4.11 TRANSFORMATIONS OF RANDOM VECTORS
The previous techniques for transforming one random variable to another random variable can be extended to multiple random variables. We will not derive all the results in detail, but instead present the main ideas behind transformations of continuous random variables, and then summarize the results for various types of transformations, including those for discrete random variables.
Consider random vector 
and its transformation to random vector 
according to 
where 
are known functions (possibly nonlinear). These functions could be strictly increasing, strictly decreasing, nonmonotonic, may contain flat regions, or may have some combination of these features over various intervals. The functions might also map to a subset of the elements of 
to 
. We first consider the case where the N functions are either strictly increasing or strictly decreasing. For ...
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