Problem
$X$ and $Y$ random variables, the common probability density function of $X$ and $Y$ is given as follows: $$f(x,y)= \begin{cases} 2e^{-x-y},&\textrm{when } x\geqslant y\geqslant 0\\ 0\;,&\textrm{otherwise } \end{cases} $$
Find the marginal probability function for $X$ and $Y$
Proposed Solution
$$ h(x) = \int_{y}^{\infty}f_{(x,y)} dy = \int_{y}^{\infty} 2e^{-x-y} dy = e^{-x} = \frac{2}{e^x}$$
$$ g(y) = \int_{0}^{\infty}f_{(x,y)} dy = \frac{2}{e^y} $$
$$ h(x) = \frac{2}{e^x} , g(y) = \frac{2}{e^y}$$
Is any of my work correct? Any feedback is much appreciated, and if you think I should add more details to my calculations, please point it out and I will edit my work accordingly.
Thank you for your time.
Update: I made a change according to k = 2
No, the work you have above is not correct. Close, but still wrong.
The support for the joint probability density function is $\{\langle x,y\rangle: 0\leq y\leq x\}$. Therefore, what are the domains of integration for the marginal probability density functions?
PS: The accepted answer to your previous question told you to find $k$ by using $\int_0^\infty\int_{\color{blue}y}^\infty k\mathrm e^{-x-y}~\mathrm d y~\mathrm d x=1$ so … think about this a little more.