Tutorial 11

Tutorial 11 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2009-10) EXAMPLE CLASS 11 1. Let X 1 ,...,X n be i.i.d. Bernoulli random variables with success probability θ . (a) Find the mle of θ . What is its approximate distribution if n is very large? (b) Construct an approximate 100(1- α )% equal-tailed confidence interval for θ based on the large-sample distribution of the mle. (c) Illustrate your answer to (b) in the special case where n = 100 ,α = 0 . 05 and the number of success is observed to be 25. 2. Consider a random sample ( T 1 ,...,T n ) drawn from an exponential distribution with rate λ > 0. It can be shown that 2 λ ∑ n i =1 T i has a chi-squared distribution with 2 n degrees of freedom. (a) Derive a size 0.05 critical region for the likelihood ratio test of H : λ = λ against H 1 : λ > λ . (b) Show how to use (a) to obtain a 95% lower confidence bound for λ ....
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Tutorial 11 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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