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Mathematical Statistics MIdterm Exam#3-2010

Mathematical Statistics MIdterm Exam#3-2010 - Mathematical...

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Unformatted text preview: Mathematical Statistics-2010 Midterm #3 6/03/2010 1. State the definitions of the following terms. (1) The minimum variance unbiased estimator. (2) Consistent estimator. (3) Bias of an estimator. (4) Mean square error. 2. Let X , X , … , X be a random sample of size n from N µ, σ . (1) Find the distribution of the sample mean,X, and the sample variance, S . (2) Show that X and S are independent. (3) Find the MSEs of σ ∑ X X and S as an estimator of σ . (4) Show that both of σ and S are consistent estimators of σ . 3. (1) State and prove the Central Limit Theorem (CLT). (2) Show by CLT that if X is a random variable having a Poisson distribution with mean λ and λ is large, the distribution of standard normal distribution. 4. Let Y Y Y be the order statistics of a random sample from exponential e ,0 x ∞. √ can be approximated with a distribution with pdf f x (1) Find the joint density of . Y and Y . (2) Find an expression for the joint density of Y and the sample range R (3) Find the sampling distribution of R. 5. Let X , X , … , X (1) Show that σ be a random sample from N 0, σ ∑ X is an unbiased estimator of σ . . Y Y. (2) Find an expression for (3) Find the Cramèr-Rao lower bound for an unbiased estimator of σ . (4) Find the variance of σ . Is σ a minimum variance unbiased estimator of σ ? ...
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