This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up to
access the rest of the document.
This is an unformatted preview. Sign up to view the full document.
View Full DocumentSuppose S = {X1 , . . . , Xn } is a simple random sample from a population with and nite variance 2 < . Show that the sample mean X is an unbiased estimator for , so is sample variance s2 for 2 . 1. Before moving to formal proofs, there are several properties regarding expectation and variance as we metioned in early chapters: E (aX + b) = aE (X ) + b. (1) This property is called linearity of expectation, which naturally implies n E (X1 + . . . + Xn ) = E (X1 ) + E (X2 + . . . + Xn ) = E (X1 ) + E (X2 ) + E (X3 + . . . + Xn ) = E (Xi ). i=1 The other properties are regarding variance, V ar(aX + b) = a2 V ar(X ), V ar(X1 + X2 ) = V ar(X1 ) + V ar(X2 ) + 2Cov (X1 , X2 ). (2) The property given in (2) implies: (a) If X1 , . . . , Xn are independent, then they are uncorrelated, i.e. Cov (Xi , Xj ), i = j . Therefore, n V ar{ n Xi } = i=1 n V ar(Xi ) + i=1 Cov (Xi , Xj ) = V ar(Xi ). i=1 i=j (b) If X1 , . . . , Xn are independent and identically distributed, n n V ar(X ) = V ar{n1 Xi } = n2 nV ar(X1 ) = Xi } = n2 V ar{ i=1 i=1 V ar(X1 ) . n 2. The following equality is helpful in many derivations, n n (Xi a)2 n{X a}2 , a. (Xi X )2 = (3) i=1 i=1 This is because n (Xi X )2 = i=1 Note that n {(Xi a) (X a)}2 = i=1 n i=1 (Xi [(Xi a)2 2(Xi a)(X a) + (X a)2 ]. i=1 a)(X a) = n(X a)(X a), the result in equation (3) follows. 3. There are two equivalent ways of expressiong variance, they are V ar(X1 ) = E {X E (X )}2 = E (X 2 ) {E (X )}2 . Now, n E ( X ) = n 1 E (Xi ) = n1 n = . i=1 E (s2 ) = E n i=1 (Xi X )2 n1 n = 1 E { (Xi X )2 } = n 1 i=1 n i=1 E (Xi )2 nE (X )2 . n1 Note that E (Xi )2 = V ar(Xi ) = 2 and E (X )2 = V ar(X ) = n1 V ar(Xi ) = n1 2 , we then have E (s2 ) = n 2 n n1 2 = 2 . n1 1 ... View Full Document
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentBinomial_examples
ch1_add_2
ch2_add_6
ch4_add_1
lec3
lec6
Chapter 8 - 54,56,57,58,62
CLT
Homework 9 solution
stat210a_2007_hw11_solutions
Chapt4-1
STAT100B_HW3S
Copyright © 2015. Course Hero, Inc.
Course Hero is not sponsored or endorsed by any college or university.