You are lucky enough to have $100 x (210 - 1) = $102, 300 in your college savings fund. You think that having some money should give you the power to make more money. To that end, you choose to gamble on coin flips with a friend who has even more money in the bank. Your strategy is the following. You will start by betting $100 that the first coin flip is heads. If it's not you will "double down" and bet $200 the next flip is heads. You will keep "doubling down" until you win a flip, at which point you will quit. Suppose the first head occurs on the k th flip. The bet at that point will be 100 x 2*-1 = 10010 x [1 00 ..0 ]2 k - 1 zeros where the 10 and 2 subscripts indicate that the 100 is ordinary base 10 while the other term in the product is binary. The total losses from the previous flips will be 100 x (2*-2 + 2*-3 + ...20) = 10010 x [ 1 1 .. 1 2. k - 1 ones By looking at the binary expressions for winnings and losses it's easy to see the difference is always $100. Your last win eclipses all your preceding losses, and you always finish with a $100 gain! However, you see a potential problem with your plan. If you are unlucky and get tails on the first 10 flips (and you are about to make your 11th flip) your losses at that point will total 10010 x [1 1 .. 1 ]2 = $102, 300. 10 ones Thus, your friend will see that you are bankrupt after 10 losses, demand to stop playing, and demand that you hand over your whole college savings account. (a) Suppose that Y is the net gain or loss from playing the gambling game. Specify the PMF of Y. (b) Calculate E[Y]