What is the multivariate analogue to point 3. in this theorem? is || in here referring to absolute value? Thanks.
2026-03-26 19:38:05.1774553885
Pdf upon transformation of multivariate random variable
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If $g$ is a one-to-one function on the support of $X$, then the pdf of $Y = g(X)$ is $$ f_Y(y) = f_X(g^{-1}(y)) |\det(J(g^{-1}(y)))|$$ for $y$ in the range of $g$, where $J(x)$ is the Jacobian matrix of $g$ at $x$.
Yes, $| \cdot |$ means absolute value.