Transforming Random Variables from Correlated to Uncorrelated
Assume that μx, σx, and Cx represent a vector of mean values, a vector of standard deviations, and covariance matrix respectively, of a set of dependent random variables x, denoted as x1, x2, …, xn. It is desired to obtain μv, σv, and Cv, where the new Cv is a diagonal matrix, i.e. off-diagonal elements are all zero.
In terms of two variables, μx, σx, and Cx will be:
where ρ is the correlation coefficient, defined as 
, and .
It is required to get
To obtain the uncorrelated ...
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