Let x and y be discrete random variables show that x and y

Question

Let X and Y be discrete random variables. Show that X and Y are independent if and only if f X,Y (x, y) = fX(x)fY (y) for all x and y.

Let X have distribution F and density function f and let A be a subset of the real line.

Let IA(x) be the indicator function for A: IA(x) =  1 x ? A 0 x /? A.

Let Y = IA(X).

Find an expression for the cumulative distribution of Y . (Hint: first find the probability mass function for Y .)