Suppose $U, W, V, S$ are four independent normal random variables with mean $0$ and variance $1$. Let $X=W+U$, $Y=2W+S$, $Z=3W+V$. What is $f(X, Y, Z)$?
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A linear transformation of (independent) normals is always normal and fully determined by its mean and variance (resp. covariance matrix in the multivariate case).
The vector $(X, Y, Z)$ will be multivariate normal. To determine its mean and covariance matrix, you need only apply the rules for the variances and covariances of linear combinations of random variables—which you should have previously studied—combined with the information that $U$,$W$, $V$ and $S$ are independent (hence, the covariance between any pair is zero) with means 0 and variances 1.