Find the joint cdf $F(x,y)$ for two random variables $X$ and $Y$ whose joint pdf is given by
$f(x,y) = \frac 1 2$ , $0 \le x \le y \le 2$.
I know that I have to integrate with respect to $x$ as well as $y$. What I am having problems with is the bounds. Can someone please explain how I am supposed to figure that out?
Best Answer
Here is a sketch of the support of the pdf and the cdf, and a geometrical computation of $F_{X,Y}(x_0,y_0)=\frac12y_0-\frac14x_0^2$ (check it, it is a good exercice).
It can help to understand why, for example concerning the marginal in $X$, we integrate with respect to $y$ (this is general) with the integration bounds $x_0$ and $2$ (this is particular):
$$f_X(x_0)=\int_{y=x_0}^{y=2}f(x,y)dy=\int_{y=x_0}^{y=2}\dfrac12dy=\dfrac12(2-x_0)$$
and a similar computation for $f_Y(y_0)=\int_{x=0}^{x=y_0}f(x,y)dy=\cdots.$