Joint Distribution, expected value correlation of a graphed triangle
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 fX,Y(x,y)=2 and 2 I got fX(x)=2−2y and fY(y)=2−2x. Then for correlation in 3 I got 112−(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.
Re: Joint Distribution, expected value correlation of a graphed triangle
For the first one this will be a conditional distribution.
You will have probabilities in the form of P(Y|X) or P(X|Y) and then you will have to use the relationship between conditional and joint distributions to get the actual distribution.
Getting this will help you with the rest of the problems.