Reference no: EM132388806

Assignment -

Problem 1 - This is called the Cupon Collector's Problem described at the introduction of chapter 1: 10 different coupons are distributed into cereal boxes, on per box, so that any randomly selected box is equally likely to have any of the 10 coupons inside. Write a program to simulate the process of buying cereal boxes until all 10 distinct coupons have been collected. For each run, keep track of how many cereal boxes you purchased to collect the complete set of coupons. Then use your program to answer the following questions.

(a) What is the probability you collect all 10 coupons with just 10 cereal boxes?

(b) Use counting techniques to determine the exact probability in (a). Hint: Relate this to the Birthday Problem.

(c) What is the probability you require more than 20 boxes to collect all 10 coupons?

(d) What is the probability of collecting all 10 coupons in fewer than average boxes (i.e., less than 29.3)?

(e) Let X = the number of cereal boxes purchased in order to obtain all 10 coupons. Use a simulation program to estimate E(X) and SD(X).

(f) Repeat (e) with 20 coupons instead of 10. Does it appear to take roughly twice as long to collect 20 coupons as 10? More than twice as long? Less?

Problem 2 - A plan for an executive traveler's club has been developed by an airline on premise that 10% of its current customers would qualify for membership.

(a) Assuming the validity of this premise, among 25 randomly selected current customers, what is the probability that between 2 and 6 (inclusive) qualify for membership?

(b) Again assuming the validity of the premise, what are the expected number of customers who qualify and the standard deviation of the number who qualify in a random sample of 100 current customers?

(c) Let X denote the number in a random sample of 25 current customers who qualify for membership. Consider rejecting the company's premise in favor of the claim the p > 0.1 if x ≥ 7. What is the probability that the company's premise is rejected when it is actually valid?

(d) Refer to the decision rule introduced in part (c). What is the probability that the company's premise is not rejected even though p =0.20 (i.e., 20% qualify)?

Problem 3 - You show up at a bus stop to wait for a bus that comes by once per hour. You do not know what time the bus came by last. Let X = the arrival time of the next bus measured in hours, X ∼ U(0, 1).

(a) Suppose you go to this bus stop once a day for work 5 days a week. Simulate a month (4 weeks) of the amount of time you waited for the bus.

(b) From your simulation, give a histogram for the amount of time in a month that you spent waiting for the bus.

(c) From your simulation, give the mean and standard deviation.

(d) Repeat (a) - (c) for a year.

(e) Determine the expected value and standard deviation for the amount of time waiting for the bus based on X ∼ U(0, 1). Compare this to the means and standard deviations derived before.

Note - Need help with the coding part of the assignment.