Self-Study Outline

[systemtrader]

You survived "a" grad school. I know people that studied literature in grad school too and are extremely, extremely intelligent people. But they can't solve a quadratic equation if their life depended on it. If you are wowed by the MATLAB, I'm afraid you don't really have what it takes to be a real quant. Everybody I've known with an ounce of mathematical talent, can learn to do some pretty complicated shit in MATLAB in just a month.

Mathematical talent is not the only or most important thing. You should know this, having worked for in finance. It's quite common for people to glamourize some other job though. I'm always surprised by how many quants secretly want to be professors, as if that's really a great job. As a former professor, I can tell you it's a crap job. Being a quant in finance is soooo much better. It helps to be brainwashed and think the opinion of the 4 other people in the universe that can actually understand your result is more important than thousands of people using your work.

It takes a certain level of competence to realize what competence is. There are a lot of people that look at quant math and think "It's only calculus and linear algebra. I got A's in those courses!" and miss the point. The reason PhDs are desired in quant roles doesn't have to do with their advanced knowledge. In fact, their basic math skills actually deteriorate (if you spend all your time thinking about what a Grothendieck category is, that's not gonna help you compute eigenvalues!). It's because they know how to think properly. I demonstrated this recently to a friend. He asked me what kind of math skills are required to be a quant. I asked him to show that e^pi > pi^e. He took one day to show it. I remember taking 10 minutes, when the problem was posed to me. This might be 5 minutes longer than would be expected at a BB interview. It takes only college freshman level math skills but somehow most people that took calculus can't manage it.

lol... you were a professor? you're the first professor ive seen who insults and discourages someone who wants to learn new stuff. maybe there's a reason you didn't like being a professor? the guy clearly said he just wants to learn some basics of quant, not become a hardcore quant. is there a rule that only math whizs and soon-to-be hardcore quants should learn quant finance? what's more funny is that you took all that time to write that essay just to insult someone who wants to learn some quantitative finance.

 

[bigbadwolf]

I asked him to show that e^pi > pi^e. He took one day to show it. I remember taking 10 minutes, when the problem was posed to me. This might be 5 minutes longer than would be expected at a BB interview. It takes only college freshman level math skills but somehow most people that took calculus can't manage it.

Going off-topic but this question has been asked at Oxford entrance interviews for undergrads. I "solved" it in a couple of minutes: take natural log on both sides and compare pi and e.ln pi. pi can be rewritten as e.(pi/e), which means we're essentially comparing pi/e and ln pi. A simple calc argument can be invoked to show that pi/e > ln pi. QED.

 

[Daniel Duffy]

Going off-topic but this question has been asked at Oxford entrance interviews for undergrads. I "solved" it in a couple of minutes: take natural log on both sides and compare pi and e.ln pi. pi can be rewritten as e.(pi/e), which means we're essentially comparing pi/e and ln pi. A simple calc argument can be invoked to show that pi/e > ln pi. QED.

Schkool maths?

 

[bigbadwolf]

School maths?

Well, if someone has really mastered "A" level math and further math, he or she should be able to do this or sketch a line of reasoning. The interviewers are trying to distinguish the sheep from the goats.

 

[Daniel Duffy]

A similar one is compute exp(5) as fast as possible with pen and paper:

http://www.wilmott.com/messageview.cfm?catid=26&threadid=79451&FTVAR_MSGDBTABLE=

 

[Daniel Duffy]

Going off-topic but this question has been asked at Oxford entrance interviews for undergrads. I "solved" it in a couple of minutes: take natural log on both sides and compare pi and e.ln pi. pi can be rewritten as e.(pi/e), which means we're essentially comparing pi/e and ln pi. A simple calc argument can be invoked to show that pi/e > ln pi. QED.

