<p>I'm almost turning into a math major even though I've been an econ major thus far. Anyhow, I have to take certain classes if I want to get into my dream graduate progam in decision theory.</p>
<p>I was just wondering, how hard is real analysis? I've done alright in your basic linear algebra and calc 1-3 courses. Also, what type of higher level stats courses would you recommend that have applications for gambling/uncertainty? I know courses vary by school, but if you know of specific topics/theorems that are of value, I can search that way.</p>
<p>Real analysis can be very challenging if you haven't had any experience with rigorous math. Here's a little self-test to see if you're well-prepared (all true/false).</p>
<ol>
<li>If a function f:R->R is continuous everywhere then it's differentiable everywhere.</li>
<li>The function f(x) = {0 if x is an odd integer; 1 otherwise} isn't Riemann integrable.</li>
<li>A function f:[a,b]->R is Riemann integrable (on [a,b]) if and only if it's continuous.</li>
<li>There exists a convergent, increasing sequence of reals that is bounded above.</li>
<li>If f<em>n : R->R are continuous function, and if lim f</em>n(x) = f(x) for all x, then f is continuous.</li>
<li>There are 'more' rational numbers than there are negative integers.</li>
<li>There are as many real numbers as there are irrational numbers.</li>
<li>The rationals are dense in the reals.</li>
<li>The series 1 + 1/2 + 1/3 + 1/4 + ... diverges. As does the series 1 + 1/2^2 + 1/3^2 + 1/4^2 + ....</li>
<li>If we are told that the series 1 - 1/2 + 1/3 - 1/4 + ... converges to S, then S = 1 + 1/3 - 1/2 + 1/5 - 1/4 + ....</li>
<li>Every re-arrangement of the above series must converge (but not necessarily to the same thing).</li>
<li>In the reals with Euclidean distance, every Cauchy sequence converges.</li>
<li>If the power series SUM a<em>n x^n has radius of convergence R, then SUM a</em>n R^n must converge.</li>
<li>Every continuous function f: (0,1)->R attains its bounds (supremum and infimum).</li>
<li>No continous function f: (0,1]-> attains its bounds.</li>
<li>Every sufficiently differentiable function has a unique Taylor series (that converges on some interval).</li>
<li>If f is Riemann integrable, then INTEGRAL f(x) dx is differentiable.</li>
<li>In the reals, the subset [0,1] is compact and connected.</li>
<li>If a<em>n is a sequence of reals such that lim a</em>(n+1) - a<em>n is finite, then lim a</em>n is finite as well.</li>
</ol>
<p>Enjoy. ;)</p>
<p>As for your other question, I have no experience with stats, but I'd guess anything to do with probability or Markov processes/chains would be useful.</p>
<p>I'm not a math major, but I like to joke that I'm a pseudo-math major (because of the amount of math I take/learn/etc) I could probably get a math minor if it weren't for unit requirements.</p>
<p>Real analysis can be a tough course. People say it's because of the fact you have to write proofs, but I'm not sure I really believe that. Proofs are just like showing your work in a regular old math problem, except you have to justify your arguments, and the steps you take to solve the problem are significantly more important than the result itself.</p>
<p>That test that devious posted looks helpful, but you have to keep in mind that some of the questions would probably look a little confusing until you take real analysis. For example, most people who don't take real analysis wouldn't realize that there were different types of integration other than Riemann integration (a la Lebesgue), so the term "Riemann integral" might seem a bit redundant.</p>
<p>I would be sure that you're completely comfortable with calculus, including multivariable. Perhaps a course like Number Theory could prepare you well for a proof-intensive course like real analysis, also.</p>
<p>As for probability, an upper-division prob. course like a junior/senior math major would take would probably be something to consider, either instead of or alongside sort of an application-oriented course like an econ major would take.</p>
<ol>
<li>There are 'more' rational numbers than there are negative integers.</li>
<li>There are as many real numbers as there are irrational numbers.</li>
</ol>
