My Math Forum Problem with transformation

 March 22nd, 2010, 11:49 PM #1 Newbie   Joined: Mar 2010 Posts: 1 Thanks: 0 Problem with transformation I have problem involving the 2-norm of a Gaussian vector. say, $X \sim N( \mathbf{x}, \ W )$ where $\mathbf{x}$ is the mean vector and $W > 0$ the covariance matrix. i can transform this to another Gaussian vector say $Y=W^{-1/2}*X \ \ \ ----> \ \ \ Y \sim N( W^{-1/2}*\mathbf{x}, \ I )=$. now $Y$ has an identity matrix as covariance matrix, and the distribution of the 2-norm $||Y||_2$ has a generalized rayleigh distribution. but the problem i am having is say suppose the original problem is finding the probability of the 2-norm of $||X||_2$ $P( \ || X ||_2 \ < \ \mathbf{a} \ ) \ < \ \mathbf{b}$ for some constant $\mathbf{a}$ and probability $\mathbf{b}$ =====> $P( \ || Y ||_2 \ < \ h(\mathbf{a}) \ ) \ < \ \mathbf{b}$ is $\mathbf{a}$ the same in both probability statements or has to be transformed? if $\mathbf{a}$ has to be transformed then how do i transform the $\mathbf{a}$ in the first probability statement for $||X||_2$ into another constant $h(\mathbf{a})$ for the probability statement of $|| Y ||_2$? i know the closed form pdf of $||Y||_2$ and the cdf as well, but i need to transform $\mathbf{a}$ by $h(\mathbf{a})$ if i have to. any help is deeply appreciated. please cite a reference(s) where i can find help on this problem. thank you.

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