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I try to derive the probability density function (PDF) of the random variable $$S=\frac{ AX+BY+CZ+D }{ EX+FY+GZ+H }.$$ We know that the random variables $X, Y, Z$ are independent and each of them satisfies Gaussian distribution. I do not know how to derive the PDF of $S$. Any kind of help would be appreciated.

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If $X,Y,Z$ are independent and Gaussian, then $P = AX+BY+CZ+D$ and $Q = EX+FY+GZ+H$ are both Gaussian, but not (necessarily) independent. $(P,Q)$ will jointly follow a bivariate normal distribution.

The problem therefore reduces to determining the distribution of the quotient of a pair following a bivariate normal distribution. This is known as a correlated non-central normal ratio. As you can see, the wiki page doesn't offer an exact solution for the general case but some approximations.

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