Detour

A post I wrote three years ago is still getting me a couple of comments/emails a day in regard to obtaining copies of solution manuals. In response I have set up a bulletin board to help facilitate the trading, begging for, and bartering of solution manuals. I am hoping that the site can become a solutions manual trading post on the net for needy graduate and undergraduate students to find eachother and offer whatever value they can to those lucky few that are in possession of the holy grail for students. Happy searching.

I was looking over a Casella & Berger, “Statistical Inference” 7th edition and have been pondering a question all night as a result.

A finite set is defined as…

A set S is finite if it has the same cardinality as some natural number n in N (The set of all natural numbers) . We then define |S| = n and say that S has n elements. A set is infinite if it is not finite.

A countable set is defined as…

A set S is countable if |S| = |N|. A set S is at most countable if |S| ≤ |N|.

Thus a set S is countable if there is a one-to-one mapping of N onto S, that is, if S is the range of an infinite one-to-one sequence.

Otherwise, the set is uncountable.

My proof skills have always been lackluster, and it has been well over a year since I last attempted a proof, so I was hoping someone who has some recent practice with proofs, set theory, or probability theory could help me figure out an example of an uncountable finite set. I am of the mind, at the moment, that there is no such thing. Casella & Berger give no example…does this mean such a set is not possible? A Google search of “finite uncountable set” and uncountable finite set” yielded less then lackluster results.

Update: Kerry mentions in the comments section that it is a bit strange that I do proofs in my spare time…Kerry prefers to bowl…Vinny likes to update his blog layout. To each their own!

The realization that I have a true disdain for mathematical proofs came to me late in my educational career. I am convinced that this could have been avoided by a formal and structured introduction to them. I sometimes wish I had paid more attention to my mathematics education in grammar school and beyond. Trying to pick up proofs in my first year of an economics PhD was a recipe for disaster. I took tons of math as an undergrad, but somehow avoided doing all but the most trivial proofs throughout my acquisition of a degree in statistics. Hell, when a proof was listed in any of my textbooks I would skip right over them…’they aren’t material’, I would think to myself. It was eye-opening for me to find out that the proofs had a structured language. I always thought that epsilon, delta, etc. were just pulled out of thin-air to confuse the reading of proofs. The thought that delta almost always means the change in just didn’t occur to me and probably explains why I never took the time to read the gibberish.

Adam Osman has a link to a graduate mathematics student named Fernando Q. Gouva written to convey what he learned by teaching real analysis to undergrads. After reading the article I am convinced that Fernando will b e an excellent professor of mathematics. Teaching and thinking about how to teach complex subjects often takes a back-seat to research prowess in the hiring and retaining of professors. So to see a graduate student actively involved in introspection about how to better teach a subject is refreshing. Fernando almost made me want to pick up some of my advanced mathematics texts and have fun for the first time in nearly a year. Cool. Check out the article if you have the time and inclination.

I don’t miss doing math for hours upon hours each day. An economics Ph.D seems to have an endless string of equations and proofs in the first year. I look at old freinds websites, such as this, and thank god I left that behind. I never found joy in math equations and proofs and will not miss that part of graduate school life.

One thing I do miss about my old life as a student is writing. I used to write everyday. Whether it was my blog, ideas in a journal, or notes in the margin of whatever book I was reading at the time. I just don’t write anywhere near the amount I used to when I was in school. I miss it.

What I can’t figure out is why this is. I have way more free time then I used to. I haven’t lost my desire to explore ideas…I just don’t do it as much anymore. I have to make a determined effort to change this.

First, I need to understand why this has happened?

Chris Silvey

My God!

Link added on the strength of one post….

Economic Majors Soliloquy

To attempt a PhD, or not to attempt a PhD:
That’s a really good question.
Whether ’tis advisable to tackle the math
And statistics from outrageous curricula,
Or to accept a lucrative job offer,
And by avoiding those rigours,
Make a good income? To play, to relax:
No more; and by finishing to say we end
The fatigue and social-isolation
That an industrius student is heir to, ’tis a relief
An object of our lust.

Good job David!

My advice…If math gives you pause, then don’t do it. Otherwise, do it.

Ian (a married PhD first year at Cornell) responds to the bottom two quotes…

I feel like I have to comment on both of these. I just attended a presentation by one of my (married) fellow students here at Cornell who has some very good data on PhD programs in the humanities. He found that married men are more likely to finish (rather than quit) a PhD and to finish early. Unfortunately, his data doesn’t include experience in the job market, so maybe married students really do write crappier dissertations, as the poster suggests. And, of course, this is for students in the humanities, where the average completion rate is like 8 years, or something. The point is that there is reason to think that married students fare as well as their single counterparts. Of course, you have to be sure that your significant other is ready to suffer with you through the privations of graduate school life; something that is all the more difficult, I think, in Ithaca.

