Question 1
Let ???????? and ???????? be discrete random variables with joint frequencies shown in Table 1 below
(a) With clear workings, evaluate (i.e. compute) the following probabilities: i) ????????(???????? >1, ???????? = 3 ), ii) ????????(???????? ≥ 2, ????????2 < 4 ).
(b) Compute
????????(???????? = ????????)????????(???????? = 1|???????? = ????????) ????????=0 and
????????(???????? = 1, ???????? = 0) + ????????(???????? = 1, ???????? = 1) + ????????(???????? = 1, ???????? = 2) + ????????(???????? = 1, ???????? = 3)
What do you observe about the two terms? Explain the rationale behind your
observation(s).
(c) Discuss the steps you would implement to calculate ????????[????????|???????? ≤ 1]. What are some practical applications of conditional expectations?
Question 2
Let ???????? and Φ be the pdf and cdf, respectively, of the standard normal variable ????????. Let ???????? = 2???????? − 3. Let ???????? be a binomial random variable, independent of ????????, with parameters ???????? = 10 and ???????? = 0.2.
(a) Can you determine ????????(???????? ≤ 1|???????? = 2)? If so, calculate this probability.
(b) Determine the probability for which ???????? is within one standard deviation of its mean.
(c) Appraise if we may determine the mean, median and variance of ???????? = ???????? + ????????. If so, compute them.
(d) Let ????????1, ????????2, … , ????????5 be 5 independent standard normal random variables. For each ???????? = 1, … ,5, define ???????????????? = 2???????????????? − 3
Let ???????? = 1 4��???????????????? + 3� 25 ????????=1
Using what you understand about normal random variables and the random variables associated with it, describe how you would compute ????????(???????? > 2). Therefore, compute ????????(???????? > 2|???????? > 2).
Question 3
You are recently hired by Corsica Worldwide Holdings (CWH) as general manager of The Capri, a five-star hotel owned by CWH. The senior management at CWH wants you to assess the hotel’s business and develop a business strategy to reduce cancellations or no-shows. You have been given a tight dateline to draft a business proposal.
As general manager, one of the things you set out to do is to analyze and interpret the hotel’s bookings data. You want to gain insights into the booking behaviors of your customers, and hopefully, identify where the cancellations are taking place and how to reduce them. The data on The Capri’s bookings can be found in ANL321_TMA_hotel.csv.
Using the appropriate statistical tools, conduct an exploratory data analysis on The Capri’s booking data. Write a short report, not exceeding 1000 words excluding figures and tables,
with the heading “A Report on The Capri and Possible Business Strategies”. Your exploratory data analysis should provide the senior management an overview of the business at The Capri as well as insights on the hotel’s guests and booking attributes, such as what the busiest months are, which market segments do the guests come from, etc.
Your report should also provide insights on cancellations at The Capri. Within your report, craft a proposal to the senior management to explain your strategies on how a reduction on cancellations can be achieved. Regression analysis is not required for this report.
Your report will be assessed based on several factors: discussion of the hotel’s business, the quality of writing (e.g. structure, coherence, argumentation, etc.), exploratory data analysis and business recommendations.
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