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  1. #101
    Senior Member LostInNerSpace's Avatar
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    Quote Originally Posted by ygolo View Post
    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.

  2. #102

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    Quote Originally Posted by LostInNerSpace View Post
    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.

    Accept the past. Live for the present. Look forward to the future.
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    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
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  3. #103
    The Architect Alwar's Avatar
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    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.

  4. #104

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    Quote Originally Posted by Alwar View Post
    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.

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  5. #105
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    Quote Originally Posted by Alwar View Post
    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.
    My wife and I made a game to teach kids about nutrition. Please try our game and vote for us to win. (Voting period: July 14 - August 14)
    http://www.revoltingvegetables.com

  6. #106
    Senior Member Scott N Denver's Avatar
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    I find Schaums guides pretty practical, if generally basic in their questions.

    There are also the "3000 solved problems in____" books

  7. #107

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    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.
    Attached Images Attached Images

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  8. #108
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    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?

  9. #109

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    Quote Originally Posted by musicheck View Post
    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

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  10. #110
    Ruler of the Stars Asterion's Avatar
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    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 :\
    5 3 9

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