Question:
A stone is thrown into a circular pond of radius 1 meter. Suppose the stone falls uniformly at random on the area of the pond. Find the expected distance of the stone from the center of the pond.
I have tried to solve this by creating a probability density function and the finding the expected value of $X^2+Y^2$. I defined 2 uniform random variables $X$ and $Y$ in $[-1,1]$ and found that the joint PDF of $X$ and $Y$ is
$$\frac14\quad,-1<x<1,-1<y<1;\quad 0\quad,\text{ otherwise }$$
After this, I found the expected value of $X^2+Y^2$ which came out to be $\frac23$ and seems to be the right answer. But my question is if my joint PDF is right to solve this question. It looks like $X$ and $Y$ are dependent here. Also the pond cannot be more than $1$ meter in radius but my function $X^2+Y^2$ takes value $2$ when $X=1,Y=1$.