Math Stat Final-2007 - fሺxሻ ൌ ଵ ୻ሺ஑ሻஒ ಉ...

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Mathematical Statistics-2007 Final Exam 6/28/2007 1. Definitions. (1) The Central Limit Theorem. (2) The Neyman-Pearson Lemma. (3) Sufficient Statistics. (4) The minimum variance unbiased estimator. 2. Let X ,X ,…,X be a random sample of size n from Nሺµ, σ . (1) Find the distribution of the sample mean, X , and the sample variance, S . (2) Show that X and S are independent. 3. (1) Show that if X~F ୬,୫ , then X ିଵ ~F ୫,୬ . (2) Show that if T~t , then T ~F ଵ,୬ . 4. Let X ,X ,X be a random sample of size 3 from uniform(0,1). Let Y ,Y ,Y be the order statistics such that Y ൏Y ൏Y . (1) Find the joint distribution of Y and Y . (2) Find the distribution of the sample range RൌY െY . 5. Let X ,X ,…,X be a random sample from Gamma distribution with pdf
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Unformatted text preview: fሺxሻ ൌ ଵ ୻ሺ஑ሻஒ ಉ x ஑ିଵ e ି ౮ ಊ , x ൐ 0 , where α, β ൐ 0 are parameters. Assume that α is known. (1) Find the MLE of β . (2) Is the MLE from(1) a sufficient statistics for β . Justify your answer. (3) Find the MLE of ଵ ஒ . 6. Show that the mean of the a random sample of size n is a minimum variance unbiased estimator of the parameter λ of a Poisson distribution. 7. Let X ଵ , X ଶ , … , X ୬ be a random sample of size n from Nሺµ, 9ሻ . (1) Construct the most powerful test for testing H ଴ : µ ൌ 0 vs H ଵ : µ ൌ 1 . (2) Show that the test is the uniformly most powerful test for testing H ଴ : µ ൌ 0 vs H ଵ : µ ൐ 0 ....
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This note was uploaded on 05/09/2011 for the course DIF 1486 taught by Professor Yow-jenjou during the Spring '11 term at National Chiao Tung University.

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