While studying for examination I came across a question that asks for finding the correlation coefficient between two random variables X and Y uniformly distributed over the upper half of a semi circle of radius 1.
Attempt:
Since the points are uniformly distributed over the area which is $\dfrac{\pi r^2}{2}$ and $r=1$ then we can find the joint pdf as:
$$f_{X,Y} (x,y) = \dfrac{2}{\pi}$$
Now I tried finding the marginal pdfs as:
$$f_X(x) = \int_{-\infty} ^{\infty} f_{X,Y}(x,y)dy = \int_0^1 \dfrac{2}{\pi}dy = \dfrac{2}{\pi}$$
and
$$f_y(x) = \int_{-\infty} ^{\infty} f_{X,Y}(x,y)dy = \int_{-1}^1 \dfrac{2}{\pi}dy = \dfrac{4}{\pi}$$
I have a feeling what I did regarding marginals is wrong and even if they're right I'm not sure how to proceed to compute the correlation coefficient $\rho $