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How to derive the multivariate normal distribution

  1. Jul 18, 2009 #1
    If the covariance matrix [itex]\mathbf{\Sigma}[/itex] of the multivariate normal distribution is invertible one can derive the density function:

    [itex]f(x_1,...,x_n) = f(\mathbf{x}) = \frac{1}{(\sqrt(2\pi))^n\sqrt(\det(\mathbf{\Sigma)}}\exp(-\frac{1}{2}(\mathbf{x}-\mathbf{\mu})^T\mathbf{\Sigma}^{-1}(\mathbf{x}-\mathbf{\mu}))[/itex]

    So, how do I derive the above?
     
  2. jcsd
  3. Jul 18, 2009 #2
    Start with a normal distribution where all the variables are independent and then do a change of variables.
     
  4. Jul 18, 2009 #3
    I was on that track before, make use of the CDF and then differentiate back to get the PDF. This is how far I get: Let Y be a standard i.i.d. Gaussian vector. Then use the transformation

    [itex]
    \mathbf{X} = \mathbf{A}\mathbf{Y} + \mathbf{\mu}
    [/itex]

    [itex]
    P(\mathbf{X} < \mathbf{x}) = P(\mathbf{A}\mathbf{Y} + \mathbf{\mu} < \mathbf{x}) = P(\mathbf{Y} < \mathbf{A}^{-1}(\mathbf{x}-\mathbf{\mu}))
    [/itex]
    Now I differentiate this to get the PDF

    [itex]
    f_{\mathbf{X}}(\mathbf{x}) = f_{\mathbf{Y}}(\mathbf{A}^{-1}\mathbf{x-\mu})\det(\mathbf{A}^{-1}) = f_{\mathbf{Y}}(\mathbf{A}^{-1}\mathbf{x-\mu})\frac{1}{\det(\mathbf{A})} = \frac{1}{(2\pi)^{n/2}\det(A)}\exp\left(\frac{1}{2}(\mathbf{x-\mu})^{T}(\mathbf{AA^T})^{-1}(\mathbf{x-\mu})\right)
    [/itex]

    So [itex]\det(\mathbf{A})}[/itex] pops out in the denominator, instead of [itex]\det(\mathbf{AA^T})}[/itex] it as it should be. Something is wrong in my differentiation here but I can't figure it out.
     
  5. Jul 19, 2009 #4
    Why do you think the denominator should be [itex]\det(\mathbf{AA^T})}[/itex].

    That would give you something analogies to the variance while the denominator of the Gaussian function is the standard deviation.

    You want:

    [itex]\sqrt{|\mathbf{AA^T}|}=\sqrt{|\mathbf{A}|}\sqrt{|\mathbf{A^T}|}=|\mathbf{A}|[/itex]
     
  6. Jul 19, 2009 #5
    Ok, so now it works out. [itex]\mathbf{\Sigma} = \mathbf{A}\mathbf{A}^T[/itex] is the covariance matrix. Thank you for your help!
     
  7. Jul 19, 2009 #6
    exactly! And, your welcome :)
     
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