Find the joint density function of $(Y_1,Y_2)$ where $Y_1=X_1+X_2$ and $Y_2=\frac{X_2}{X_1+X_2}$, and the joint density function

Question

Let the joint density function of $(X_1, X_2)$ is given by \begin{align*} f_{X_1,X_2}(x_1,x_2) = \begin{cases} x_1e^{-x_1-x_2}, & x_1,x_2 > 0 \\ 0, & \text{otherwise} \end{cases}. \end{align*}

Is it true that the formula of joint density function of $(Y_1,Y_2)$ given as follows: $$f_{Y_1,Y_2}(y_1,y_2) = f_{X_1,X_2}(x_1,x_2) |J(x_1,x_2)|^{-1},$$ where $J$ is the Jacobian matrix? If yes, I got the answer, but I didn't know why the double integral over $(0,\infty)$ was diverges?

Any helps? Thanks in advanced.

Answer 1

Here $x_1=y_1(1-y_2)$ and $x_2=y_1y_2$ so the Jacobian is $$\begin{vmatrix} \frac{\partial x_1}{\partial y_1} & \frac{\partial x_1}{\partial y_2}\\ \frac{\partial x_2}{\partial y_1} & \frac{\partial x_2}{\partial y_2} \end{vmatrix}= \begin{vmatrix} 1-y_2& -y_1 \\ y_2&y_1 \end{vmatrix} = (1-y_2)y_1 +y_1y_2=y_1.$$ It follows that the joint density of $(Y_1,Y_2)$ is $$ f_{Y_1,Y_2}(y_1,y_2)=\begin{cases} (1-y_2)y_1^2e^{-y_1},& 0<y_1<\infty, 0<y_2<1\\ 0,& \text{otherwise}. \end{cases} $$