Let $(X,Y)$ be a pair of random variables with joint pdf $f_{XY}$. Let $(U,V)$ be two random variables obtained from $(X,Y)$ by $U = u(X,Y)$ and $V = v(X,Y)$ where $u$ and $v$ are, say, nice invertible functions. Then there is a method to compute the pdf of $(U,V)$ using Jacobians, as stated, for instance, as Theorem 2.9 here: https://www.utstat.toronto.edu/mikevans/jeffrosenthal/book.pdf
I have a confusion when applying this method and haven't been able to figure out what I am missing. Obviously something very basic.
Suppose in $(X,Y)$, the second random variable $Y=X$ and is thus "spurious"; so the pdf of $f_{XY}$ is simply the pdf $f_X$. More precisely, $f_{XY}(x,y) = 0$ when $x\neq y$ and equal to $f_X(x)$ when $x=y$.
Now suppose $(U,V)$ is defined as $(aX, aY)$ for some constant $a > 0$. Once again, the "$V$" is spurious, and so the pdf of $(U,V)$ should simply be the pdf of $aX$; that is $f_{UV}(x,y) = 0$ when $x\neq y$ and $f_{UV}(x) = \frac{1}{a}\cdot f_X(\frac{x}{a})$ when $x=y$.
However, if we look at the Jacobian approach, then we get $f_{UV}(x,y) = \frac{1}{a^2}f_{XY}(x/a, y/a)$ where we got the $\frac{1}{a^2}$ as the determinant of the $2\times 2$ diagonal matrix with entries $1/a$ on the diagonal.
What am I missing in the application of the "Jacobian method"?
A clarification would be greatly appreciated. Thanks!
PS: This question arose when I was trying to figure out the pdf of a "scaled" Dirichlet distrubution. More precisely, say $(X_1, \ldots, X_k) \sim \mathrm{Dir}(\alpha_1, \ldots, \alpha_k)$ be a vector on the $k$-dimensional simplex, what is the pdf of $(aX_1, aX_2, \ldots, aX_k)$ for some scalar $a>0$. My answer and what I believe is true is off by a factor $1/a$...and I think this is because the sum of the $X_i$'s is $1$ (which means $X_k$ is determined by the others). This led to the question above. Thanks!
