Question $3)$

The random variables $x$ and $Y$ have joint probability density function $(pdf)$ given by

$f_{x,Y}(x,y)=\{\begin{array}{ccc}c{e}^{-(x+y)},& x0,& y0\\ 0,& \text{e}\text{l}\text{s}\text{e}& \end{array}$

for constants $>0,>0.$

a$)$ Find $c$ in terms of $$ and $$

b$)$ Find the marginal probability density functions of $x$ and $Y,$ respectively.

c$)$ Find the conditional probability density function of $x$ given $Y(f_{x_{\frac{?}{Y}}}(\frac{x}{y}))$

d$)$ Are $x$ and $Y$ independent?