# Problem with transformation

• Mar 22nd 2010, 11:16 PM
mathflete
Problem with transformation
I have a problem involving the 2-norm of a Gaussian vector.

say,$\displaystyle X \sim N(\ \mathbf{x}, \ W \ )$ where $\displaystyle \mathbf{x}$ is the mean vector and $\displaystyle W > 0$ the covariance matrix.
i can transform this to another Gaussian vector say $\displaystyle Y = W^{-1/2}*X \ \ \ ----> \ \ \ Y \sim N( \ W^{-1/2}*\mathbf{x}, \ I \ )$.
now $\displaystyle Y$ has an identity matrix as covariance matrix, and the distribution of the 2-norm $\displaystyle ||Y||_2$ 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 $\displaystyle ||X||_2$

$\displaystyle P( \ || X ||_2 \ < \ \mathbf{a} \ ) \ < \ \mathbf{b}$ for some constant $\displaystyle \mathbf{a}$ and probability $\displaystyle \mathbf{b} \ \ \ ----> \ \ \ P( \ || Y ||_2 \ < \ h(\mathbf{a}) \ ) \ < \ \mathbf{b}$

is $\displaystyle \mathbf{a}$ the same in both probability statements or has to be transformed?

if $\displaystyle \mathbf{a}$ has to be transformed then how do i transform the $\displaystyle \mathbf{a}$ in the first probability statement for $\displaystyle ||X||_2$ into another constant $\displaystyle h(\mathbf{a})$ for the probability statement of $\displaystyle || Y ||_2$?

i know the closed form pdf of $\displaystyle ||Y||_2$ and the cdf as well, but i need to transform $\displaystyle \mathbf{a}$ by $\displaystyle h(\mathbf{a})$ if i have to.

any help is deeply appreciated.

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

thank you.