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Unformatted text preview: April /08 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Example Class 8 1. A random variable X follows Gamma distribution with probability den- sity function f ( x ) = x θ- 1 exp(- x ) / Γ( θ ), where x > 0 and θ > 0 is an unknown parameter. Suppose X 1 ,...,X n are i.i.d. with the distribution of X . ( C9(e), Assignment 4 ) (a) Show that the parametric family of distribution has monotone likelihood ratio with respect to the hypothesis H : θ ≥ θ against H 1 : θ < θ , where θ is a fixed constant. (b) Describe the critical region of the uniformly most powerful test of H against H 1 as specified in (a) 2. A survey of 223 phone boxes found 53 of them to be out of order. A similar survey of 202 phone boxes found 45 of them out of order one month later. It is assumed that each phone box has a fixed probability p of being out of order during the first survey, and a fixed probability q of being out of order during the second survey. The telephone company reports that the condition of the phone boxes is improving. Test H : no change in condition against H 1 : the condition may have changed. ( D1, Assignment 4...
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