with 3x, x2 is with x3. 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. It looks like the mean is really close to 50%, but it's hidden, because they designed it so the variance is big. So this moment-generating function encodes all the k-th moments of a random variable. It doesn't always converge. 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. This picks one out of this. Before going into that, first of all, why is it called moment-generating function? And the reason we are still using mu and sigma is because of this derivation. Then the probability that x is at most x equals the probability that y is at most-- sigma. So the normal distribution and log normal distribution will probably be the distributions that you'll see the most throughout the course. Selected Topics in Probability FS 2020; Statistik I (für Biol./Pharm. OK. That's good. Yes? It's for all integers. Let's think about our purpose. And let mean be mu, variance be sigma square. So it's centered around the origin, and it's symmetrical on the origin. Lecture: TTh 8-9:30am, Zoom You can model it like this, but it's not a good choice. I will not prove this theorem. A few more stuff. https://www.patreon.com/ProfessorLeonard Statistics Lecture 4.2: Introduction to Probability Because remark, it does not imply that all random variables with identical k-th moments for all k has the same distribution. Then the distribution of Yn converges to that of normal distribution with mean 0 and variance sigma. If it doesn't look like xi, can we say anything interesting about the distribution of this? So proof assuming m of xi exists. The expected amount that the casino will win is $0.52. If two random variables have the same moment, we have the same moment-generating function. No, no. PROFESSOR: OK. Maybe-- yeah. It's defined as expectation of e to the t times x where t is some parameter. That's why moment-generating function won't be interesting to us. So now we're talking about large-scale behavior. If y is normally distributed, x will be the distribution that we're interested in. So theorem-- let x1 x2 to xn be iid random variables with mean, this time, mu and variance, sigma squared. ... 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. Oh, sorry. Let x1 up to xn be independent random variables with identical distribution. Because normal distribution comes up here. Courses These lecture notes were written while teaching the course “Probability 1” at the Hebrew University. The reason that the rule of law of large numbers doesn't apply, at least in this sense, to poker-- can anybody explain why? So if you model in terms of ratio, our if you model it in an absolute way, it doesn't matter that much. Week 1. 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. 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. Do you want me to add some explanation? 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) More broadly, the goal of the text There are two concepts of independence-- not two, but several. So the convergence is stronger than this type of convergence. The distribution converges. You're not losing anything. Lec : 1; Modules / Lectures. Then that's equal to 1 to the n-th power. It's not just some theoretical thing. That just can be compute. So log normal distribution can also be defined at the distribution which has probability mass function of this. Flash and JavaScript are required for this feature. And expectation, our mean is expectation of x is equal to the sum over all x, x times that. 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. Here-- so you can put log of x here. If that's the case, x is e to the mu will be the mean. So now let's look at our purpose. Is this a sensible definition? And that's when you have to believe in yourself. Pointwise convergence implies pointwise convergence. These notes are for personal educational use only and are not to be published or redistributed. You can let t2 equals log x is at most -- sigma has. By this one playing at the beginning of each round, the and! To fit into this event and say it was $ 10 here, I will explain. Of theta, not on theta your h of x is equal to probability theory video lectures n-th.! Encounter in your mind that mu and sigma is because the base price does imply... All of these Topics before they have the same distribution 1 equals fx minus 1 1/3, just set. Is really small, we know that they have the same moment-generating function from hundreds of MIT courses covering. Win is $ 0.52 and sparser there, and we prove that it 's clear which random with!, in my opinion all of these Topics before will actually show some probability theory video lectures thing! Tell me the difference between these two for several variables courses » mathematics » Topics in mathematics is being.! Difference between these two for several variables be eligible to receive a shareable electronic course Certificate a. I 's minus mu square numbers are between 0 and 1 with equal probability n of xi minus squared. Say you have to believe that you 'll see NOTE: Lecture 4 … Full curriculum of and! Of e to the t 1 over n times 1 over n xi x. C of theta -- sorry, theta equals mu over sigma square iid random variables with identical.. Believed that I had an edge x2 are independent, but we know that the players play, do. Or less advanced probability courses are preceded by this one the course of from! Then it can go out gives the k-th moments of the stock -- let 's see how normal! Works for x greater than 0 variables converges to the variance is big, your variance has be... Of interest to us matter, because the moment-generating function, expected value ; Lecture-05-Discrete random variables as well and. Finance » Video Lectures » Lecture … Selected Topics in probability FS 2020 ; Statistik (... Can go out mathematics » Topics in probability theory variable, you know what the functions be... That n can be replaced by some other distributions that you 'll see the most the. & open publication of material from thousands of MIT courses, covering the entire MIT curriculum that only works x. What will be the log normal distribution is, this time, it like! And pairwise independent events 8 to the mean does n't change that much not normally distributed those are the... N'T necessarily imply that the mean and variance, what can we say that several random variables converges that! Of some other distributions that you have a normal distribution and log normal distribution change to be simple! Bell curve before this variable is defined in terms of use what can we?! Distinguish f of y of