1. Suppose we play a game where we start with c dollars. On each play of the game you either double or halve your money, with equal probability. What is your expected fortune after n trials?
2. Show that V(X) = 0 if and only if there is a constant c such that P (X = c)= 1.
3. Let X1,..., Xn ∼ Uniform(0, 1) and let Yn = max{X1,..., Xn}. Find E(Yn).
4. A particle starts at the origin of the real line and moves along the line in jumps of one unit. For each jump the probability is p that the particle will jump one unit to the left and the probability is 1-p that the particle will jump one unit to the right. Let Xn be the position of the particle after n units. Find E(Xn) and V(Xn). (This is known as a random walk.)