{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Mathematical Statistics MIdterm Exam#3-2010

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ? ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online