Let X and Y be continuous random variables with joint PDF
Chapter , Problem 32(choose chapter or problem)
Let X and Y be continuous random variables with joint PDF Ix,Y . Suppose that for any subsets A and B of the real line, the events {X E A} and {Y E B} are independent. Show that the random variables X and Y are independent. Solution. For any two real numbers x and y, using the independence of the events {X x} and {Y y}, we have Fx.y (x, y) = P(X x, Y y) = P(X x) P(Y y) = Fx (x)Fy (y). Taking derivatives of both sides, we obtain 82 Fx. y 8Fx 8Fy fx.y (x,y) = 8x8y (x,y) = 8x (x) 8y (y) = fx (x)fy (y) , which establishes that X and Y are independent.
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