Question 1

Let 𝑋𝑋 and 𝑌𝑌 be discrete random variables with joint frequencies shown in Table 1 below

(a) With clear workings, evaluate (i.e. compute) the following probabilities: i) 𝑃𝑃(𝑋𝑋 >1, 𝑌𝑌 = 3 ), ii) 𝑃𝑃(𝑋𝑋 ≥ 2, 𝑌𝑌2 < 4 ).

(b) Compute

𝑃𝑃(𝑋𝑋 = 𝑗𝑗)𝑃𝑃(𝑌𝑌 = 1|𝑋𝑋 = 𝑗𝑗) 𝑗𝑗=0 and
𝑃𝑃(𝑌𝑌 = 1, 𝑋𝑋 = 0) + 𝑃𝑃(𝑌𝑌 = 1, 𝑋𝑋 = 1) + 𝑃𝑃(𝑌𝑌 = 1, 𝑋𝑋 = 2) + 𝑃𝑃(𝑌𝑌 = 1, 𝑋𝑋 = 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 𝐸𝐸[𝑌𝑌|𝑋𝑋 ≤ 1]. What are some practical applications of conditional expectations?

Question 2

Let 𝜙𝜙 and Φ be the pdf and cdf, respectively, of the standard normal variable 𝑍𝑍. Let 𝑋𝑋 = 2𝑍𝑍 − 3. Let 𝑌𝑌 be a binomial random variable, independent of 𝑋𝑋, with parameters 𝑛𝑛 = 10 and 𝑝𝑝 = 0.2.

(a) Can you determine 𝑃𝑃(𝑋𝑋 ≤ 1|𝑌𝑌 = 2)? If so, calculate this probability.

(b) Determine the probability for which 𝑋𝑋 is within one standard deviation of its mean.

(c) Appraise if we may determine the mean, median and variance of 𝑅𝑅 = 𝑋𝑋 + 𝑌𝑌. If so, compute them.

(d) Let 𝑍𝑍1, 𝑍𝑍2, … , 𝑍𝑍5 be 5 independent standard normal random variables. For each 𝑗𝑗 = 1, … ,5, define 𝑋𝑋𝑗𝑗 = 2𝑍𝑍𝑗𝑗 − 3

Let 𝑆𝑆 = 1 4��𝑋𝑋𝑗𝑗 + 3� 25 𝑗𝑗=1

Using what you understand about normal random variables and the random variables associated with it, describe how you would compute 𝑃𝑃(𝑆𝑆 > 2). Therefore, compute 𝑃𝑃(𝑌𝑌 > 2|𝑆𝑆 > 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.