Pillai Probability quotTwo Functions of Two Random Variablesquot

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How to find the joint probability density function of two functions of two random variables X and Y, from the joint probability density function of X and Y is discussed here. In particular, when X and Y are independent and jointly Gaussian random variables, their magnitude and phase functions are shown to be independent, whereas the independence of the magnitude and phase functions is no longer true when X and Y are correlated Gaussian random variables. As usual, all relevant density functions are derived 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