Let (x, y) be a 2 dimensional stochastic variable with density function, where the bounded region denotes the set A.
$f(x,y) = \frac{3}{2}xy1_A (x,y)$
- Find the marginal distributions.
Solution
To find the marginal distributions of the variables, integrate the function with respect to each variable, no problem there. But I'm not sure about the limits. Clearly the region is bounded from from 0:2 for both y and x but I can't find a good argument as to why it would be 2. Should it be 2-x?
\begin{equation} f_X(x) = \int_{0}^{2} \frac{3}{2}xy dy = 3x \end{equation}
The same can be done for the Y distribution but again, same problem for limits.
Part 2
What is the distribution of X+Y and X-Y
No the limits are not correct. You integrated over a rectangle and not over the triangle
$$f_X(x)=\int_0^{2-x}f(x,y)dy=\frac{3}{4}x(x-2)^2\cdot\mathbb{1}_{(0;2)}(x)$$
and similarly for the other marginal rv
$$f_Y(y)=\int_0^{2-y}f(x,y)dx$$
To find the distribution of $U=X+Y$ and $V=X-Y$ there are severale possible ways, for example
definition of CDF
Jacobian
...which method are you more familiar with?
the more efficient way is to set
$$\begin{cases} u=x+y\\ v=x-y \end{cases}$$
Calculate the jacobian and derive the joint density
$$f_{UV}(u,v)=\frac{3}{16}(u^2-v^2)$$
Now all you have to do is to derive it w.r.t. $du$ to get $f_V$ or derive it w.r.t. $dv$ to obtain $f_U$
It remains to you the difficulty to understand the region of $(U,V)$ support in order to set the correct integral bounds...
...after a simple reasoning you will find that the joint domain $(U,V)$ is
$$|V|<U<2$$
and thus
$$f_U(u)=\frac{u^3}{4}\cdot\mathbb{1}_{(0;2)}(u)$$
$$f_V(v)=\frac{4-3v^2+|v^3|}{8}\cdot\mathbb{1}_{(-2;2)}(v)$$
Further details about the solution
Starting from the initial system you get
$$\begin{cases} x=\frac{u+v}{2}\\ y=\frac{u-v}{2} \end{cases}$$
With jacobian $|J|=\frac{1}{2}$
At this point, by simple substitution in $f(x,y)$ you get the joint density $f(u,v)$
Now observing that
$0<\frac{u+v}{2}<2$
$0<y=\frac{u-v}{2}<2$
$0<u<2$
you get that the joint domain $(U,V)$ is the triangle
$$|v|<u<2$$
The rest follows by integration