<p>How difficult it is depends on how difficult your school makes it, and how much experience you have with proofs. At least at my school, there's an "advanced calculus" course (basically introductory analysis) that you are required to take before real analysis (if you don't do the Honors Math sequence which is a whole 'nother level of rapeage). Then there's the introductory graduate level real analysis where you develop measure theory and lebesgue integration and whatnot. You'd definitely want to make sure you took something like introductory analysis first so that you're familiar with proofs and how things in analysis work. </p>
<p>As far as statistics courses, you'd probably want to do a general advanced probability course. After that, I guess either a game theory course, or maybe something with stochastic processes.</p>
<p>"As my friend told me, Real Analysis is best taken as the only math class in a semester."</p>
<p>Lies. Real Analysis+Combinatorial Game Theory+Algebraic Topology+Representation Theory of Finite and Compact Groups+Statistical and Thermal Physics+Faust (damn lit class...).</p>
<p>To prove the first you would need to exhibit a one-to-one and onto map (a.k.a. bijection) from the rationals onto the negative integers. One way to do this is to use the classical bijection from the rationals onto the positive integers (see <a href="http://mathforum.org/library/drmath/view/52830.html)%5B/url%5D">http://mathforum.org/library/drmath/view/52830.html)</a>, then use the map n <-> -n to get the bijection from the positive integers onto the negative integers. The composition of these two bijections will give us what we need.</p>
<p>Now, the proof that there are more real numbers than there are rational numbers is due to Cantor, and one of his proofs is rather popular: Cantor's</a> diagonal argument. Using this, we can conclude that the irrationals are uncountable. To show that there are as many irrationals as reals, we have to use a small result from cardinal arithmetic.</p>
<p>For more info on this, google for countability or cardinality.</p>
<p>yeah, I researched a little more, and I got at least a couple more courses before real analysis. I'm pretty much ready for advanced probability and stats though...</p>
<p>Do you think if your simply "good" at math but not amazing you can still do reasonably well up through adv. undergad level math, or will my brain crap out at a certain point? I do feel like my brain has adapted to the sort of mathematical logic/reasoning required of me as a function of time and effort...I wonder if the brain is "plastic" enough to develop new neural pathways to allow for expanding math ability. I guess I just doubt myself because I didn't care about math in high school.</p>
<p>I don't need to be a pure math genius, as what I'm looking to do is more applied math/computer-based decision modelling than super-high level theoretical math...much of what I will do actually involves behavioral psych. However, this does mean that I need to be comfortable Reimann integration, convergents/divergents, as well as constrained/unconstrained optimization, Lagranges theorem (which i've had some exposure to through econ), Khun-Tucker, basic measure theory, experience with Mathematica, etc.</p>
<p>also, I have to admit some of my motivation is financial and not academic. I want to go into consulting, and decision modelling is highly lucrative, PLUS I'm very interested in it.</p>
<p>decision theory/behavioural economics. I read "Rational Choice in an Uncertain World" and a little bit of "Heuristics and Biases" and became very interested in the field...I also met a Decision theory consultant from MIT who basically studies gambling all day and gets paid mass amounts of cash.</p>
<p>As others have said, a lot depends on how hard your school makes it, and how much proof experience you have. If you've done almost no proofs, I would actually recommend it as the only math class in a semester if you can afford that.</p>
<p>Also, where do you go to school, dilksy? There can't be that many that offer classes in combinatorial game theory.</p>
<p>University of Michigan, though we only offer it intermittently. The professor who developed the course retired, so somebody else decided to keep it alive as a university class (the original professor still teaches it at Michigan Math and Science Scholars camp). It's always been an Inquiry Based Learning course (ie, figure crap out on your own), and this semester it counts as an upper level writing course. Only prereq is two years of high school math, so the stuff we do isn't terribly advanced. I was hoping more of the pure math majors would take it, but I guess I'm the only one. It kinda sucks when you're always the only person to figure stuff out.</p>