Results 1 to 3 of 3

Math Help - Why is the normal distribution considered unique?

  1. #1
    Member
    Joined
    Sep 2009
    Posts
    242
    Thanks
    1

    Why is the normal distribution considered unique?

    As I have been told, the normal distribution is special because of 4 criteria:

    1) The errors do not depend on the coordinate system.
    2) Errors in perpendicular directions are independent.
    3) Large errors are less likely than small errors.
    4) The sum of the distribution the (CDF) over the infinite domain equals 1.

    My question is: Aren't there a lot of bell-shaped curves out there that meet these criteria?

    For example \frac{1}{\pi(1+x^2)} meets the three first assumptions plus

    \int_{-\infty}^{\infty} \frac{1}{\pi(1+x^2)}=1

    This one is also a lot more tractable. So why is the normal distribution considered so unique and special?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,383
    Thanks
    1318
    The real importance- and "uniquness"- of the Normal distribution is this: if we have a large number, n, of trials from any probability distribution (with finite moments) the probability distribution for the mean of the trials is approximately Normal with the same mean and with standard deviation equal to the standard deviation of the original distribution over the square root of n. That is, if you take a large enough sample the Normal distribution will work no matter what the "underlying" distribution is.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Sep 2009
    Posts
    242
    Thanks
    1
    That's the CLT you're referring to if I'm not mistaken.

    You say "approximately normal." Could you just as well have said "approximately bell-shaped"? Or is there something special about 1/exp(-1/2 etc.) that makes the CLT true?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. normal distribution prior and posterior distribution proof
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: March 9th 2011, 06:12 PM
  2. Replies: 2
    Last Post: March 29th 2010, 02:05 PM
  3. Replies: 3
    Last Post: January 22nd 2009, 05:55 PM
  4. Replies: 1
    Last Post: April 20th 2008, 06:35 PM
  5. Using skewed distribution vs Standard Normal Distribution
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: September 26th 2007, 05:22 PM

Search Tags


/mathhelpforum @mathhelpforum