Let X and Y be two independent random variables. Let us define two functions as follows: $$U:=\lambda_1 X+ \lambda_2Y$$ $$V:=\lambda_3\sqrt{XY}$$ What is the joint density function of U and V i.e., $f_{UV}(u,v)$ ?
Some more descriptions:
X and Y are iid exponential random variables with unity mean. The joint density function of X and Y is defined as below $$f_{XY}(x,y)=e^{-(x+y)} ;x \geq 0, y \geq 0$$
Using density transformation, we can have the joint density function of U and V i.e., $f_{UV}(u,v)$ as follows-$$f_{UV}(u,v)=\sum_{i}f_{XY}(x_i,y_i)|J_{g_i}(x_i,y_i)|^{-1} \label{eqn_7}$$.
where the solution pair $(x_i,y_i)$ is the i'th solution of above U and V equations and $J_g(x_i,y_i)$ denotes the Jacobian of the function $g(x,y)$.
Am I in write track? If one can solve the rest, I can cross-check my derivation with them.