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Math Help - three i.i.d. Gaussian random variables - joint pdf

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    three i.i.d. Gaussian random variables - joint pdf

    1) you have a 3x1 vector x of three i.i.d. Gaussian random variables.
    you have a matrix A which is k x 3, where k=2, 3, 4, and in all cases the
    coefficients of A are such that its rank is the minimum between k and 3.
    Write the joint p.d.f. of y=Ax for al the values of k.
    (Note: k=2,3 are easy but k=4 requires some thought)

    2) Let x be a vector of variables, A a matrix and y a given vector. What is the solution of min_x ||Ax-y||^2
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  2. #2
    Grand Panjandrum
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    Re: three i.i.d. Gaussian random variables - joint pdf

    Quote Originally Posted by mabdulahad View Post
    1) you have a 3x1 vector x of three i.i.d. Gaussian random variables.
    you have a matrix A which is k x 3, where k=2, 3, 4, and in all cases the
    coefficients of A are such that its rank is the minimum between k and 3.
    Write the joint p.d.f. of y=Ax for al the values of k.
    (Note: k=2,3 are easy but k=4 requires some thought)
    The covariance matrix of X=[x_1,x_2,x_3]' is the diagonal matrix \Sigma with \sigma_1^2, \sigma_2^2, \sigma_3^3 down the diagonal.

    \text{CoVar}(y)=\text{E}((y-\overline{y})(y-\overline{y})')=\text{E}(A(x-\overline{x})(x-\overline{x})'A')=A \Sigma A'

    CB
    Last edited by CaptainBlack; November 13th 2011 at 06:39 AM.
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    Re: three i.i.d. Gaussian random variables - joint pdf

    but what is the joint probability for k=2 , 3 and 4??? can you give detail solution?
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    Grand Panjandrum
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    Re: three i.i.d. Gaussian random variables - joint pdf

    Quote Originally Posted by mabdulahad View Post
    but what is the joint probability for k=2 , 3 and 4??? can you give detail solution?

    The joint distribution is the multivariate normal with mean:

    A\overline{x}

    and covariance matrix:

    R=A \Sigma A'

    In the case k=4 the multivariate normal distribution is degenerate, see: Multivariate normal distribution - Wikipedia, the free encyclopedia

    CB
    Last edited by CaptainBlack; November 13th 2011 at 06:48 AM.
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