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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2010-11) EXAMPLE CLASS 8 1. Let X 1 ,...,X n denote a random sample from the distribution that has the p.d.f. f ( x | θ ) = 1 √ 2 π exp ‰- ( x- θ ) 2 2 ,-∞ < x < ∞ . Find the critical region for the most powerful size α test of H : θ = 0 vs H 1 : θ = 1. 2. A random variable X follows Gamma distribution with probability density function f ( x | θ ) = x θ- 1 e- x / Γ( θ ) , x > . The θ is an unknown parameter. Suppose X 1 ,...,X n are i.i.d. with the distribution of X . (a) Give an expression for the p-value of the likelihood ratio test of H : θ ≥ θ against H 1 : θ < θ , where θ is a fixed constant. (b) Describe the critical region of the size α likelihood ratio test of H against H 1 as specified in (a). 3. Let X 1 ,...,X n be i.i.d. sample from an exponential distribution with p.d.f....
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- Spring '11