wouldnt the correct way to prove that to be use sum formulas for a geometric series, while also showing the the infinite sum is 1?
Can anybody explain to me exactly what the difference is between a probability density function and a probability distribution function? This is proving to be a major roadblock to my study. I understand that you get from a PDF to a CDF via integration and vice versa, but I mainly have no idea what the distribution function actually means.
The CDF of a random variable X is the function F such that F(t) is the probability that X is less than or equal to t. So, that should be pretty easy to interpret. Formally, the density f of a random variable X is defined to be any function such that when you integrate f over a set A you get the probability that X is in A. In particular, if you integrate over (-infinity, t] you get the probability that X is less than or equal to t, i.e. you get F (and now you can apply the FToC to get that F' = f, leaving out some annoying details). But you could also (say) take a set like A = the rational numbers, and integrating f over A in this case would give the probability that X is a rational number which in the continuous case happens to be 0.
There is no such formal thing as a 'probability distribution function' - it is a term which can refer to either the probability mass function for discreet variables, or the density function for continuous variables. There is a lot of semantic confusion over distribution/density/cdf. Often when people say distribution function they mean the cumulative distribution function (this is the case in the S and R languages which confused the hell out of me at first).
In probability theory the probability distribution function is always the CDF. The probability distribution (without the word function) refers to the function P that assigns probabilities to sets. The pmf and pdf refer to neither of these things but are Radon-Nikodym derivatives of P wrt some dominating measure (counting and Lebesgue, respectively). Of course, all of these creatures determine the others hence the sloppiness in some usage.
I think I got a job as an Algebra II Honors/Analysis of Functions teacher next Fall. Yay, for my first full time teaching gig not being boring algebra I or geometry. I think I will stick with teaching HS for 2 or 3 years and go on to grad school (I just got my B.S. in HS Math Ed). I do want to save up a good amount of money (I am pretty frugal so teaching/tutoring will help me save a decent amount). I talked with yorost (I think some of you guys remember him from back in the day) and he was telling me about operations research. The college I would probably go to would be FIT about an hour away, and I would want to work as a TA while taking classes. I just don't feel satisfied ending my math education here; I've always loved math since I was a little kid, and learning it more and more makes me fall in love with it more. I think, doctorate-wise, FIT offers PhD's in operations research and differential equations. Of course, I am really not sure where to go from here quite yet.
I'm actually not much into applied stuff, but it makes sense to go that route. Unless...the "pure math" route would land me a potential job as a math prof, which I would love...
You have to be remarkably gifted to do anything worth doing in pure math. Not to say you can't make some contributions of course. Although I have high standards for what I consider "worth doing" although not as high as guys like Gauss or Von Neumann who didn't publish stuff that most people would gleefully publish.
I'm planning on getting a Ph.D LOL but I don't know in which field yet atm I'm really interested in combinatorics+optimization tho, I'm taking linear programming now and network flows next term. But I'm also taking complex analysis + group theory now, and real analysis + polynomials/rings next term, cuz my major is pure math. But after real analysis I have to take measure theory for my degree, and right now complex analysis is a drag for me. I love algebra though. Measure theory is the last analysis course I'm required to take, but topology sounds cool and I think I'll take set theory, logic and more algebra in my last year. Other than pure math I'm thinking of taking lots of combinatorics and then I guess I'll figure out what to do after my undergrad after I take courses in these various topics EDIT: if any of you nerds are interested I'll post a list of courses I am am taking or am seriously considering taking during my undergrad and hopefully y'all can tell me what you think of these topics if you've studied them before * I've starred the courses I'm required to take Complex analysis * Group theory * Linear programming * Real analysis * Polynomials, rings and finite fields * Network flow theory * Combinatorial enumeration * Graph theory * Measure theory and Fourier analysis * First-order logic Differential geometry * Galois theory Rings, modules and representations Representations of finite groups Commutative algebra Topology Set theory Algebraic enumeration Matroid theory Combinatorial designs Integer programming holy shit I just realized that's a lot of courses to want to take before the end of my undergrad.
You must go to a good school if they are making you take Measure Theory during undergrad. At my university I don't think they offer an undergrad version.
The required course for my degree is called "Lebesgue integration and Fourier analysis", and its sequel (not required) is just "Measures and integration".
