Assignment submission consists of both an excel report and a written report. The Excel report is an excel file showing all calculations and relevant data analysis output from Excel. In the written report, you should provide written answers to all assignment questions presenting any tables or graphs as required. The written report should be submitted via Turnitin, the university’s antiplagiarism software, as a word document (.docx) or PDF file (.pdf) file. Note that any suspected plagiarism will forwarded to the academic honesty team for investigation. For the written report you should apply the following page set-up and formatting:

Font: 12pt Times New Roman or Calibri

Margins: 2.5 cm in all sides

Line spacing: 1.5 lines

Page limit: maximum 10 pages including any tables, graphs, appendices and references

Rounding: keep your final solution with 4 decimal places

Failure to follow any of the above formatting instructions will result in an immediate penalty of 5 marks (out of 45). Markers will not read material beyond the 10-page limit.

There are no specific formatting requirements for the Excel file apart from that all calculations used for the written report must be clearly presented and labelled according to the section of the written report for which the output was used.

1. Rolling three die. Consider a game where a player rolls three standard 6-sided die. The final score the player receives is the sum of the two highest numbers on the three die. For example, if the player rolled three die and got scores of 1, 3 and 5, the final score would be 8. In the case if all three number are the same on each dice, the player receives a final score of zero. For example, if the player rolled three die and got scores of 6, 6 and 6, the final score score would be 0.

(a) (2 points) Provide a plot of the probability distribution for the player’s final score from a singlegame.

(b) (2 points) Calculate the mean and variance of the distribution of the player’s final score frompart (a).

(c) (3 points) Assuming the game is played identically and independently for two times, find andplot the distribution of the average final score across both games. Compare the distribution for the average final score across two games with the distribution for a single game found in part

(a).

(d) (3 points) Find the mean and variance of the distribution in part (c) for the average of finalscores from two games and compare them with the mean and variance found in part (b).

(e) (4 points) If the original game is played repeatedly 100 times, and the average final score acrossall 100 games is calculated, discuss with justification how the distribution of this average may be approximated, and any assumptions/conditions required. Plot this approximate distribution and discuss how it compares with the distributions in parts (a) and (c).

(f) (2 points) Imagine you were to play 100 games and then take the average of the final scoresfrom all 100 games, and then repeat the process many times. Use the approximate distribution in e) to estimate an interval within which you would expect 90% of these averages of final scores across 100 games to lie.

2. Event Management. An events manager is planning to hold a festival in Sydney for the entire month of July in 2022 and asks the weather expert for some advice on the likely number of days it could rain which could impact the number of people attending the festival. The weather expert notes that historically 21% of all days are rainy days for the month of July in Sydney. The weather expert believes that the number of rainy days in July in 2022 could be modelled with a Binomial distribution.

(a) (2 points) Discuss the requirements of the Binomial distribution and whether you feel it isappropriate in the current scenario.

(b) (3 points) Assuming the Binomial distribution is appropriate for modelling the number of rainydays in July 2022, determine the probabilities of the following:

i Exactly 8 rainy days in July next year

ii Less than 9 rainy days of rain in July next year

(c) (2 points) What is the expected number of rainy days in Sydney during July next year andwhat is the variance of the number of rainy days?

The events manager is also concerned about the possibility of serious transport disruptions to the Sydney rail network, which can seriously impact the number of people able to attend the festival. The events manager contacts Transport for NSW who inform her that serious disruptions typically occur at a rate of 1 per 40 weekdays. On weekends, disruptions are less frequent occurring at a rate of 1 per 80 days.

(d) (2 points) The events manager believes the number of disruptions on weekdays and weekendscan be separately modelled as Poisson random variables. Discuss whether you feel the choice of the Poisson distribution is appropriate for modelling the number of disruptions on either weekdays or weekends.

(e) (3 points) Assuming the Poisson distribution is appropriate for modelling these disruptions,determine the probabilities of the following:

i No more than 4 transport disruptions on weekdays during July next yearii Zero transport disruptions on weekends during July next year

(f) (2 points) Find the mean and variance for the number of weekday disruptions in July next year.(g) (2 points) Find the mean and variance for the number of weekend disruptions in July next year.

(h) (4 points) The events manager believes the level of profit from the festival is linked to both the amount of rain and the number of transport disruptions in the following way:

P = 500 − 24R− 36D− 40E

where P is the amount of profit measured in thousands of dollars, R is the number of rainy days, D is the number of weekday transport disruptions, and E is the number of weekend transport disruptions all in July next year. Calculate the expected profit for the festival, as well as the standard deviation of profit, stating any required assumptions.

3. Website Speed. The speed at which you can log into a website through a smartphone is an important quality characteristic of that website. In a recent test, the mean time to log into the INFORMS OR/MS Tomorrow website was 6.734 seconds. Suppose that the download time is normally distributed, with a standard deviation of 2.7 seconds. What is the probability that a download time is

(a) (2 points) less than 2 seconds?

(b) (2 points) between 1.8 and 2.4 seconds?

(c) (2 points) 99% of the download times are slower (higher) than how many seconds?

(d) (2 points) 95% of the download times are between what two values, symmetrically distributedaround the mean?

(e) (1 point) Suppose the download times are uniformly distributed between 1 to 13 seconds. What is your answer to 3(b)?