(a) If the knowledge that an event A has occurred implies that a second event B cannot occur, then the events A and B are said to be A. collectively exhaustive. B. the sample space. C. mutually exclusive. D. independent. (b) If event A and event B are as above and event A has probability 0.2 and event B has probability 0.2, then the probability that A or B occurs is ____

In a carnival game, a player spins a wheel that stops with the pointer on one (and only one) of three colors. The likelihood of the pointer landing on each color is as follows: 61 percent BLUE, 21 percent RED, and 18 percent GREEN.

(a) Suppose we spin the wheel, observe the color that the pointer stops on, and repeat the process until the pointer stops on BLUE. What is the probability that we will spin the wheel exactly three times?

(b) Suppose we spin the wheel, observe the color that the pointer stops on, and repeat the process until the pointer stops on RED. What is the probability that we will spin the wheel at least three times?

(c) Suppose we spin the wheel, observe the color that the pointer stops on, and repeat the process until the pointer stops on GREEN. What is the probability that we will spin the wheel 2 or fewer times?

9. Police plan to enforce speed limits by using radar traps at four different locations within the city limits. A person who is speeding on his/her way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations (location 1, location 2, location 3, and location 4, respectively). The radar traps at each of the locations (location 1, location 2, location 3, and location 4, respectively) will be operated 40%, 30%, 20%, and 30% of the time. If a randomly selected person received a speeding ticket on his/her way to work, what is the probability that he/she passed through the radar trap located at location 1? (10 points)

Hint: Consider the following five events:

A: a speeding person receives a speeding ticket (note: a speeding person will receive a speeding ticket if the person passes through the radar trap when operated.)

B₁: a speeding person passing through location 1,

B₂: a speeding person passing through location 2,

B₃: a speeding person passing through location 3,

B₄: a speeding person passing through location 4.

Problem 15-05 (Algorithmic)

Consider the following time series data.

1. Develop a three-week moving average for this time series. Compute MSE and a forecast for week 7. Round your answers to two decimal places.

   Week	Time Series Value	Forecast
   1	16
   2	13
   3	18
   4	11
   5	15
   6	14

   
   MSE: ——————
   
   The forecast for week 7: ————————
   
2. Use  = 0.2 to compute the exponential smoothing values for the time series. Compute MSE and a forecast for week 7. Round your answers to two decimal places.

   Week	Time Series Value	Forecast
   1	16
   2	13
   3	18
   4	11
   5	15
   6	14

   
   MSE: ————————
   
   The forecast for week 7: ——————
3. Use trial and error to find a value of the exponential smoothing coefficient  that results in a smaller MSE than what you calculated for  = 0.2. Find a value of  for the smallest MSE. Round your answer to three decimal places.
   
   alpha = ——————————

Consider the following IS-LM model with a banking system:

Consumption:

C = 7 + 0.6Y_D

Investment:

I = 0.205Y − i

Government expenditure:

G = 10

Taxes:

T = 10

Money demand: M^d / P = Y / i

Demand for reserves:

R^d = 0.375D^d

Demand for deposits:

D^d = (1 − 0.2)M^d

Demand for currency:

CU^d = 0.2M^d

This says that consumers hold 20% (c = 0.2) of their money as currency and the required reserve ratio is 37.5% (θ = 0.375). Demand for central bank money (H^d) is the total amount of currency being demanded plus the total demand for reserves. Suppose the price level is P = 1 and that the initial supply of central bank money is $100.

1.Solve for the money multiplier. Explain your work.

2.Solve for equilibrium output and the equilibrium interest rate at the initial supply of central bank money (ie. $100).

3.Suppose that the central bank sells $80 worth of bonds using open market operations. Solve for the new equilibrium output.

4.Solve for the the new equilibrium interest rate after the open market operations and use an IS-LM graph to explain what happened.

Consider a residential network that connects to the internet with a DSL link that has a download rate of 4 Mb/s. Assume that there are three UDP flows sharing the link and the remote hosts are sending at rates of 1 Mb/s, 2 Mb/s and 3 Mb/s. Assume that the ISP router has a link buffer that can hold 300 packets (assume all packets have the same length).

a) For each flow, what fraction of the packets it sends are discarded?

b) For each flow, about how many packets does it have in the queue.

c) Now, suppose the queue at the ISP router is replaced by three queues that can each hold 100 packets and that the queues are scheduled using weighted-fair queueing, where the weights are all 0.33. In this case, what fraction of packets are discarded from each flow?

d) How many packets does each flow have in the queue?

e) Now, suppose the weights are 0.2 for the first flow, 0.6 for the second and 0.2 for the third. In this case, what fraction of packets are discarded from each flow?

f) How many packets does each flow have in the queue?

