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FINAL EXAMACSC12/71-200MATHEMATICAL STATISTICS

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FINAL EXAM

ACSC12/71-200

MATHEMATICAL STATISTICS

Question 1: (8 Total Marks)

Indicate whether each statement is TRUE or FALSE. [2 marks each] a) If two independent random variables, 𝑋 and 𝑌, have distributions with cgfs 𝑘_𝑋(𝑡) and 𝑘_𝑌(𝑡), respectively, then the distribution of 𝑍 = 𝑋 + 𝑌 has cgf 𝑘_𝑍(𝑡) = 𝑘_𝑋(𝑡) + 𝑘_𝑌(𝑡).

1. If 𝑋 and 𝑌 are independent normally distributed random variables, then their sum, 𝑆 = 𝑋 + 𝑌, and their difference, 𝐷 = 𝑋 − 𝑌, are also independent.
2. If 𝑁 is a discrete random variable with support {0,1,2 … } and 𝐸(𝑁) = 𝑛, a positive integer, and 𝑋₁, 𝑋₂, … are independent and identically distributed random variables all independent of 𝑁, then the random sum 𝑆 𝑋_𝑖 and the fixed sum 𝑇 = ∑^𝑛_𝑖=1 𝑋_𝑖 must have 𝑉𝑎𝑟(𝑆) ≥ 𝑉𝑎𝑟(𝑇).
3. If 𝑇₁ and 𝑇₂ are estimators of a parameter 𝜃 based on the same data, and 𝑇₁ is unbiased but 𝑇₂ is

                             not, then the variance of 𝑇₁ must be lower than the variance of 𝑇₂.

Question 2: (10 Total Marks)

Let 𝑋₁, … , 𝑋₁₀₀ be iid 𝑊𝑒𝑖𝑏𝑢𝑙𝑙(2,1/𝜃) random variables with 𝐸 𝜃 and pdf: 2     −𝑥2

𝑓_𝑋

𝜃

Suppose their observed values satisfy .

1. a) Find the method of moments estimator, 𝜃̂_𝑀𝑜𝑀, and the maximum likelihood estimator, 𝜃̂_𝑀𝐿𝐸, for

𝜃 based on this data. [NOTE: Recall  [4 marks] b) Find an approximate 95% confidence interval for 𝜃 based on 𝜃̂_𝑀𝐿𝐸. [3 marks]

1. c) Find an approximate 95% confidence interval for 𝜃 based on 𝜃̂_𝑀𝑜𝑀. [3 marks]

Question 3: (9 Total Marks)

The route I walk from my Gold Coast apartment to Bond has 4 street crossings. I have determined I independently encounter road traffic at a crossing with probability 0.4. Thus, the number of crossings I must stop and wait at during a trip is 𝑇 ~ 𝐵𝑖𝑛(4,0.4). Also, the lengths of time I have to stop are iid random variables, 𝑅_𝑖, so the extra time added to my trip due to traffic is 𝑋 = ∑^𝑇_𝑖=1 𝑅_𝑖. Finally, I have discovered my walking time without any stops, 𝑊, is normally distributed with mean 8 minutes and standard deviation 2 minutes.

1. a) If the 𝑅_𝑖’s are independent of 𝑇 and 𝐸(𝑅_𝑖) = 2 min, 𝑆𝐷(𝑅_𝑖) = 0.5 min and 𝐶𝑜𝑟𝑟(𝑋, 𝑊) = 0.5, find the mean and standard deviation of my total trip time, 𝑋 + 𝑊. [4 marks] b) Assume 𝑅_𝑖 has mgf 𝑚_𝑅(𝑡) and consider new random variables 𝑅_𝑖^′ defined by 𝑅_𝑖^′ = 0 if I don’t have to stop at the i^th crossing and 𝑅_𝑖^′ = 𝑅_𝑖 if I do. Show 𝑋′ = ∑⁴_𝑖=1 𝑅_𝑖^′ has the same distribution as 𝑋 and comment. [HINT: Find mgfs of 𝑋 and 𝑋′ and compare.] [2 marks] c) Now, suppose 𝑅_𝑖 ~ 𝑁(2,0.5) and 𝐶𝑜𝑟𝑟(𝑅₁, 𝑊) = 0.8. Calculate 𝐸(𝑊|𝑅₁ = 3). [3 marks]

Question 4: (9 Total Marks)

You have recently been appointed Deputy Defective DooDad Detective at WhatsItWorks Corp and you have determined the proportion of defective DooDads (called DooDuds) from the current DooDad Delivery Device is 10%. Management is considering upgrading to the DoctorDoo2000, advertised as making fewer defectives, but say it is only worth it if you confirm the lower defective rate (at 𝛼 = 0.05).

1. You have a source on the DoctorDoo2000 development team who tells you its true defective rate is 𝑝 = 9%. Based on this, what size sample will have a power of detection of 90% (i.e., how big does 𝑛 need to be so that the chance of rejecting 𝐻₀: 𝑝 = 0.1 is 90%)? [2 marks]
2. You take a sample of 7500 DooDads from a DoctorDoo2000 and observe 691 DooDuds, should you recommend the upgrade to management or not? [4 marks]
3. Your source now informs you DoctorDoo2000s may be inconsistent, individual machines having varying defective rates. This concern is based on samples from two DoctorDoo2000s, each of size 5000, the first had 455 DooDuds and the second 595. Do you agree this is statistically significant evidence (at 𝛼 = 0.05) of a difference in defective rates between machines? [3 marks]

Question 5: (9 Total Marks)

I am a huge fan of superhero comic books and movies; however, I have always liked Marvel characters better than DC characters. To assess whether my views are consistent with the wider superhero-loving public, I gathered data about the box-office opening weekend earnings (in millions of US dollars) for a number of recent superhero movies from both companies and summarised the data:

1. a) Using this information, calculate a CLT-based 95% confidence interval for the mean difference in first weekend earnings for Marvel versus DC movies. [4 marks] b) Earnings are generally not symmetrically distributed, so there is concern about using CLT-based intervals with such small samples. To deal with this, a bootstrap-t procedure is implemented. From 𝐵 = 10,000 bootstrap samples, we calculate values of the form:

                                                                   𝑡_𝑏∗ =  𝑀𝑎𝑟𝑣,𝑏) − (𝑋̅𝐷𝐶 − 𝑋̅𝑀𝑎𝑟𝑣)

𝑠

and find that 𝑡 and 𝑡.  Use this information to construct a bootstrap-t confidence interval for the mean difference in weekend earnings. [2 marks]

Suppose overall earnings of superhero movies are independent and 𝐿𝑁(𝜃, 1), so that, 𝑋_𝑖, the earnings of a random superhero movie has expectation 𝐸(𝑋_𝑖) = 𝑒^𝜃+0.5 = 𝜏(𝜃).  Define estimators 𝑇₁ = 𝑋̅ = 𝑒^ln(𝑋̅) and 𝑇₂ = 𝑒^{̅ln̅̅̅(̅𝑋̅̅̅)} (i.e., “𝑒 to the average log value” instead of “𝑒 to the log of the average value”).  

10. c) Clearly, 𝑇₁ is an unbiased estimate of 𝜏(𝜃). Show 𝑇₂ is a biased and inconsistent estimator of 𝜏(𝜃), but is still more accurate than 𝑇₁ when the sample size is less than 10. [3 marks].