structural:summarization
Vector to vector

Suppose x is an n -dimensional random variable with joint density f. If y = G (x), where G is a bijective, differentiable function, then y has density p: p Y (y) = f (G − 1 (y)) | det | {\displaystyle p_{Y}(\mathbf {y})=f{\Bigl (}G^{-1}(\mathbf {y}){\Bigr)}\left|\det \left\right|} with the differential regarded as the Jacobian of the inverse of G (⋅), evaluated at y.

For example, in the 2-dimensional case x = (x, x), suppose the transform G is given as y = G (x, x), y = G (x, x) with inverses x = G (y, y), x = G (y, y). The joint distribution for y = (y, y) has density p Y 1, Y 2 (y 1, y 2) = f X 1, X 2 (G 1 − 1 (y 1, y 2), G 2 − 1 (y 1, y 2)) | ∂ G 1 − 1 ∂ y 1 ∂ G 2 − 1 ∂ y 2 − ∂ G 1 − 1 ∂ y 2 ∂ G 2 − 1 ∂ y 1 |. {\displaystyle p_{Y_{1},Y_{2}}(y_{1},y_{2})=f_{X_{1},X_{2}}{\big (}G_{1}^{-1}(y_{1},y_{2}),G_{2}^{-1}(y_{1},y_{2}){\big)}\left\vert {\frac {\partial G_{1}^{-1}}{\partial y_{1}}}{\frac {\partial G_{2}^{-1}}{\partial y_{2}}}-{\frac {\partial G_{1}^{-1}}{\partial y_{2}}}{\frac {\partial G_{2}^{-1}}{\partial y_{1}}}\right\vert.}