The Heaton Electronics Company produces SD memory cards in sizes ranging from 2GB to 32GB. The management team are looking at options for growing the company in the future. The company has identified three options for growth.

• Keep the product range the same but increase the factory capacity to produce more of the products
• Expand the product range to include larger memory cards (up to 124 GB)
• Invest in new technology for manufacturing cards with extremely large memory sizes.

The management team believe that the profit (after investment) over the next three years will depend on whether the demand for the larger memory card sizes grows slowly, moderately or at a rapid rate. The potential profit under each market condition is shown below

Table: Potential profit (in £millions)

Part a)

i) What is the best decision using the Maximax rule?

ii) What is the best decision using the Maximin rule?

[2 marks]

Part b)

What is the best decision using the Minimax Regret rule (show your working, include your opportunity losses in your answer)

[4 marks]

Part c)

The company decide to use a Hurwicz criterion of alpha (  ) = 0.6. Calculate the best decision using the Hurwicz criterion. (show your working in your answer)

[4 marks]

Part d)

Are the company being optimistic or pessimistic using an alpha of 0.6 for the Hurwicz criterion? Briefly explain your answer.

[2 marks]

Part e)

The company narrow down the choice to either expanding the product range or investing in new technology. Both projects generate cash flow at different stages of the project. The cash flow generated at each stage is shown below. Using a discount rate (r) of 2.5%, calculate the Net Present Value (NPV) for both projects and determine the best decision based on the NPV.

Table: Cash flow comparison for two projects

[6 marks]

1. Chain Rule and the Markov Rule

1. You as a sports analyst are modeling the free throw success of Lebron James. You are asked to predict the probability that Lebron will make four free throws in serial succession during an important game that takes place in the opposite team’s venue.

Based on your data of Lebron’s previous free throws, you estimate that the median probability (0.5 below and 0.5 above) of a single free throw by Lebron under a wide range of expected conditions is 0.75. Also, estimate that the probability of free throw 2 given he made free throw 1 is P(2|1) = 0.8. Also you estimate that the Pr of free throw 3 given he made free throw 2 with kudos is P(3|2) = 0.9. But because of the extreme conditions at the important game in the opposite team’s site, you estimate that the Pr he makes free throw 4 given he made free throw 3 is 0.7.

Using these data, write the expression for and predict to 2 sd the Pr of success of four free throws in succession. Assume that the Pr he makes the first free throw is his median value of 0.75. As in b) assume the Markov Rule in which the primary dependency is with the previous free throw and further back free throws are less influential and considered independent for this analysis. Also, calculate the mean value and variance, and standard deviation of the probabilities where each probability value is equally weighted. Finally write an expression and calculate the numerical value of P(1,2,3,4) to two standard deviations, which represents uncertainty management of estimating the epistemic uncertainty of the probability values.

               Mean Probabilities=

Variance =

                Standard Deviation =  

P(1,2,3,4) =      to two standard deviations, which represents uncertainty management of epistemic uncertainty in the estimated probability values.

1. b. Now predict the probability to 2 sd of Lebron’s making 4 free throws in series if each free throw is considered independent of all of the other free throw attempts. Use Lebron’s median Pr value of success = 0.75.

P(1,2,3,4) =

Assume the same standard deviation calculated in a:

P(1,2,3,4) =        to two standard deviations exhibiting an estimate of epistemic uncertainty in the analysis. Do the results for c) and d) agree within the combined estimated uncertainties in the two values?

.

Q: During a televised, title prize fight at a Las Vegas hotel, Luis threw a punch which hit Michael in the face. As a result:

a: This would ordinarily be an act of negligence, however since Michael consented to the fight, he has no viable legal action against Luis. INCORRECT

b: This would ordinarily constitute strict liability for engaging in an inherently dangerous activity, however since Michael consented to the fight, he has no viable legal action against Luis.

c: This would ordinarily be an action of intentional infliction of emotional distress, however since Michael consented to the fight, he has no viable legal action against Luis.

d: This would ordinarily be a battery, however since Michael consented to the fight, he has no viable legal action against Luis.

Problem 8-21 (Algorithmic)

Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:

Type I rooms do not have wireless Internet access and are not available for the Business rental class.

Round Tree's management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 140 rentals in the Super Saver class, 60 rentals in the Deluxe class, and 40 rentals in the Business class. Round Tree has 125 Type I rooms and 135 Type II rooms.

1. Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types.
   

   Variable	# of reservations
   SuperSaver rentals allocated to room type I
   SuperSaver rentals allocated to room type II
   Deluxe rentals allocated to room type I
   Deluxe rentals allocated to room type II
   Business rentals allocated to room type II

2. How many reservations can be accommodated in each rental class?
   

   Rental Class	# of reservations
   SuperSaver
   Deluxe
   Business