So there are that say 1, a2, a3, for which this does not hold. PROFESSOR: OK. Maybe-- yeah. So that's all about distributions that I want to talk about. So using this formula, we can find probability distribution function of the log normal distribution using the probabilities distribution of normal. So that disappears. So I can take it out and square my square. So you will see something about this. It looks like the mean is really close to 50%, but it's hidden, because they designed it so the variance is big. Lec : 1; Modules / Lectures. I like this stuff better. Massachusetts Institute of Technology. When I first saw it, I thought it was really interesting. There is a hole in this argument. It might be mu. So remember that theorem. And continuous is given by probability distribution function. 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 say it was $10 here, and $50 here. ... with Applications in Finance » Video Lectures » Lecture … A few more stuff. And this part is well known. Be careful. And expectation of y is the integral over omega. Other questions? Because normal distribution comes up here. Want to be 99% sure that x minus mu is less than 0.1, or x minus 50 is less than 0.1. Is it mu? And then it can go up to infinity, or it can go down to infinity eventually. So be careful. » f sum x I will denote. All the more or less advanced probability courses are preceded by this one. So you will parametrize this family in terms of mu with sigma. Our second topic will be we want to study its long-term our large-scale behavior. So let's do that. Is this a sensible definition? Yes? OK. 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. That will be our first topic. So first of all, just to agree on terminology, let's review some definitions. Topics in Mathematics with Applications in Finance. 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. So it contains all the statistical information of a random variable. Log x is centered at mu, but when it takes exponential, it becomes skewed. That's just totally nonsense. So that's good. OK. Any questions? The reason this inequality holds is because variances x is defined as the expectation of x minus mu square. So plug in that, plug-in your variance, plug in your epsilon. Mathematics as a subject is vast and with these online tutorials, we have tried to segregate some major topics into distinct lectures. The moment-generating function of Yn is equal to expectation of e to t Yn. overview. We will mostly just consider mutually independent events. Before proving it, example of this theorem in practice can be seen in the Casino. PROFESSOR: OK, so good afternoon. So let's start with our first topic-- the moment-generating function. Pointwise convergence implies pointwise convergence.