Joint CDF of $f(x,y)$

Question

I would appreciate a little help here.

Let $X$ and $Y$ be continuous random variables with a joint pdf of the form $$f(x,y)=K(x+y)$$ $$0< x <y<1$$

Find $K$. I solved it. It turns out to be $2$.

Find the marginals. Again, I found those Find the joint CDF, $F(x,y)$. This is the part I am stuck on. I used the CDF definition but I still get answers that are different from the book.

Thank you

Answer 1

The CDF will come in several parts:

For $0 \le x \le y \le 1$ you have $$F(x,y) = \int_{a=0}^x \int_{b=a}^y 2(a+b) \,db \,da = x(y^2+xy-x^2).$$

For $0 \le x \le 1 \le y$ you have $F(x,y)=F(x,1) = x+x^2-x^3.$

For $0 \le y \le 1$ and $y \le x$ you have $F(x,y)=F(y,y) = y^3.$

For $x\ge 1$ and $y \ge 1$ you have $F(x,y)=F(1,1)=1$.

For $x\le 0$ you clearly have $F(x,y)=0$. Similarly for $y\le 0$ you have $F(x,y)=0$.