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Tutorial 4

# Tutorial 4 - 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 4 1. Let X 1 , X 2 , . . . , X n be a random sample from the normal distribution N (0 , θ ), 0 < θ < . What is the sufficient statistic for θ . 2. Let X 1 , X 2 , . . . , X n be a random sample from the normal distribution N ( θ, σ 2 ), -∞ < θ < , where the variance σ 2 is known. Show that ¯ X is a sufficient statistic for the mean θ . 3. Let X 1 , X 2 , . . . , X n denote a random sample from a distribution with p.d.f. f ( x | θ ) = θx θ - 1 , 0 < x < 1 0 , otherwise , where θ > 0. Use the factorization criterion to show that Q n i =1 X i = X 1 X 2 · · · X n is a sufficient statistic for θ . 4. Let X 1 , X 2 , . . . , X n be a random sample of size n from a geometric distribution. Find the sufficient statistic for θ . 5. If X 1 , X 2 , . . . , X n constitute a random sample from the population given by f ( x ) = e - ( x - δ ) , x > δ 0 , otherwise . (a) Show that ¯ X is a biased estimator of δ .

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