Problem 5. (20 pts.) You are looking into buying a new cell phone plan. You would like to buy a plan with the lowest expected payment. You narrow things down to two plans. Plan A: Has a flat rate of $0.25 per minute used. Plan B: You can purchase monthly talk time at the rate of $0.20 per minute; for each additional minute (beyond the purchased talk time) the rate is $0.50 per minute (for example, if you contract for 400 minutes, you first pay 400x$0.20=$80 at the beginning of the month. If you end up using 410 minutes during the month, you will have to pay additional (410 - 400)$0.50=$5); all unused minutes at the end of the month are lost. Assume that you don't pay attention to your monthly statements and you will not modify your phone usage behavior based on the bills. From your previous bills you estimated the following discrete probability of your phone usage: Minutes Probability 250 0.2 350 0.25 600 0.4 700 0.1 800 0.05 (a) (4 pts.) What is the expected monthly payment under Plan A? (b) (4 pts.) What is the optimal number of monthly minutes that should be contracted under Plan B? (c) (4 pts.) What is the expected monthly payment under Plan B if you contract for the minutes you computed in part (b)? (d) (2 pts.) Which plan, A or B, would you purchase? (e) (6 pts.) How would your answer change if the Phone Company changes their plan B, where it would provide 50% discount applied to next month's bill on any unused contracted minutes at the end of the previous month. For example if your contract is for 400 minutes and you used only 300 minutes this month, the next month discount will be $0.200.5100=$10.