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14. If x has the probability distribution f(x) = 12x for x = 1,2,3,…, show that E(2X) does not exist. This is famous Petersburg paradox, according to which a player’s expectation is infinite (does not exist) if he is to receive 2x dollars when, in a series of flips of a balanced coin, the first head appears on the xth flip. 17. The manager of a bakery knows that the number of chocolate cakes he can sell on any given day is a random variable having the probability distribution f(x) = 16 for x = 0,1,2,3,4, and 5. He also knows that there is a profit of $ 1.00 for each cake which he sells and a loss (due to spoilage) of $0.40 for each cake he does not sell. Assuming that each cake can be sold only on the day it is made, find the baker’s expected profit for a day on which he bakes

a. 3 of the cakes;

b.4 of the cakes;

c.5 of the cakes.

18. If a contractor’s profit on a construction job can be looked upon as a continuous random variable having the probability density For -1 < x < 5

elsewhere

(x+1)

f(x) = { 118 0

22. Mr. Adams and Ms. Smith are betting on repeated flips of a coin. At the start of the game Mr. Adams has a dollars, Ms. Smith has b dollars, at each flip the loser pays the winner one dollar, and the game continues until either player is “ruined.” Making use of the fact that in an equitable game each player’s mathematical expectation is zero, find the probability that Mr. Adams will win Ms. Smith’s b dollars before he loses his a dollars.