Suppose $(X, Y )$ is uniformly distributed over the set $\{(x, y) : 0 < y + x < 2, 0 < x < 2\}$.

Find the joint density of $(X,Y)$ and marginal density of $F_Y(y)$.

I am having a tough time starting this problem as I don't know what to do with that function above. I know marginal density involves a double integral, but I cant even find joint density because the equation is throwing me off.

How should one begin this particular problem?


closed as off-topic by Did, Daniel W. Farlow, graydad, HK Lee, kjetil b halvorsen Apr 17 '15 at 7:50

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    $\begingroup$ Uniformly distributed over a set $S$ means a density $f$ constant on $S$ and $0$ elsewhere. Try to find out how the set $S$ looks like here. Draw a picture. $\endgroup$ – drhab Apr 16 '15 at 18:30
  • $\begingroup$ The marginal density function involves a single integral, $f_Y(y)=\int_{?}^{?} f_{X,Y}(s,y)\operatorname d s$. The marginal cumulative distribution function is the double integral: $F_Y(y)=\int_?^y\int_{?}^{?} f_{X,Y}(s,t)\operatorname d s\operatorname d t$ . Be sure of which you want. $\endgroup$ – Graham Kemp Apr 16 '15 at 19:32

The joint density is $$f_{X,Y}(x,y)=\begin{cases} 1,& \text{ if } 0\le x,y \le 2 \text{ and } x+y \le 2 \\ 0, & \text{ otherwise }\end{cases}$$ as shown below

By definition, the marginal density of $Y$ is

$$f_Y(y)=\int_{-\infty}^{+\infty}f_{X,Y}(x,y)\ dx=\int_0^{2-y}\ \ \frac{1}{2}\ \ dx=\begin{cases}\frac{2-y}{2}&,\text{ if } 0\le y \le 2\\ 0&, \text{ otherwise }.\end{cases}$$

Then the marginal cumulative distribution function of $Y$ is

$$F_Y(y)=\int_{-\infty}^y f_Y(u) \ du=\begin{cases}0&, \text{ if }y<0\\ \int_0^y\frac{2-u}{2}du=y-\frac{y^2}{4}&,\text{ if }0\le y <2\\ 1&, \text{ if } y\ge2. \end{cases}$$


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