Dunno, that's the way it's set up at my school. Here are the course descriptions. The first is a prerequisite for the second. PMATH 450 LEC 0.50 Lebesgue Integration and Fourier Analysis Lebesgue measure on the line, the Lebesgue integral, monotone and dominated convergence theorems, Lp-spaces: completeness and dense subspaces. Separable Hilbert space, orthonormal bases. Fourier analysis on the circle, Dirichlet kernel, Riemann-Lebesgue lemma, Fejer's theorem and convergence of Fourier series. PMATH 451 LEC 0.50 Measure and Integration General measures, measurability, Caratheodory Extension theorem and construction of measures, integration theory, convergence theorems, Lp-spaces, absolute continuity, differentiation of monotone functions, Radon-Nikodym theorem, product measures, Fubini's theorem, signed measures, Urysohn's lemma, Riesz Representation theorems for classical Banach spaces.
Well, I meant you would probably get some measure theory tools before studying Fourier analysis in earnest. I guess they just go get Lebesgue measure and leave more general measures to the sequel. If the sequel wasn't optional they might not do it that way I guess.
At a glance, the problem you state seems to be an overdetermined system, that it, there is no unique solution to get through simple linear algebra. What you are looking is basically a linear transformation that simulates the effect of perspective on a bitmap. The exact solution probably has something to do with using a perspective transformation on B, treating it as if it were a three dimensional sheet of paper. Those weird constraints would then have to be treated in some other way. Needless to say you will be putting your linalg skills to use here. Or just google "nintendo mode 7 matrix" because thats what the snes do anyway. Computationally, the transformation is pretty simple: where x and y are the original pixel coordinates, x' and y' are the transformed pixel coordinates, x0 and y0 is the offset and (a,b,c,d) is the coefficient matrix. The challenge lies in finding just what coefficients to use.
I want to teach myself some complex analysis this summer. I got a book from Amazon called "Visual Complex Analysis," which seems to be a very nice icebreaker for the subject. I also have a rigorous book on the subject, but I would wait on that one. For complex analysis, what other math should I be knowledgeable in? Calc 3 and linear algebra?
yeah thats good theres also a free online book that is p good i think from some georgia guy it has good proofs on the hand-wave level
If it's this one (http://people.math.gatech.edu/~cain/winter99/complex.html), I found it a while ago actually. The VCA book is very nice as well, plenty of geometric interpretations, which I think i would benefit from greatly
Jamesman, you should know a lot of multivariable calculus, and some basic linear algebra might be helpful but not too important. Vector analysis might not be necessary, but you can derive some important complex analysis theorems using Stokes's theorem (or so I've heard). Personally I haven't learned vector calculus and I was okay learning complex analysis. Personally, I also think real analysis is helpful because although you don't really use the results, you'll have a useful scope for differentiable functions. Basically, complex-differentiable ("holomorphic") and real-differentiable functions appear very similar, but some crazy magic happens when you work with functions of a complex variable. It's really an amazing topic, and there are a lot really cute formulas and results. For example: a non-constant holomorphic function will never have a local maximum on the interior of its domain, a bounded function holomorphic on the entire plane is necessarily constant, uniform limit of holomorphic functions is holomorphic, and many others. So having experience with real-differentiable functions will help you appreciate the strange results which occur for complex-differentiable functions. For books, I'd recommend Serge Lang's complex analysis. That's the book I learned out of when I took the course. Also google George Cain's complex analysis notes, you can get PDFs for free online and they're very useful for intuitive, example-based discussion. Geometric interpretation is awesome, but I think symbolic descriptions were easiest for me to learn. It's good to draw a picture of sets in the plane, but understanding the image of a holomorphic function can be pretty confusing. But the pictures look pretty awesome when you get a computer to generate them: check this out. EDIT: oya I just noticed you already posted Cain's notes ... They were really helpful for me when learning proofs and understanding concepts. Great notes. It's important to be precise and think about technical details, but I think it's also necessary to understand the ideas of proofs, and he really gives a good presentation of the concepts.
I see. I do need to brush up on my multivariable calc...I may do that from now through May. I already took it but I haven't used it in so long that some of the finer details are probably forgotten now. So maybe this would be a good and logical order: Multivariable calc refresher (textbook/khan) Linear algebra (khan) Real Analysis Complex Analysis The only thing...real analysis. I have an idea of what the subject is about, but I guess I should teach that to myself as well (never learned it). Any recs for doing so? I suppose a solid text supplemented with some online lectures would be perfect, even if to get a basic understanding of RA. Then I can work on CA. From what I understand, RA is a lot of rigorous mathematics, and I guess it's just calculus but all proofs (or so I heard). Btw, I'm doing all this because 1) I am just very interested in how complex numbers behave, they've always fascinated me and 2) I want to go to grad school for math and would like to have a good head start before taking these classes for real, because I would probably be working at the same time. I appreciate your responses, y'all!
