How to find CDF of PDF

Solution 1:

The joint cdf is given by

$P(X\leq u,Y\leq v)$ . As $f(x,y)>0$ for $x<y<1$. We have to look at whether $u<v$ or $u>v$ and separate the cases .

if $u\geq v$ . Then

$P(X\leq u, Y\leq v) = P(X\leq v, Y\leq v) = \int_{0}^{v}\int_{0}^{y}2\,dx\,dy=v^{2}$.

If $u<v$ . Then we have $P(X\leq u,Y\leq v)=2(uv-\frac{1}{2}u^{2})$

So $$F(x,y)= \begin{cases} y^{2}\cdot\mathbf{1}_{\{x\geq y,\,0\leq y\leq 1\}}\\2(xy-\frac{x^{2}}{2})\cdot\mathbf{1}_{\{x<y,\,0\leq y\leq1\}}\\2(x-\frac{x^{2}}{2})\cdot\mathbf{1}_{\{y>1\,,\,0\leq x\leq 1\}}\\1\cdot \mathbf{1}_{\{x>1,y>1\}} \\0,\,\text{elsewhere} \end{cases}$$