Solution
= Solution
By <independence of random variables>, the <joint probability density> is the product of the two <probability density functions> of the rate-one <exponential distributions>:
$$
\boxed{f_{X,Y}(x,y)=e^{-(x+y)}\mathbf1_{\{x>0,\ y>0\}}.}
$$
Its values on the boundary are immaterial, because the boundary has zero area.