MBTIc Math thread

[LostInNerSpace]

4) The central limit theorem, linear regression and other reasons the Gaussian is so "natural"

Useful but often used inappropriately. Ygolo, I recall you mentioning options trading. I'm sure you are aware one of the assumptions of Black Scholes is that financial prices are normally distributed. This has been shown to be incorrect. The distribution of prices might be leptokurtic but that does not mean they are normally distributed. People often try to compensate for that leptokurtosis with the log normal distribution. But the pervasiveness of Black Scholes itself probably makes it useful for valuing option premiums. According to Benoit Mandelbrot the Levy distribution might be more appropriate for financial prices. Probability and statistics is the only area of math for which I have had a real practical need. That will change before too long. The plans I have for my software will motivate me to study Math in more depth.

The way I work is to identify something that could be useful. I look for an overview to get a sense for what I can do with it. Then once I learn it I often find more uses and it leads me to look into other related things. It is a painstakingly slow process because I am forever going off on tangents.

 

[ygolo]

Useful but often used inappropriately. Ygolo, I recall you mentioning options trading. I'm sure you are aware one of the assumptions of Black Scholes is that financial prices are normally distributed. This has been shown to be incorrect. The distribution of prices might be leptokurtic but that does not mean they are normally distributed. People often try to compensate for that leptokurtosis with the log normal distribution. But the pervasiveness of Black Scholes itself probably makes it useful for valuing option premiums. According to Benoit Mandelbrot the Levy distribution might be more appropriate for financial prices. Probability and statistics is the only area of math for which I have had a real practical need. That will change before too long. The plans I have for my software will motivate me to study Math in more depth.

The way I work is to identify something that could be useful. I look for an overview to get a sense for what I can do with it. Then once I learn it I often find more uses and it leads me to look into other related things. It is a painstakingly slow process because I am forever going off on tangents.

I haven't been trading in a while. I know you mentioned a site that lets you do program trading.

But right now, I need a site or service that will allow me programatic access to past option prices for a vast array of underlying securities. I want to run automated back-tests on strategies I think of, and maybe fine tune some strategies on particular stocks that have high enough volume.

 

[Alwar]

Do any of you know of a program or web site that generates random math problems so you can practice? I am relearning math from the ground up and it would be helpful if I could just print out like 100 problems to work on.

 

[ygolo]

Do any of you know of a program or web site that generates random math problems so you can practice? I am relearning math from the ground up and it would be helpful if I could just print out like 100 problems to work on.

I don't know about a math problem generator, but Math is all around you, you can pose yourself problems from almost all life situations.

What seat gives you the biggest viewing angle in a movie theater?
What is the shortest path from a parking spot to your building?
What is the optimal path (in terms of time) to take to beat a pacman level?
etc.

Any time you want to find an optimum, you can turn it into a math problem.

----

In Good Will Hunting, there was a math problem that supposedly took the prof. 2 years to solve.

B.S.

The actual problem on the board was:
"Draw all homeomorphically irreducible trees with n=10"

This is actually quite easy. But I put it up as a challenge for people.

I'll scan my solution in after a little while. Probably tomorrow.

 

[The_Liquid_Laser]

Do any of you know of a program or web site that generates random math problems so you can practice? I am relearning math from the ground up and it would be helpful if I could just print out like 100 problems to work on.

I'd probably just get an old textbook and start working problems.

 

[Scott N Denver]

I find Schaums guides pretty practical, if generally basic in their questions.

There are also the "3000 solved problems in____" books

 

[ygolo]

I would also recommend The Art and Craft of Problem Solving, by Paul Zeitz.

Excellent book to improve your puzzle solving skills, and generally enhancing your imaginative powers when it comes to Math.

-----

Oh and the solution to the Good Will Hunting problem.

I could have made a mistake, but the approach is rather straight-forward --build upon trees of smaller n and avoid duplication just because the trees are drawn differently.

 

[musicheck]

Here's a question I've been thinking a bit about recently.

In a linear regression, the least squares estimator is also the maximum likelihood estimator when we assume the error is normally distributed. In a lot of financial work, a cauchy distribution has a better fit than a normal. What are the properties of the maximum likelihood estimator of a regression coefficient under the assumption of cauchy distributed errors, and in what cases will this lead to an answer that's noticeably different than least squares?

 

[ygolo]

Here's a question I've been thinking a bit about recently.

In a linear regression, the least squares estimator is also the maximum likelihood estimator when we assume the error is normally distributed. In a lot of financial work, a cauchy distribution has a better fit than a normal. What are the properties of the maximum likelihood estimator of a regression coefficient under the assumption of cauchy distributed errors, and in what cases will this lead to an answer that's noticeably different than least squares?

These are some interesting questions and very open ended.

It may be illustrative, first, to look at the Gaussian distribution case:

...and now, I'll analyze the Cauchy distribution case:

If you click on the images, you'll have the option of looking at bigger versions.
Gaussian Discussion
Cauchy Discussion

 

[Asterion]

This should be fairly simple (hopefully). I haven't been able to solve it after thinking about it and asking a few people:

Simplify and rationalize:
(((4+h)^.5) - 2) / h

I've acquired the answer from the book, but I can't figure out what they did to solve it :\

 

[The_Liquid_Laser]

This should be fairly simple (hopefully). I haven't been able to solve it after thinking about it and asking a few people:

Simplify and rationalize:
(((4+h)^.5) - 2) / h

I've acquired the answer from the book, but I can't figure out what they did to solve it :\

Multiply the expression by ((4+h)^.5 + 2)/((4+h)^.5 + 2). After simplifying the expression you should get your answer.

