Math 302, assignment 4 Due Oct. 8

Note: there are several questions on WebWork for this week as well.

1. a. In a game, a player draws three cards from a deck. He wins $1 for each heart card chosen. What isthe expected amount won?b. If additionally the player gets $2 for each queen (so the queen of hearts is worth $3), what is theexpected amount won?

2. For a geometric random variable X, show that P(X > n+m|X > n) = P(X > m). (We say that thegeometric random variable has no memory.)

3. a. Prove that if EX = 0 then f(t) = E(X t)2 is minimized at t = 0.b. Prove that if EX = then f(t) = E(X t)2 is minimized at t = .

Extra practice problems (do not hand in):p. 163 4 problems 21, 23, 27, 28, 30,37, 38, 42Assume X and Y are independent random variables each taking on the values 1 and 1 with probability 12 .Let Z = XY . Show that X,Y and Z are not independent random variables but are pairwise independent(that is, each pair of them are independent).