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# Example Class6 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2010-11) EXAMPLE CLASS 6 1. The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: f ( x | x 0 , θ ) = θx θ 0 x - θ - 1 , x x 0 , θ > 1 . Assume that x 0 > 0 is known and that X 1 , X 2 , . . . , X n is an i.i.d. sample. (a) Find the method-of-moments estimator of θ . (b) Find the mle of θ , quote the theorem for the large-sample properties of mle, and find its large-sample mean and variance. 2. Let X 1 , . . . , X n is an i.i.d. sample from a Rayleigh distribution with θ > 0: f ( x | θ ) = x θ 2 e - x 2 / 2 θ 2 , x 0 . (a) Find the method-of-moments estimator of θ . (b) Find the mle of θ , quote the theorem for the large-sample properties of mle, and find its large-sample mean and variance. 3. Specify the null and alternative hypotheses in terms of unknown parameters for the following problems. (a) A property developer suggests to build a private residential estate by the lake if no more than 21,000 fish live in the lake. This project is expected to be highly profitable. However, an environmentalist objects strongly to the project as the living environment of the fish will be seriously damaged. Give your answer i. if you were the property developer; ii. if you were the environmentalist. 1

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(b) A coin has a probability p to turn up a head when tossed. You want to determine if the coin is fair.
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