My question is if we could define, for example, a diffeomorphism $g: A \rightarrow B$ in the following manner: $X_{1},X_{2},X_{3}$ are independent random variables where $X_{1},X_{2} \sim N(0,1)$ and $X_{3} \sim \exp(\lambda)$. Can we now define $U=X_{1}+X_{2}+X_{3}$ and $\frac{X_{2}+X_{3}}{\sqrt{X_{1}^{2}}}$ consider a transformation defined as $g(x_{1},x_{2},x_{3}) = (u,v)$ where $(u,v) = \Big( x_{1}+x_{2}+x_{3},\frac{x_{2}+x_{3}}{\sqrt{x_{1}^{2}}} \Big)$? If not, is there another way to determine the joint distribution $f_{U,V}(u,v)$? How would one go about this? Can we use a diffeomorphism and a Jacobian matrix?
2026-03-29 06:02:28.1774764148
Determining a joint distribution of 3 random variables using transformation
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