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Example Class 9

Example Class 9 - The University of Hong Kong Department of...

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Page 1 of 2 The University of Hong Kong Department of Statistics and Actuarial Science STAT1801 Probability and Statistics: Foundations of Actuarial Science (10-11) Example Class 9 1. Based on a random sample of 2 observations, consider two competing estimators of the population mean 𝜇𝜇 : 𝜇𝜇̂ 1 = 1 2 𝑋𝑋 1 + 1 2 𝑋𝑋 2 , 𝜇𝜇̂ 2 = 1 3 𝑋𝑋 1 + 2 3 𝑋𝑋 2 Which one is more efficient? 2. Let 𝑋𝑋 1 , 𝑋𝑋 2 , … , 𝑋𝑋 𝑛𝑛 be i.i.d. Bernoulli random variables such that 𝑋𝑋 𝑖𝑖 = 1 0 , 𝑝𝑝 , 1 − 𝑝𝑝 Find the maximum likelihood estimator of 𝑝𝑝 . 3. A random sample of 5 men and 5 women had the following cholesterol levels in their blood (grams per liter): Men: 3.0 1.8 2.1 2.7 1.4 Women: 2.7 2.2 1.9 1.4 1.5 Assume the cholesterol levels are normally distributed with population mean 𝜇𝜇 𝑥𝑥 and population variance 𝜎𝜎 𝑥𝑥 2 for men, population mean 𝜇𝜇 𝑦𝑦 and population variance 𝜎𝜎 𝑦𝑦 2 for women. The two samples are assumed to be independent.
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