Let $f_{UV}(u,v)=\frac{1}{2}\lambda_1\lambda_2 e^{-\frac{1}{2}\left((\lambda_1 +\lambda_2)u + (\lambda_2 - \lambda_1)v \right)}$ the joint density of U and V random variables. I have to decide if X and Y are independent, by calculating marginal density. However I'm getting in trouble, because I didn't find that U and V are independent. Would you help me please?
The problem is that I couldn't identify the marginals as any density known.
$X\sim Exp(\lambda_1)$ , $Y\sim Exp(\lambda_2)$, where $U=X+Y$ and $V=X−Y$. I calculated the joint density of $(U,V)$, using Jacobian Method. Now I'm struggling to show if $U$, $V$ are independent.
$f_X(x) = \lambda_1 e^{-\lambda_1 x}, f_Y(y) = \lambda_2 e^{-\lambda_2 y}$
$f_{XY} (x, y) = \lambda_1 \lambda_2 e^{-(\lambda_1 x + \lambda_2 y)}$
$U = X + Y, V = X-Y$ and using,
$x = \frac{u+v}{2}, y = \frac{u-v}{2}$ ...(i)
Joint pdf of $U$ and $V$, $f_{UV}(u,v)=\frac{1}{2}\lambda_1\lambda_2 e^{-\frac{1}{2} \big((\lambda_1 +\lambda_2)u + (\lambda_1 - \lambda_2)v \big)}$
Now to find marginal pdf of $U$ and $V$,
Please note from $(i)$ that $-u \leq v \leq u$ or $|v| \leq u$
So, $f_U(u) = \displaystyle \int_{-u}^{u} f_{UV}(u, v) \ dv$
$f_V(v) = \displaystyle \int_{|v|}^{\infty} f_{UV}(u, v) \ du$
Can you take it from here?