Consider two discrete random variables X and Y which take on values from the set {1, 2, 3, ….,K}. Suppose we construct an n ˟ n matrix ρ whose elements comprise the joint PMF of the two random variables. That is, if is the element in the i th row and th column of ρ, then p i, j = P X, Y (I, j) = Pr (X=I, Y=j).
(a) Show that if and are independent random variables, then the matrix ρ can be written as an outer product of two vectors. What are the components of the outer product?
(b) Show that the converse is also true. That is, show that if ρ can be factored as an outer product, the two random variables are independent.