# [SOLVED] Difference between a discrete probability distribution and a continuous probability distribution..

## [SOLVED] Difference between a discrete probability distribution and a continuous probability distribution..

FACULTY OF ENGINEERING AND THE BUILT ENVIRONMENT
INDIVIDUAL ASSIGNMENT
DEPARTMENT OF QUALITY AND OPERATIONS MANAGEMENT

PROGRAMME : DIPLOMA
OPERATIONS MANAGEMENT & MAN SERV

SUBJECT : OPERATIONS MANAGEMENT TECHNIQUES 2A

CODE : OPT22A2

TOTAL MARKS : 100 WEIGHT : 25%
INSTRUCTIONS
⦁ This assignment is composed of TWO sections.
⦁ Answer ALL the questions in both sections of the assignment using information from the provided learning material and other sources where relevant. ALL the sources should be cited using Harvard referencing technique.
⦁ The cover page should indicate the following;
⦁ NAME AND SURNAME
⦁ STUDENT NUMBER
⦁ MODULE NAME AND CODE
⦁ QUALIFICATION
⦁ Submit the typed answers by uploading a document on the Turn-it in link titled “Individual Assignment Submission Link” created on blackboard. ALL PAGES SHOULD BE NUMBERED EXCEPT COVER PAGE
SUBMISSION DETAILS
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NB: Once the set time for turn-it-in link to be active expires, the system will not allow you to make a submission. Thus, it is IMPERATIVE that you submit this assignment within the set time. The acceptable similarity percentage is 30%.
NO CONNECTIVITY ISSUES EXCUSES WILL BE CONSIDERS AND THE WILL BE NO SUBMISSION EXTENSION GRANTED FOR THIS ASSIGNMENT.
SECTION A
QUESTION 1 [15 marks]
⦁ What is the difference between a discrete probability distribution and a continuous probability distribution? Give your own example of each. (4)
⦁ What is the expected value, and what does it measure? How is it computed for a discrete probability distribution? (3)
⦁ What is the variance, and what does it measure? How is it computed for a discrete probability distribution? (3)
⦁ Give an example of a good decision that you made that resulted in a bad outcome. Also give an example of a bad decision that you made that had a good outcome. Why was each decision good or bad? (5)

QUESTION 2 VALLEY SWIM CLUB [25 Marks]
The Valley Swim Club has 300 stockholders, each holding one share of stock in the club. Ashare of club stock allows the shareholder’s family to use the club’s heated outdoor pool during the summer, upon payment of annual membership dues of \$175. The club has not issued any new stock in years, and only a few of the existing shares come up for sale each year. The board of directors administers the sale of all stock. When a shareholder wants to sell, he or she turns the stock in to the board, which sells it to the person at the top of the waiting list. For the past few years, the length of the waiting list has remained relatively steady, at approximately 20 names.
However, during the past winter, two events occurred that have increased the demand for shares in the club. The winter was especially severe, and subzero weather and heavy ice storms caused both the town and the county pools to buckle and crack. The problems were not discovered until maintenance crews began to ready the pools for the summer, and repairs cannot be completed until the fall. Also during the winter, the manager of the local country club had an argument with her board of directors and one night burned down the clubhouse. Although the pool itself was not damaged, the dressing room facilities, showers, and snack bar were destroyed. As a result of these two events, the Valley Swim Club was inundated with applications to purchase shares. The waiting list suddenly grew to 250 people as the summer approached.
The board of directors of the swim club had refrained from issuing new shares in the past because there was never a very great demand, and the demand that did exist was usually absorbed within a year by stock turnover. In addition, the board has a real concern about overcrowding. It seemed like the present membership was about right, and there were very few complaints about overcrowding, except on holidays like Memorial Day and the Fourth of July. However, at a recent board meeting, a number of new applicants had attended and asked the board to issue new shares.
In addition, a number of current shareholders suggested that this might be an opportunity for the club to raise some capital for needed repairs and to improve some of the existing facilities. This was tempting to the board. Although it had set the share price at \$500 in the past, the board could set it at a much higher level now. In addition, any new shares sold would result in almost total profit because the manager, lifeguard, and maintenance costs had already been budgeted for the summer and would not increase with additional members.
Before the board of directors could make a decision on whether to sell more shares and, if so, how many, the board members felt they needed more information. Specifically, they would like to know the average number of people (family members, guests, etc.) that might use the pool each day during the summer. They would also like to know the number of days they could expect more than 500 people to use the pool from June through August, given the current number of shares.
The board of directors has the following daily attendance records for June through August from the previous summer; it thinks the figures would provide accurate estimates for the upcoming summer:
139 380 193 399 177 238
273 367 378 197 161 224
172 359 461 273 308 368
275 463 242 213 256 541
337 578 177 303 391 235
402 287 245 262 400 218
487 247 390 447 224 271
198 356 284 399 239 259
310 322 417 275 274 232
347 419 474 241 205 317
393 516 194 190 361 369
421 478 207 243 411 361
595 303 215 277 419
497 223 304 241 258
341 315 331 384 130
291 258 407 246 195

