Question

1. In class, we have discussed the following theorem: "When the domains of r and y are not dependent, then X and Y are independent if and only if fx,y(x, y) -g(x)h(y) for all r, y for some functions g and h." In this exercise, we will explore what happens if the domains of r and y are dependent Let fxy(x, y) = 6x for 0 < x < y < 1. (a) Sketch the region 0 < x < y < 1. The domains of x and y are dependent. Explain what this means (b) Show that we can write fx.y ()-())fr ll y for some functions g and h (c) Are X and Y independent? Check this by deriving the marginal densities fx(x) and fy (y), and checking whether fx.y(x,y)-fx(x) x fy(y). Hint: To check whether a marginal density you derived is correct, you can check for yourself whether it integrates to 1

Answer 1