Suppose we have some region $R$ in $\mathbb{R}^2$, for simplicity lets say its compact in the usual topology. Suppose that random variables $X$ and $Y$ are both uniformly distributed on $R$.
I seek $E[XY]$, which is defined here in the continuous case as $$\int_{R} x \cdot y \cdot p_{x,y}(x,y) \, dx \, dy $$ where I assume $p_{x,y}(x,y)$ is the joint PDF for $XY$. My issue is in finding such a joint distribution.
Given the scenario above, I would find the pdf of $X$, $Y$ to be $$p_x(x) = p_y(x) = \left[\int_{R} 1 \cdot d_{A} \right]^{-1} = c.$$
Can I use this to find $p_{xy}(x,y)$? I think I am misunderstanding some things.
The explicit problem that raised this quesion took the region $R$ to be the area bounded by $1-x^2$ and the positive axes. I found $c$ to be 1.5 but am not sure how to proceed.
Edit: After receiving some comments that I am unable to answer I am going to post the specific problem that gave rise to my questions for clarity:
“Random variables $X$ and $Y$ are uniformly distributed on the region bounded by the $x$ and $y$ axes and the curve $y = 1-x^2.$
Calculate $E[XY]$”
In their solution they find $c=1.5$ and then perform the integration with integrand $xy1.5$.