Let $X$ and $Y$ have a joint density function given by
$$
f(x, y) =
\begin{cases}
1, & \text{for } 0\leq x\leq2,\;\max(0,\,x-1)\leq y\leq \min(1,\,x), \\
0, & \text{otherwise}.
\end{cases}
$$
Determine the joint and marginal distribution functions.
I know that $F_{XY}(x, y) = \int_{-\infty}^x\int_{-\infty}^y f(u, v)\;dudv$. But I have no idea how to apply this fact.
The obvious case that I can find is that if $x > 2$, then $F_{XY}(x, y) = 1$.
Some other bounds are e.g. $0<x<1$ and $0<y<x$, but how do I find the CDF for this case? I can find some of the bounds, but don't know how to use them for the integrals.
Best Answer
Revised correction, based on conversation with Maxim!
Summary for all $(x,y)$. For $x\le 0$ or $y\le 0,\ F(x,y)=0$.
For $0\lt x\le 1,\ 0\le y\le 1$, two parts:
1) $0\lt y\le x,\ F(x,y)=xy-\frac{y^2}{2}$,
2) $x\lt y\le 1,\ F(x,y)=\frac{x^2}{2}$,
For $1\lt x,\ 0\le y\le 1$, two parts: [part 2) vacuous for $x\gt 2$]
1) $0\le y\le x-1,\ F(x,y)=y$,
2) $x-1\lt y\le1,\ F(x,y)=xy-\frac{y^2}{2}-\frac{(x-1)^2}{2}$,
For $1\lt y,\ F(x,y)=F(x,1)$.