The joint pdf of two random variables is given as

$f_{x,Y}(x,y)=\{\begin{array}{ccc}C{e}^{-2x-y},& ifx0,y0,\text{a}\text{n}\text{d}& x+y=1\\ 0,& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

for some CinR. Define the random variable $Z=x-Y$

$($a$)$ Find $C.$

$($b$)$ Are $x$ and $Y$ independent? Explain.

$($c$)$ Find the distribution of $Z.$