Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find $f_{X,Y}(x,y)$ and the marginals $f_X(x)$ and $F_Y(y)$.
My attempt: Since the random vector is uniform it will have a constant joint density of the form $\frac{1}{c}$. I also figured that $-1 \leqslant X\leqslant1$ and $0 \leqslant Y \leqslant \sqrt{(1-X^2)}$ as these bounds define the semicircle. When I integrate the joint density $\frac{1}{c}$ over the bounds I just mentioned and equating it to $1$, I find that the joint density must be $$f_{X,Y}(x,y)=\frac{2}{\pi}$$ But I do not think this is correct, as when I compute the marginals and then compute conditional probablities using this joint density, I get illogical results. Thanks for any help.