Tutorial 6

Tutorial 6 - 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 (2009-10) EXAMPLE CLASS 6 1. Take a random sample X 1 ,...,X n from the distribution f ( ·| θ ) given below. Derive the mle ˆ θ n and, quoting the theorem for the large-sample properties of ˆ θ n , ﬁnd its large-sample mean and variance. (a) f ( x | θ ) = θ - 1 e - x/θ , x > 0; (b) f ( x | θ ) = e - θ θ x /x !, x = 0 , 1 , 2 ,... . 2. 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. (b) A coin has a probability p to turn up a head when tossed. You want to determine if the coin is fair. (c) Assume the number of bus breakdowns on a single day is Poisson ( λ ) ran- dom variable. The company claimed that the probability that there is no bus breakdown on a single day is bigger than 3/4.

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This note was uploaded on 05/04/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.

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Tutorial 6 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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