Problem with transformation

I have a problem involving the 2-norm of a Gaussian vector.

say, where is the mean vector and the covariance matrix.

i can transform this to another Gaussian vector say .

now has an identity matrix as covariance matrix, and the distribution of the 2-norm 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

for some constant and probability

is the same in both probability statements or has to be transformed?

if has to be transformed then how do i transform the in the first probability statement for into another constant for the probability statement of ?

i know the closed form pdf of and the cdf as well, but i need to transform by if i have to.

any help is deeply appreciated.

please cite a reference(s) where i can find help on this problem.

thank you.