Let X and Y be the coordinates of a point chosen uniformly at random from the triangle that joins the points (0,1), (0,0), and (1,0).
1) Find the joint distribution of X and Y.
2) Determine the expected value of X and the expected value of Y (these are the expected coordinates of a point chosen at random).
3) Find the correlation between X and Y.
4) If the original units were measured in inches, would there be a different correlation if the units were changed to centimeters? Justify your answer mathematically?
For 1, I got $f_{X,Y}(x,y) = 2$ and 2 I got $f_X(x) = 2-2y$ and $f_Y(y) = 2-2x$. Then for correlation in 3 I got $\frac{1}{12} - (4-4x-4y+4xy)$. I feel like I am forgetting something however and this should be different. For 4, I know this is unchanged but I am not sure how to show this. Any help is appreciated.