Let $X$ and $Y$ have a joint density function given by $$ f(x, y) = x * y, [0,2] × [0,1] $$ Determine CDF.
I know that $F_{XY}(s, t) = \int_{-\infty}^s\int_{-\infty}^t f(x,y) \;dydx$. But I have no idea how to apply this fact. Also i know that i have to check some bounds for $s$ and $t$, but how can i get them?
The density function you are given is zero everywhere outside the small rectangle given: $0 \leq x\leq 2$ and $0 \leq y \leq 1$. In order to get the right answer, you have to account for that. When a function is a different formula in different places, you have to integrate it piecewise. If the $x$ you are asking about is less than $0$, then $F = 0$. If you are in the nonzero part of the function, then you have to integrate $xy$ from $0$ up to the value of interest to you.
Imagine continuing the edges of the nonzero domain into lines that go forever. Those four lines cut the entire plane into 9 pieces like a tic-tac-toe board. With $+x$ pointing right and $+y$ pointing up as usual, that means that in the bottom left "square", $F_{X,Y}(x,y) = 0$. Likewise the center left and bottom center zones, and the top left and bottom right zones. In the top right zone, can you tell the value of $F_{X,Y}(x,y)$? It's a number.
In the last $3$ zones we have to integrate. For the top center, $y$ is integrated over the entire nonzero area, and gains no more going above. Suppose $y=5$. Then your $y$ integral would be in $3$ pieces: $\int_{-\infty}^0$, then $\int_{0}^{1}$, then $\int_1^5.$ The first integral and the last integral are $0$ because the density is $0$ there. Likewise for $x$ in the center right zone. The most interesting part is the center rectangular zone, where both integrals go up to just the specified point for upper limits.
For your x integral, you have to integrate $\int_{-\infty}^{0} 0 dx$ and then $\int_{0}^x xy dx$ and likewise for y. In effect, for $0 \leq x\leq 2$ and $0 \leq y \leq 1$, $F_{X,Y}(x,y) = \int_{0}^x \int_{0}^y xy dy dx$.
