I have a hard time thinking about these kinds of problems. This is Problem 20 from chapter 1 of "Intemediate course in Probability" but Gut.
The random variables in $X_1$ and $X_2$ are independent and equidistributed with density function: $$f(x) = \begin{cases} 4x^3 & \space \mathrm{for} \space\space \space 0\leq x \leq 1 \\0 &\ \mathrm{for} \space\space\space x\gt 1 \end{cases} $$
Let $Y_1=X_1 \sqrt {X_2}$ and $Y_2=X_2 \sqrt {X_1}$. Determine the joint density function of $Y_1$ and $Y_2$.
So this is what I have:
I figure I can set up: $f_{X_1,X_2}=(4x_1^3, 4x_2^3)$. Inverting the new variables $Y_i$ gives me $X_1=(Y_1^{4/3}/Y_2^{2/3})$ and $X_2=(Y_2^{4/3}/Y_1^{2/3})$, and Jacobian:
$$\begin{vmatrix} 4/3(Y_1/Y_2^2)^{1/3} & -2/3(Y_1/Y_2)^{4/3} \\ -2/3(Y_2/Y_1)^{4/3} & 4/3(Y_1/Y_2^2)^{1/3} \\ \end{vmatrix} = \lvert\frac{16}9(\frac{1}{Y_1Y_2})^{1/3}-\frac49\rvert$$
Using the transformation theorem I get $f_{Y_1,Y_2}=f_{X_1,X_2}(g(X_1),g(X_2))\cdot|J|$, where g is the inverse above. This does not give me the correct answer however, have I got it wrong or are there mistakes in my calculation?