The board has developed the following criteria for making a decision on whether to issue new shares:
⦁ The expected number of days on which attendance would exceed 500 should be no more than 5 with the current membership.
⦁ The current average daily attendance should be no more than 320.
⦁ The average daily weekend (Saturday and Sunday) attendance should be no more than 500. (Weekend attendance is every sixth and seventh entry in each progression of seven entries in the preceding data.)
If these criteria are met, the club will issue one new share, at a price of \$1,000, for every two average attendees between the current daily average and an upper limit of 400.
Should the club issue new shares? If so, how many will it issue, and how much additional revenue will it realize?
QUESTION 3 [10 Marks]
The owner of Gilley’s Ice Cream Parlor has noticed that she sells more ice cream on hotter days during the summer, especially on days when the temperature is 85° or higher. To plan how much ice cream to stock, she would like to know the average daily high temperature for the summer months of July and August. Assuming that daily temperatures are normally distributed, she has gathered the following data for the high temperature for 20 days from a local almanac:
86° 92°
85 94
78 83
91 81
90 84
92 76
83 78
80 78
69 85
74 90

Compute the mean and standard deviation for these data and determine the expected number of days in July and August that the high temperature will be 85° or greater.

SECTION B
QUESTION 1 [25 Marks]
EVALUATING R&D PROJECTS AT WESTCOM SYSTEMS PRODUCTS COMPANY
West-Com Systems Products Company develops computer systems and software products for commercial sale. Each year it considers and evaluates a number of different R&D projects to undertake. It develops a road map for each project, in the form of a standardized decision tree that identifies the different decision points in the R&D process from the initial decision to invest in a project’s development through the actual commercialization of the final product.
The first decision point in the R&D process is whether to fund a proposed project for 1 year. If the decision is no, then there is no resulting cost; if the decision is yes, then the project proceeds at an incremental cost to the company. The company establishes specific short-term, early technical milestones for its projects after 1 year. If the early milestones are achieved, the project proceeds to the next phase of project development; if the milestones are not achieved, the project is abandoned. In its planning process the company develops probability estimates of achieving and not achieving the early milestones. If the early milestones are achieved, the project is funded for further development during an extended time frame specific to a project. At the end of this time frame, a project is evaluated according to a second set of (later) technical milestones. Again, the company attaches probability estimates for achieving and not achieving these later milestones. If the later milestones are not achieved, the project is abandoned.
If the later milestones are achieved, technical uncertainties and problems have been overcome, and the company next assesses the project’s ability to meet its strategic business objectives. At this stage, the company wants to know if the eventual product coincides with the company’s competencies and whether there appears to be an eventual, clear market for the product. It invests in a product “prelaunch” to ascertain the answers to these questions.
The outcomes of the prelaunch are that either there is a strategic fit or there is not, and the company assigns probability estimates to each of these two possible outcomes. If there is not a strategic fit at this point, the project is abandoned and the company loses its investment in the prelaunch process. If it is determined that there is a strategic fit, then three possible decisions result: (1) The company can invest in the product’s launch, and a successful or unsuccessful outcome will result, each with an estimated probability of occurrence; (2) the company can delay the product’s launch and at a later date decide whether to launch or abandon; and (3) if it launches later, the outcomes are success or failure, each with an estimated probability of occurrence. Also, if the product launch is delayed, there is always a likelihood that the technology will become obsolete or dated in the near future, which tends to reduce the expected return.
The following table provides the various costs, event probabilities, and investment outcomes for five projects the company is considering:
Project
Decision Outcomes/Event 1 2 3 4 5
Fund 1 year \$200,000 \$350,000 \$170,000 \$230,000 \$400,000
P(Early milestones, Yes) .70 .67 .82 .60 .75
P(Early milestones, No) .30 .33 .18 .40 .25
Long-term Funding \$650,000 780,000 450,000 300,000 450,000
P(Late milestones, Yes) .60 .56 .65 .70 .72
P(Late milestones, No) .40 .44 .35 .30 .28
Prelaunch Funding \$300,000 450,000 400,000 500,000 270,000
P(Strategic fit, Yes) .80 .75 .83 .67 .65
P(Strategic fit, No) .20 .25 .17 .33 .35
Invest, Success \$7,300,000 8,000,000 4,500,000 5,200,000 3,800,000
P(Invest, Success) .60 .65 .70 .75 .80
Invest, Failure \$2,000,000 3,500,000 1,500,000 2,100,000 900,000
P(Invest, Failure) .40 .35 .30 .25 .20
Delay, Success \$4,500,000 6,000,000 3,300,000 2,500,000 2,700,000
P(Delay, Success) .80 .70 .65 .80 .85
Delay, Failure \$1,300,000 4,000,000 800,000 1,100,000 900,000
P(Delay, Failure) .20 .30 .35 .20 .15

