Catherine Kellar MTH 320

Difficult: Why is x_l = l(delta)? (top of pg 293). Also, I think that we could have used just a bit more explanation on what DFT/FFT's are. Application: In general, the DFT seems really cool based on what I've seen of Michael's pictures from Physics and such, but I can't seem to understand how exactly they work.

Difficult: The hardest part was following all of the integrals going on. The idea of fourier transforms is interesting, but I don't know how well I'll actually be able to follow it. Also, how is having a basis for only a subset of F make a suitable basis for all of the functions/signals we care about? Application: I've seen some of what DFT/FFTs can do, and it's super cool. I'm excited to see how we can apply this to audio and visual conversions!

Difficult: I don't really get the whole polar coordinates thing. I get that there are polar representations of all complex numbers, but I don't fully understand how to make that happen. Also, on the homework, problems 4 and 5 I don't understand fully how to do, and especially for 5 I don't see where that was explained. Application: Why do we always use such weird scripts for things? Like the symbols for the real/imaginary parts of an imaginary number. But in general, I still have a lot to learn about imaginary numbers.

difficult: There was just a lot of information in this section. Following the rejection sampling algorithm was particularly difficult. Conceptually, I guess they make sense, but actually using/computing things with the concepts will be difficult. Application: I think I've seen these kinds of samplings used before, and I know I've seen them in stats class before, but it's been a while. I believe you when you say they're useful.

Difficult: I'm afraid I don't really get what a Monte Carlo method is. I get that it's important, and that it provides a good model of the real world, but besides the formulas that they provide us with, I don't get where they come from or how they're used. Application: Well, just from the name, the Monte Carlo method being named for a casino makes me laugh. I wonder if that's because the casino makes money from it, or people make money from the casino using these? Also, being able to approximate an integral with a sum, I think, would be much more efficient, helping with a lot of computation times and such.

For Wednesday November 15, as you study for the exam, write responses to the following questions. There is a study guide available  here . Which topics and ideas do you think are the most important out of those we have studied? As always, the theorems and definitions, most especially those associated with each distribution that we need to know. What kinds of questions do you expect to see on the exam? I exist to see at least one in Bayesian stats, several on calculating the pdf, cdf, and pmf, wine excited value and variance, and I hope not too many on bias or estimates. What do you need to work on understanding better before the exam? The are a few things/defs that I don't know yet, and I need to work through the practice problems.

Difficult: I feel like this section was rather poorly written, honestly. There were a lot of very big formulas flying around, most of which I have no idea where they came from, and more importantly I have absolutely no idea what the prior or posterior distributions are, or rather where we found them. No idea. Application: Well, to be fair I think this section is talking about taking a set of iids and making them into a probability distribution, or something like that. Given that that is the case, it would seem that, much like with the LOLN, we can use these new distributions to tell us things about the (perhaps incomplete) information that we already had.

Difficult: Basically this whole section. If I understand it right, it's saying that, given any random distribution, I can sum several random variables from that distribution, and eventually the probability of those events all happening converges to the normal distribution. But I'm not sure quite how, or why. I can see what was done but not really follow it. Application: I think that the central limit thm is used in a lot of practical applications- like they said, they expect grades, snowfalls, etc, though I rather like the example they give of rolling dice.

Difficult: I'm trying to understand what the inequalities mean, and I seem to run into a bit of a hiccup when we say things about the indicator random variable. It is interesting that it comes back up, but I will admit that I don't fully get it. Also, trying to keep track of inequality signs is going to be difficult in some of these things. I'm trying to understand what exactly the inequalities mean, but the proofs in some ways only seem to make them more confusing. Application: The law of large numbers seems intuitive, so it's interesting that we have an actual proof for it. That's cool, though. I guess it just goes to show that sometimes proving intuitive things is important, too.

Difficult: In this particular section, just understanding what is being talked about took me a while. I don't know that I really understand how these things work. But more or less. Really, I just want to say that we just spent, in Ch 3 (or was it 4..?) a lot of time on data structures and graphs, which were complicated but not that bad, and literally only 2 weeks to learn all of probability- and now we're going into statistics. We have gone so fast that I feel like I have learned nothing, and couldn't use this stuff if I tried. Which I've been doing. It's quite unnerving to realize that we have a midterm on this stuff coming up, and I feel like I have no idea what we just learned about. At all. Application: There are tons of applications for statistics. We see them every day. I did appreciate the quote at the beginning, though, about needing a better experiment. I wonder if he ever ran any?

difficult: I really don't have an intuitive idea of what covariance means, and hence understanding the calculations dealing with it was also a bit difficult, especially understanding the matrices full of variables that were being shown. Application: Well, I suppose that this has a lot of applications. Most systems that we model aren't single, or even bi-, variate, but multivariate, so understanding how/why these things work will help us to model things further down the line in cool ways.