1. Determine each of the following is true or false? If false,provide a counterexample.
(a) Let X be a continuous random variable which has the pdf fX.Then, for each x, 0 ≤ fX(x) ≤ 1.
(b) Any two independent random variables have ÏXY = 0.
(c) Let X and Y be random variables such that E[XY ] = E[X]E[Y]. Then, X and Y are independent.
2. Ann plays a game with Bob. Ann draws a number X1 ∼ U(0,1) andBob draws a number X2 ∼ U(0,1). Assume X1 and X2 areindependent.
(a) Calculate the conditional probability of Ann winning the gamegiven Ann draws x1 ∈ [0,1].
(b) Calculate the probability of Ann winning the game. Hint:this is equal to P(X1 > X2). You may calculate this directlyusing integration; An alternative way is to use a geometricintuition. If x-axis represents x1 and y-axis represents x2, whatdoes the set of (x1,x2) such that x1 > x2 look like? What is thesize of this set?