Which brings me to my second point, in response to Doug, who is worried about what to do between now and math camp. My advice would be not to get too freaked out about the work at this point. There’ll be time for that when you get there. I am only in my first year (at Cornell), so I haven’t seen a lot, and I don’t know what it’s like at the higher-ranked schools, but plenty of people make it through the first year of an econ PhD, even if they aren’t math gods. Being a math god helps, but the reality is that if you work hard, and stay happy, you can learn what you need to learn to do what you want to do. I think Chris’ experience is instructive, in that he left grad school for the reason I think most people do: unhappiness. Sure, some people fail out, but most people who quit do so because they decide they don’t like what they are doing. Studying dynamic programming or constrained optimization for the next six months isn’t going to make you like grad school more, so don’t do it. Spend the next six months enjoying life. If you want to do something related to economics, start reading the paper, watching TV, and talking to friends to find out what makes things tick. Write down every idea you have about economics, because once you start a program, you won’t have time to think about it. Then the next thing you know, you have to start writing papers, and if you don’t have a good idea, it’s going to be haaard.

John Morrow, a first year at Wisconsin, is developing quite a nice site for the econ grad community. He has a large collection of solutions manual’s (including the ever elusive digital copies of Mas-Colell, Whinston, Green Microeconomics Solutions Manual - YEAH!). Go to his site and grow the economics PhD community to an international movement.

Good job John! I just wonder how long it will take for the publishers, authors, or prof’s to try and clamp down on the solutions trade. I hope I haven’t helped that process by announcing the availability.

I will no longer be pursuing a PhD in economics.

The withdrawal papers were signed today and I have resigned to the thought that I will never be called Prof., Dr., or have the letters PhD in any way associated with my name. There are many reasons for this decision and I make it with a heavy heart. The main reason for my decision is the fact that my wife is miserable in Ithaca, New York and didn’t want to come back after the winter break. Upon a thorough discussion we decided that five years of poverty in Upstate New York just isn’t worth the cost of postponing family and happiness. I will write more about the tertiary reasons for my decision. For now, I thought I would let you know why I haven’t been very prolific lately.

Don’t underestimate the hardship that pursuing a PhD inflicts on spouses and other immediate family. Although it is surely possible to achieve your goals, make sure your partner is happy and in some way connected to a life of sporadic solitude and abandonment as Kuhn-Tucker, Euler, and whoever invented Log-Linearization steals your loved one.

Update: There is a good discussion going on over at testmagic about whether a Phd is worthwhile or just for dreamers. I think the answer varies and is dependent on many extremely personal criteria. Check out the discussion and give your own input.

If the link does not take you directly to the discussion, ou can find it at…

TestMagic Forum > University Admissions Forum > University Admissions: Graduate > Why do you guys do a PhD?

Update: David Tufte has some words about graduate school success…

Condolences. I have three contributions. Also, see my post on what determines success in graduate school.
Something like this can happen at any time in your career. It happened to me the year I went up for tenure: fixer-upper house, surprise pregnancy, administrative position, family health problems. The upshot is that a research program that was really starting to take off is just about dead. All the work on my serious publications was finished by summer of 1997. I’m not making a drastic life-change like you are, but the career I loved is mostly collecting dust in file cabinets while I struggle to find time to teach classes, go to meetings, and occasionally make a post. You may be better off going through this now.
When we eloped to Lake Tahoe (in my first tenure-track year) the casino had a wedding photographer on retainer. He offered one of the best lines about life I have ever heard: “In another lifetime I was a (botany) professor.” He was 50-ish, and had no regrets about either having been a professor, or having walked away from it all. So, chalk it up as another entry on your CV.
I grew up in upstate New York (suburban Buffalo). I both love it, and recognize that it is an appalling place for 7 months out of the year. I didn’t give up my career as the botany professor did (although I’m sorely tempted sometimes), but I did make a conscious choice to move to where I wanted to live (the rural intermountain southwest) instead of where I wanted to work (a research school). Having done that, I feel like I where a teflon shield about job problems: disappointing students (no problem, go for a hike), nutty colleagues (no problem, go for a drive), useless administrative tasks (no problem, go up in the mountains). I think you’re making the right decision (but beware if your wife is someone who just has trouble being happy).

Law of the Unconscious Statistician

Def.: If E|g(X)| = ∞ then Eg(X) does not exist.
Cauchy Mean is an example of a random variable whose expected value does not exist.
f(x) = (1/pi)(1/(1+ x^2), -∞ < x < ∞
E(x) = ∞

I wish every law, theorem, and lemma in math and economics were named in such a memorable way. I would have a much easier time remembering what Farka’s Lemma was if it was named the vector idiotic tradition lemma or some such other inane title.

Update: Cecilia Cotton has a multpile choice test to determine if you understood the above law.

“I’m not an outlier, I just haven’t found my distribution yet.” Nice.

Update: Jacqueline thinks objects in multivariable calculus have such funny names.

Adam @ SUNY Stony Brook has started blogging about his journey toward becoming a bona fide economist. Good Luck!