Math 462/562 - Part 3 - Assignment 2
Due:Wednesday, June 15. Nothing accepted afterMonday,June 20. 10% points off for being late. Please work by yourself. See me if you need help.
1.Redo Example 1.1 in section 1.1 of Meerschaert with the following additions. Suppose that on any given day there is a probability of 0.1% that the pig will catch some disease and dies. Find the optimal time t to sell the pig so as to maximize the expected profit. Make the following assumptions and use the following notation. Assume prices, costs, etc are in dollars.
a.Treat time as discrete. Count the days with today being day zero. Thus day n = 0 is today, day n = 1 is tomorrow, etc. Assume the pig is alive today. On each day there are several possibilities. One is that the pig will have died since the previous day. If the pig was alive on one day then there is a probability of 99.9% that the pig will be alive the next day and a probability of 0.1% that the pig will die before he can be sold the next day. If the pig is alive on a certain day then you can either sell the pig on that day or wait until the next day in which case the pig may have died during that period.
b.wn = weight of the pig on day n assuming the pig survives that long = 200 + 5n
cn = cost of keeping the pig until day n assuming the pig is alive on day n-1.
= 0.45n. This applies even if the pig should die before you can sell it on day n.
sn = selling price of the pig on day n assuming the pig survives until day n = 0.65 – 0.01n
rn = revenue from selling price the pig on day n assuming it survives until day n= snwn
fn = profit from selling the pig on day n assuming the pig survives until day n = rn – cn
c.If the pig dies between days n-1 and day n, there is no cost other than keeping the pig for n days which is cn.
d.If you don't sell the pig on day n-1 the probability of 0.1% that the pig will die before you can sell it on day n still applies.
As you do the problem find the following.
1.(2 points) Find the probability that the pig lives from day 0 at least until day n.
2.(1.5 points) Find the probability that the pig lives from day 0 until day k-1 and then dies between day k-1 and day k.
3.(3.5 points) Suppose you plan to sell the pig on day n if it survives that long. Let F be the profit considered as a random variable. Thus F = fn if the pig survives until day n or F = - ck if the pig should die between day k-1 and day k for some k in the range k = 1, …, n-1, or n. Find a formula for the yn = expected value of F. One or both of the following formulas should be useful
1 + x + x2 + ... + xn = 1 + 2x + 3x2 + ... + nxn-1 =
4.(3 points) Find n so as to maximize yn.
2.The Large Charge Battery Company sells their batteries in packages of 5. They have a special guarantee on their 5 packs. If one of the 5 is a dud, the customer gets their money back. If 2 of the batteries is a dud they get double their money back. If 3 or more of the batteries is a dud they get triple their money back. Suppose 5% of the batteries are duds and whether any one battery is a dud is independent of whether any other battery is a dud.
a.(3 points) Suppose the company charges three dollars for a 5 pack. What is their expected income per pack taking into account the customers who get money back due to duds?
b.(2 points) What is the expected number of packs you must buy until you get one with 3 or more duds?