I have a 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.