For some $r>0$, the distribution of pair $(X, Y)$ of real random variables has the joint density function $f(x,y):=c1_{\{(s,t)|s^2+t^2\leq r^2\}}(x,y).$ Are $X$ and $Y$ independent? And how to determine the constant $c$ and the density of $X$ and $X-Y?$
I have managed to find the constant $c:$ $$ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) dxdy = 1$$ $$ \int\int_{\{ x^2+y^2\leq r^2 \}} c = 1$$ $$ cr^2\pi = 1 \Rightarrow c = \frac{1}{r^2 \pi} $$
And I also managed to get marginal distributions by integration $f(x,y)$ with respect to each variable and I get $$f_X(x)=2c\sqrt{r^2-x^2}=\frac{2}{r^2\pi}\sqrt{r^2-x^2}$$ $$ f_Y(y)=2c\sqrt{r^2-y^2}=\frac{2}{r^2\pi}\sqrt{r^2-y^2} $$
I should check if the two random variables are independent and I was going to see if $$f(x,y) = f_X(x)f_Y(y)$$ holds and my guess is, that it doesn´t since I cannot see how this product yields 1 but I need more formal mathematical argument for it.
I am also supposed to find the density of $X-Y$. So I thought of using the formula: $$ f_{X-Y}(z) = \int_{-\infty}^{\infty} f_X(z+y)f_Y(y)dy$$ Is this the best way to proceed and also if someone could tell me if pdfs are correct?
I would appreciate any kind of help!