In the image provided there are 6 different shapes:
If (X,Y) is uniformly distributed inside the area of each shape, are X and Y independent?. For how many of these shapes the answer is yes? For the first shape the area is the union of the areas of the 4 rectangles. At first, I assumed that except for the circle and the ellipse the answer for the rest is yes but I was wrong. Any hints?
Intuitively, independence says that knowing $X$ doesn't give you any information about $Y$. So, for each possible value of the $X$ coordinate, you need all $Y$ values that were originally possible, to still be possible, with the same original probabilities. To use the circle as an example, most $X$ values (any but right in the middle) rule out some $Y$ values, so $X$ and $Y$ can't be independent.