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Math Help - Multivariate normal Distrubution

  1. #1
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    Multivariate normal Distrubution

    We all know about the pdf of a multivariate Gaussian distribution which is written using covariance matrices.

    Kindly somebody give me a referece where I can find a derivation of that pdf or atleast a mathematical explanation of how the generalization to the multivariate case is made ..

    thanks

    UJJWAL
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  2. #2
    Moo
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    Hi,

    I guess the origins of the pdf are not that mysterious. It's just a generalization of what happens in the unidimensional case.
    For some hints on how to remember the pdf in the multivariate case :
    - If you ou replace the square root of the standard deviation by its equivalent : the square root of the determinant of the covariance matrix, because it has to be a unidimensional value and it kind of summarizes what happens in this covariance matrix.
    - \sqrt{2\pi} appears every time there is a normal distribution, so you will multiply \sqrt{2\pi} as many times as there "are" normal distributions in the Gaussian vector.
    - Usually, we have (x-\mu)^2. If you have studied a bit quadratic forms, you will know that the square is a "special case" of the product of transposes.


    As for a (more) formal approach (mine obviously isn't ), you may want to have a look at the book mentioned here : http://www.gbv.de/dms/goettingen/229762905.pdf (section 18.1), although I couldn't find it elsewhere than amazon.com.
    And this : Multivariate normal distribution: Encyclopedia - Multivariate normal distribution looks nice, but I haven't seen any further because there are many links to read...
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    Hi,
    With due respect, I don't have any problems in memorizing the distribution. What I want to know is that how do we generalize from the univariate to the multivariate case ? And how does the final distribution come like this ?
    You may understand that it is not very clear from the univariate distribution, that how and why the multivariate distribution should look like that ...
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  4. #4
    Moo
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    Oh sorry, I thought you didn't know how it could relate

    Well I looked for formal explanations, and the links I put above are the ones that seemed the most interesting The first reference seems to be the best.

    And I also found this : http://www.stanford.edu/class/cs229/...-gaussians.pdf, which explains kindly what gaussian vectors are, basic properties, some links with the univariate, but unfortunately, no history.
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