We can write the inequality conjecture as:

pi/log(pi) > e

Some students might say:

1. Examine the function f(x) = x/log(x).
2. Prove f(x) has min or max at x = e (df/dx = 0) [d^f/dx^2 looks inconclusive?] (this result is needed for TChebychev's Theorem on f(x)).
3. Prove that f(x) is monotonically increasing for x >= e
4. e < pi
5. QED

But check the steps 1-4 once again. (a la Georg Polya style).

6. exp(x) > x^exp(1) for all x >= exp(1).

7. And "C sanity check"

8. For large x, x/log(x) is the Prime Number Theorem, i.e. the number of primes less than or equal to x
http://en.wikipedia.org/wiki/Prime_Number_Theorem

9. If you are at a loose end over the Christmas period, you could try writing down the proof of Goldbach's Conjecture http://en.wikipedia.org/wiki/Goldbach's_conjecture

long N = 10000;
double h = 1.0/double(N);
double e = 2.71828182846;
double x = e;
for (int n = 1; n <= N; ++n)
{
    if (x/std::log(x) <= e)
    {
        std::cout << "oops\n";
    }
 
    x += h;
}

 

[Daniel Duffy]

DKoir,
Apologies for the drift.

 

[Daniel Duffy]

I asked him to show that e^pi > pi^e. He took one day to show it. I remember taking 10 minutes, when the problem was posed to me. This might be 5 minutes longer than would be expected at a BB interview. It takes only college freshman level math skills but somehow most people that took calculus can't manage it.

You can view it as a calculus problem and you are done. Very arid.
Or as a deeper problem in mathematical analysis that test your powers of abstraction.

 

[bigbadwolf]

We can write the inequality conjecture as:

pi/log(pi) > e

Some students might say:

1. Examine the function f(x) = x/log(x).
2. Prove f(x) has min or max at x = e (df/dx = 0) [d^f/dx^2 looks inconclusive?] (this result is needed for TChebychev's Theorem on f(x)).
3. Prove that f(x) is monotonically increasing for x >= e
4. e < pi
5. QED

It can be done more easily the way I'm suggesting, where pi/e and ln pi have to be compared. Just take the two corresponding functions x/e and ln x. Note that both coincide in value at x = e. Observe that the derivative of x/e is 1/e and derivative of ln x (for x > e) is 1/x; observe that for x> e, 1/e > 1/x, and. Follows that x/e > ln x for x > e and in particular pi/e > ln pi, and so e^pi > pi^e.

 

[Daniel Duffy]

It can be done more easily the way I'm suggesting, where pi/e and ln pi have to be compared. Just take the two corresponding functions x/e and ln x. Note that both coincide in value at x = e. Observe that the derivative of x/e is 1/e and derivative of ln x (for x > e) is 1/x; observe that for x> e, 1/e > 1/x, and. Follows that x/e > ln x for x > e and in particular pi/e > ln pi, and so e^pi > pi^e.

In your opinion, of course

Another answer is that x/logx is well-known in analytic number theory and the question is a special case thereof. But that might come across as cheeky in a job interview.

 

[bigbadwolf]

There is another more controversial way of solving it. A state legislature (Indiana? Illinois?) attempted in the late 19th century to legislate the value of pi as 3. Suppose they had also legislated the value of e as 2. The question would then be to determine whether 2^3 was greater than 3^2.

 

[Daniel Duffy]

There is another more controversial way of solving it. A state legislature (Indiana? Illinois?) attempted in the late 19th century to legislate the value of pi as 3. Suppose they had also legislated the value of e as 2. The question would then be to determine whether 2^3 was greater than 3^2.

Cool. This could be learned by rote.

 

[DKoir]

Daniel - No worries, I think I got mostly what I needed.

Systemstrader - looked at the price tag for the CQF . 19K seems excessive, 10K ish would have been more reasonable. I'll be paying off all my student loans this year so I think I will self study for the next 12 months and see where I'm at. Depending on where I get on my own, my bonus and what my cost of living ends up being over this next year; I might possibly do it next year.