If two random variables are uniformly distributed over a region, how do you in general find the joint PDF of those random variables?
For example, if $(X,Y)$ is distributed uniformly over the region $-2\leq x\leq 2$, $0\leq y\leq 1-x^2$, how could you derive the joint density function of $X$ and $Y$?
I believe you have to integrate, but with what integrand? Thank you.
the easiest way is to calculate the Area of the domain region and, the joint pdf is its reciprocal
In the example you posted, the domain area is the following
$$\int_{-1}^{1}[1-x^2]dx=4/3$$
thus
$$f_{XY}(x,y)=\frac{3}{4}\mathbb{1}_{[-1;1]}(x)\cdot\mathbb{1}_{[0;1-x^2]}(y)$$
you can set also $x \in \mathbb{R}$ but thea area does not changes due to the fact that $y \ge 0$
why is the joint pdf the reciprocal of the area?
observing the drawing of the joint domain, you get that
$$C\int_{-1}^{1}\left[ \int_0^{1-x^2}dy \right]dx=1$$
that is $C=3/4$
Now observe that the above double integral is, geometrically, the volume of a solid figure with base the green region and constant height C that is your uniform density.
thus
$$V=1=C\cdot A$$
that means
$$C=\frac{1}{A}$$
