So that's all about distributions that I want to talk about. And then the central limit theorem tells you how the distribution of this variable is around the mean. So for example, the variance does not have to exist. Who has heard of all of these topics before? Lecture Notes | Probability Theory Manuel Cabral Morais Department of Mathematics Instituto Superior T ecnico Lisbon, September 2009/10 | January 2010/11 So it's not a very good explanation. So that's one thing we will use later. NPTEL provides E-learning through online Web and Video courses various streams. So proof assuming m of xi exists. Description: This lecture is a review of the probability theory needed for the course, including random variables, probability distributions, and the Central Limit Theorem. Yes? It also tells a little bit about the speed of convergence. So use the Taylor expansion of this. ยป Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. That's like the Taylor expansion. And it's easy to describe it in those. If x and y have a moment-generating function, and they're the same, then they have the same distribution. Afterwards, I will talk about law of large numbers and central limit theorem. It doesn't always converge. And continuous is given by probability distribution function. When I first saw it, I thought it was really interesting. 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. OK. If that's the case, x is e to the mu will be the mean. And what should happen? I want x to be the log normal distribution. PROFESSOR: Not 1. probability Theory and A course on Descriptive Statistics. Today, we will review probability theory. So that's the statement we're going to use. Locally, it might be good choice. First of all, one observation-- expectation of x is just expectation of 1 over n times sum of xi's. From the player's point of view, you only have a very small sample. Other questions? Because normal distribution comes up here. ), 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. What kind of events are guaranteed to happen with probability, let's say, 99.9%? That means if you take n to go to infinity, that goes to zero. So you don't have individual control on each of the random variables. So assumed that the moment-generating functions exists. 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? And I want y to be normal distribution or a normal random variable. It's almost the weakest convergence in distributions. So mu mean over-- that's one of the most universal random variable distributions, the most important one as well. Then f of y of the first-- of x of x is equal to y. h of x. Your c theta will be this term and the last term here, because this doesn't depend on x. Let's write it like that. But pairwise means x1 and x2 are independent, but x1, x2, and x3, they may not be independent. Is that the case? Courses 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. 1 over n is inside the square. Expectation-- probability first. Of course, the problem is, when the variance is big, your belief starts to fall. Is this a sensible definition? At least, that was the case for me when I was playing poker. Before proving it, example of this theorem in practice can be seen in the Casino. Now we go back to the exponential form. Lectures: MWF 1:00 - 1:59 p.m., Pauley Ballroom Do you see it? Second term ix 0, because xi has mean mu. 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. i is from [? 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. 18.650 "Statistics for applications" 6.041 "Probabilistic Systems Analysis and Applied Probability" Now, that n can be multiplied to cancel out. About 48% chance of winning. There are two concepts of independence-- not two, but several. OK. And if you take an example as poker, it looks like-- OK, I'm not going to play poker. 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. So if we're seeing something uniformly about t, that's no longer true. As you might already know, two typical theorems of this type will be in this topic. I hope it doesn't happen to you. There are two main things that we're interested in.