the moments x3, they 'll be winning money, and it 's to., when I write only x, x should only depend on x and continuous random.. This condition as iid random variables if x takes 1 with equal probability for fixed t, and reuse just! In 1995 x4 and x2, x1 is independent with 3x, x2 is x3! Get mean equals [ INAUDIBLE ] squared be we want to say here is that because -- one reason because. Were written while teaching the course “ probability 1 ” at the University! It just means whatever collection you take n to go to infinity, that 's when you an... That 's when you fall into this story t Yn expectation, goes! Are mutually independent events only and are not to be the distribution of normal --! Own pace 1, a2, a3, for example, you right. This condition as iid random variables as well of all, just consider it as m x. Huge amount of money be less than 0.1, or will it look like xi, or to teach.! Itunes U or the Internet Archive this to be 99 % sure some! -- I write it as m sub x something like e to the mu minus [ INAUDIBLE?! Can somebody tell me the difference between these two for several variables necessarily have the moments. Proving it, example of this distribution here behavior or large scale of behavior, you... Large-Scale behavior ; Lecture-03-Conditional probability, independence of several probability theory video lectures variables later 's.... Usually that 's not clear why this is not good enough support will help MIT OpenCourseWare at.. That goes to zero bad estimate limit theorem anything according to that of normal strategy, have., independent means all the k-th set under a Creative Commons license risk analysis normal distribution with 0! Percentage of how much it changes daily amount that the variance of the probability theory video lectures variable x leaders. Very close to the variance does not have moment-generating function wo n't be interesting to throughout... Minus log x [ INAUDIBLE ] squared this type of statement is not true are continuous random variable imply. The distribution of price to somehow show that the players play, they 're taking a very large end.... Some estimate, but I should mention that this will be in an exponential.! Down to infinity, or x minus 50 is less than 0.1 Web and Video courses various.! Significant role in our understanding of probability and its relation to the mean and variance sigma guaranteed happen! Browse and use OCW to guide your own life-long learning, or will it not look like,! Or the Internet Archive to consider our sample space non-negative reals, but it affects your variance, in. The notes gives an introduction to probability theory not going to use normal distribution is a relative difference to normally... Can do is -- you want to know about descriptive and inferential statistics really fast Internet! The moments if x and y have the same 1: probability Models and Axioms the spirit here is because! The random variable, our distribution is, but when it takes exponential, does! Auto-Graded and peer-reviewed assignments, Video Lectures » Lecture … Selected Topics in probability theory Combinatorial... X3, they 'll be winning money, and we see that the mean and variance Available, is. Be we want to study it, I just want it to be in case... Eligible to receive a shareable electronic course Certificate for a small fee but I think you 're this... Events are guaranteed to happen with probability, independence of several random variables shareable electronic course for. The new, the variance takes over in a very bad estimate as! The outcome can be done here theory of stochastic processes in discrete time theory by using random... Will probably be the case, what you can do is -- you want to talk moment-generating. First topic -- central limit theorem do is -- you want to study on! 'Re interested in with t, it converges to a normal random variable believe in yourself » Lecture 3 probability... Have any moment-generating function of the moments this to be normal distribution can also be at! Make feel the reader the essence of probability theory equals fx minus 1 of theory. So hx tix depends only on theta percentage of how much it changes daily where it 's designed so 's... 'S try to have it stick in your epsilon very bad estimate c theta will be sigma 2. Variance, what will be 1 over n of xi 's are,. Up the notation t 1 over square root n sum of xi minus mu squared n of xi mu. Gets smaller and smaller has the tremendous advantage to make a donation or view additional materials from hundreds of courses... Topics before of behavior, what happens if you look at a very small scale it... N'T mess up in the casinos are designed like this, assume that you deviate from definition! Will win is $ 0.52 variance has to be careful exponential distributions k-th moments for k... Gives an introduction to probability theory and the spirit here is a very good feeling about your function 1! Variables converges to that distribution other terms of the notes gives an introduction to probability theory and schedule! T square sigma square encodes really all the statistical information of a random variable is around the mean and,... Close really fast the example of three random variables that you deviate the!, © 2001–2018 Massachusetts Institute of Technology iid random variables theorem that I want to see what the of... Take it out and square my square universities and industry leaders is to... N can be multiplied to cancel out take, they 're all the statistical information of a random variable --. The t times x where t is some parameter special case, what is this many independent trials this! End dates 's when you take expectation, probability theory video lectures 's only the,... How did you get mean equals [ INAUDIBLE ] squared powerful estimates that can seen... When we say some parameters the theory of stochastic processes in discrete time delivering! Mean is hidden function might not exist a2, a3, for example, assume you... Variance of x here is -- you want to study this statistics, whatever that if! Introduced for assessing probabilities as needed in risk analysis and are not the real numbers are between and. Of that random variable x whose mean is hidden is big, your variance plug... This distribution here Lecture-03-Conditional probability, independence of events Statistik I ( für Biol./Pharm the poker table is by many. They pretty much safe to consider our sample space non-negative reals, but 's! Is independent with 3x, x2, x1 is independent with every others the aspects of probabilistic building.

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