Determine the expected value for each project and then rank the projects accordingly for the company to consider.
This case was adapted from R. K. Perdue, W. J. McAllister, P. V. King, and B. G. Berkey, “Valuation of R and D projects Using Options Pricing and Decision Analysis Models,” Interfaces 29, no. 6 (November-December 1999): 5774.
QUESTION 2 [25 Marks]
Since the operations analyst comes across different types of problems, it is a good idea to classify them into some groups. This will make it easier to select models and criteria to use in the analysis. There can be two ways of classifying problems: by the degree to which the outcome is uncertain, and by the degree to which the decisions are independent.
UNCERTAINTY OF OUTCOMES
When we know for sure what the outcome of each decision will be, we are dealing with a problem under control of certainty. When a decision has more than one possible outcome and we know the likelihood of each outcome, we are dealing with a problem under conditions of risk. Finally, when a decision has more than one possible outcome and we do not know the likelihood of each outcome, we are dealing with a problem under conditions of uncertainty. Some examples may clarify these conditions of certainty, risk, and uncertainty.
⦁ A chain of supermarkets is going to open a new store at one of four possible locations. Management wishes to select the location that will maximize profitability over the next ten years. An extensive analysis was performed to determine the costs, revenues, and profits for each alternative. The results are shown below.

Management has a high degree of confidence in these figures. The decision criterion (profit) has been explicitly identified and accurately calculated for each alternative. Advise management on the optimal location to select to achieve the identified objective.

⦁ Further analysis of the supermarket chain’s problem reveals that the profit associated with each location is not known for sure. Management is convinced that the ten-year profitability of each location will depend upon regional population growth. Therefore, management cannot predict the outcome with certainty. Three possible rates of population growth were identified: low, medium, and high. The profitability (\$ millions) associated with each location and each rate of population growth was calculated, as shown below.

The figures at the bottom of the table gives the probability (likelihood) of each rate of population growth. Decision strategy in this situation is more difficult than it is under conditions of certainty. The choice is not so easy. It is not known which location will be best because the rate of future population growth is not known for certain. Prepare a brief report for management to determine which alternative is the best.

⦁ Even further analysis has cast doubt on the probability of the rates of population growth. New management doesn’t know the probabilities of low, medium, or high growth, and is faced with a problem under conditions of uncertainty. Obviously, strategy is much harder to come by in this case. Discuss the approaches from a set of several options that analysts use in the situation of uncertainty.

GOOD LUCK

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