My joint PDF function of (x,y) is given \begin{cases} 2e^{-x-y}, & 0\leq x\leq y\leq \infty\\0, & otherwise \end{cases} How do I find joint CDF from this PDF? I know that my CDF is zero when both x and y are less then zero and that I'm supposed to integrate PDF on a given region, but there is a solution in my book that says I'm supposed to calculate CDF on this area too:
$$ 0\leq y\leq x, y>0 $$
and it equals $$e^{-2y} - 1$$
Why do I do this? Shouldn't CDF be zero in this case? Also, isn't it redudant to say that y>0 in that area?
Note that this is the cumulative distribution function so $F(x,y)=\mathbb P(X\le x, Y \le y)$ is an increasing function of $x$ and of $y$
So when $0 \le y \le x$ you have
$\mathbb P(X\le x, Y \le y) = \mathbb P( X\le y, Y \le y) + \mathbb P( y \lt X\le x, Y \le y)$
where you can find $\mathbb P( X\le y, Y \le y)$ from your previous calculations for the CDF
while $\mathbb P( y \lt X\le x, Y \le y) = 0$ since the density is zero in this region