Real analysis is no joke, and probably not nessecary for elementary complex analysis. It is more-or-less a rigorous treatment of calculus, at least at the introductory level. You would be quite gifted if you could teach it to yourself, and doing so adequately would require a huge time investment. Canonical text is Rudin's Principles of Mathematical Analysis, which should be easy to find online, but it's terse on top of the subject being difficult. I would recommend attempting every problem therein if you want to really learn it, solving most of them. If you can you will see immediate improvements in your abilities.
I wouldn't say real analysis is necessary to know before studying complex analysis ... But again, it would help you achieve a certain scope of appreciation for the theorems in CA. My first real analysis course starts next week lol. I was trying to say this: my prof would show us these proclaimed "cool" results on holomorphic functions, but I couldn't appreciate that when because I didn't know the corresponding deal for real-valued functions. For example, we used Morera's theorem to prove that the uniform limit of holomorphic (complex-differentiable) functions is again a holomorphic function, and my prof claimed that this was amazing: the Stone-Weierstrass theorem says that any continuous real-valued function on a closed (real) interval is the uniform limit of polynomials. But continuous functions are not necessarily differentiable, so this situation does not have as cute results as the analogous situation for holomorphic functions. My prof mentioned the Stone-Weierstrass theorem when he proved this complex analysis result in class, but I think I'd be able to appreciate the results more if I knew more about real analysis. But theodds is right, you don't need real analysis to study complex analysis at an elementary level. At my school, one needs to take CA before RA, so RA is really considered "more difficult". Like calculus and linear algebra, complex analysis can be taught in a cookie cutter style so you focus on calculating in simple cases instead of proving statements. I think both rigorous understanding and calculation ability are important though, so it's good to flip between Cain's CA notes and another book which covers CA rigorously. Really, check out Lang's CA book, it was really helpful for me. Another CA book with lots of problems and clear explanations is "Function Theory of One Complex Variable" by Greene and Krantz: I find that Greene/Krantz is to CA as Dummit/Foote is to algebra. For real analysis, use Rudin's "Principles". Books we used in my multivariable calc class were Spivak's "Calculus On Manifolds", and "Analysis By Its History" by Hairer and Wanner. Also PLEASE do not learn linear algebra from Khan It's helpful for learning cookbook lin alg, but if you want to study the (abstract) theory of vector spaces (which is really linear algebra), you should pick up a good book ... We used Friedman/Insel/Spence "Linear Algebra" for my two lin alg courses, which is a good intro. For my second lin alg class we used Hoffman/Kunze "Linear Algebra", which is an awesome reference and very rigorous treatment of the material. I'd recommend you check out the latter for learning Linear Algebra.
Ahhhh I see, so I can hold off on RA then. RA seems interesting but I don't feel ready yet to take it on. I have a linear algebra textbook that I can study, though I'll check out those books on Amazon and see how the prices look. I had started watching the khan videos for LA a while ago (never finished, dunno why), it seemed like a good elementary intro to the subject...maybe I can watch the videos and then delve into a book?
There's a chance I'll be a lecturer at my college for either Algebra or Precalculus this summer. Anyone here have experience lecturing math classes and can give good advice about how to plan a course, write up exams/homework, etc.
Yeah, first look at what the students should walk away from the class knowing (the core ideas). Also anything in particular that they should know for their next math classes so they are prepared. So then you space out and allot time for each concept, using some common sense. Graphing inequalities should take less time than conics, for example. It's always good to create your test first, and then set up your lessons to meet the standards/ideas of the test. This way your students will be adequately prepared for tests and you don't have to worry about accidentally not covering something. You have a goal in mind and you can guide your students to that goal. That isn't to make it easy on them, but it is to enrich their education. Test generators are nice (at least partially), and if it is just algebra/precalc, bookwork should be plenty enough to give them the practice they need (especially for college students). No need to give yourself more work when there are problems (with solutions on your end) already there for them. More and more math classes use computer programs for homework as well, though from my experience working at a college tutoring students, they HATE computer programs, though mostly because they don't get the syntax right.
http://www.huffingtonpost.com/2012/...lws-main-bb|dl4|sec1_lnk3&pLid=156624#s919741 (watch the second video lower down the page) laughed uncontrollably at the part where he "sees the pythagorean theorem everywhere" basically it's exactly what you'd expect from somebody who has never had any exposure to actual advanced mathematics
During the tests, two specific areas of his brain lit up -- the area that controls math and mental imagery
Pretty sensationalist story about that dude. I don't think that skill is all that uncommon though really.
