Consider the points A=(0,0), B=(1,0), C=(0,1), Now choose a random point (X,Y) uniformly from the union of the line segments AB and AC. Find Cov(X,Y)

Question

This question is from my recommended exercises (not to be marked). I am having trouble approaching this question.

Thus far, I have drawn the geometry of the triangle that the points A,B,C construct. I am not entirely sure where to go from here. Typically for these types of problems, we take the following expected values EX, EY, and EXY to find the covariance, using multivariable calculus. However, that is only if the region we are concerned with is the whole area of the triangle.

However, the question states that (X,Y) are uniform along $AB \cup AC $. I am not sure how to approach this question from here.

I could define $AC:=\{(x,y): 0 \leq y \leq 1 , x= 0 \}$

$AB:=\{(x,y): 0 \leq x \leq 1 , y= 0 \}$

$AB \cup AC := \{(x,y): 0 \leq x \leq 1, 0 \leq y \leq 1 \}$

Note: The question gives a hint "Look at XY first"

Answer 1

Hint:

You can describe the distribution of $(X,Y)$ in the following way. Let $U$ be uniformly distributed on $[0,1]$. With probability $1/2$, $(X,Y)=(U,0)$. With complementary probability (i.e. $1/2$), $(X,Y)=(0,U)$. (Can you see why?)

Can you now calculate $XY$? What about $E[X]$ and $E[Y]$? If you can, then you should be able to find $\operatorname{Cov}(X,Y)$.