 

[Asterion]

Multiply the expression by ((4+h)^.5 + 2)/((4+h)^.5 + 2). After simplifying the expression you should get your answer.

lol, thanks, I thought of that at one point, but decided that it wouldn't work and didn't actually try it

 

[ygolo]

I wish I could speed teach Calculus, Linear Algebra and Differential Equations... so the Quantum Mechanics could be made plain. It is much simpler than I think people realize.

Anyone have ideas on how to do that?

Of course, I could be biased in that I learned classical mechanics first. Teaching that may actually be the more difficult task. This too requires Calculus, Linear Algebra and Differential Equations.

For those brave enough to attempt a crash course in the three subjects:
Calculus:
Definition of a Limit: Limit -- from Wolfram MathWorld
Definition of a derivative (requires understandning limits): Derivative -- from Wolfram MathWorld
Definition of an integral (requires understandning limits): Riemann Integral -- from Wolfram MathWorld Wolfram Mathematica Online Integrator
The fundamental Theorem of Calculus: Fundamental Theorems of Calculus -- from Wolfram MathWorld
The Second Fundamental Theorem of Calculus: Second Fundamental Theorem of Calculus -- from Wolfram MathWorld
Taylor and Maclaurin Series: Taylor Series -- from Wolfram MathWorld Maclaurin Series -- from Wolfram MathWorld

Linear Algebra:
Matrices: Matrix -- from Wolfram MathWorld
Vector Spaces:Vector Space -- from Wolfram MathWorld
Inner Product Spaces:Inner Product Space -- from Wolfram MathWorld
Orthonormal Bases: Orthonormal Basis -- from Wolfram MathWorld
Eigenvalues and Eigenvectors: Eigenvalue -- from Wolfram MathWorld Eigenvector -- from Wolfram MathWorld
Linear Transformations: Linear Transformation -- from Wolfram MathWorld

Differential Equations (very much cookbook in it's approach, just follow the recepies):
ordinary differential equations: Ordinary Differential Equation -- from Wolfram MathWorld
partial differential equations: Partial Differential Equation -- from Wolfram MathWorld
Fourier Series: Generalized Fourier Series -- from Wolfram MathWorld
Fourier and Laplace Transforms: Fourier Transform -- from Wolfram MathWorld Laplace Transform -- from Wolfram MathWorld
Euler Method (Numerical Solution method): Euler Forward Method -- from Wolfram MathWorld
Runge-Kutta Method (More sophisticated numerical solution method): Runge-Kutta Method -- from Wolfram MathWorld

 

[The_Liquid_Laser]

Lol I think you put at least three semester's worth of knowledge into one post.

 

[nemo]

For QM, I'd add a dash of complex numbers to that list.

 

[The_Liquid_Laser]

I've never really looked into QM even though I apparantly have the math background. I guess I'll have to check it out when I get the time.

 

[Ozz]

Lol I think you put at least three semester's worth of knowledge into one post.

It's 5 courses worth of math.
Calculus I, II, III
Linear Algebra
Differential Equations.

Of course, no sane person can take them all at the same time.

 

[BlueScreen]

I wish I could speed teach Calculus, Linear Algebra and Differential Equations... so the Quantum Mechanics could be made plain. It is much simpler than I think people realize.

Anyone have ideas on how to do that?

This is quite true. People have a habit of wowing and mystifying things.

I'll try to contribute to this later in the week. I think I can simplify most of it for people, if I find some motivation. Haven't used my brain properly for a while.

 

[The_Liquid_Laser]

It's 5 courses worth of math.
Calculus I, II, III
Linear Algebra
Differential Equations.

Of course, no sane person can take them all at the same time.

It's closer to three, because under the Calculus section the topics really only describe the first semester of Calculus plus some infinite series thrown in.

 

[Scott N Denver]

This is quite true. People have a habit of wowing and mystifying things.

I'll try to contribute to this later in the week. I think I can simplify most of it for people, if I find some motivation. Haven't used my brain properly for a while.

To quote a math grad student "quantum mechanics is pretty easy, its all just a linear [vector?] space". I heard a QM prof say essentially the same thing.

1) infinite-dimenionsal linear vector space [ie Hilbert space]
2) solve a modified wave equation for the particular potential [V] under question. the solution is your wave function psi, or at least a part of it [add spin or other things as needed]. this will amount to PDE, ODE, BVP type problems dealing with the Schrodinger equation
3) the construct psi star psi serves as your probability density. As a mathematical note from probability theory we assert normalization, ie the integral of psi star psi over all relevant space =1. That way when we multiply two ormore things together we still get 1 x 1...=1.
4) to compute any quantity of physical interest we compute the expectation value, [which from probability theory?] is integral [psi star * relevant operator representing quantity of interest* psi] integrated over all of relevant space. note: the allowed energy levels are an exception to the above, they come out of the SE itself. but x, x^2, p, p^2, E, etc are computed as expectation values

5) there are mathematical nuances that are relevant and can be shown, but are kinda just details for this discussion.

Overall operating motto "1) identify the potential V, 2) solve the relevant schrodinger equation to find the wave function psi 3) compute all physical quantities of interest via <psi, A, psi> where A is the operator for the physical quantity of interest

I don't know if I made things easier or harder, but there you go. :hi: