100BHWS3

# 100BHWS3 - STAT 100B HW III solution Problem 1: Suppose X1...

This preview shows pages 1–2. Sign up to view the full content.

STAT 100B HW III solution Problem 1: Suppose X 1 ,...,X n f ( x ) independently. Let μ = E[ X ], and σ 2 = Var[ X ]. Let ¯ X = 1 n n X i =1 X i , and s 2 = 1 n - 1 n X i =1 ( X i - ¯ X ) 2 . Prove E[ ¯ X ] = μ , and E( s 2 ) = σ 2 . A: E[ ¯ X ] = E[( n i =1 X i ) /n ] = ( n i =1 E[ X i ]) /n = nμ/n = μ . E[ s 2 ] = E[ 1 n - 1 n X i =1 ( X i - ¯ X ) 2 ] (1) = 1 n - 1 E[ n X i =1 ( X i - μ ) 2 - n ( ¯ X - μ ) 2 ] (2) = 1 n - 1 [ n X i =1 E( X i - μ ) 2 - n E( ¯ X - μ ) 2 ] (3) = 1 n - 1 ( 2 - n ( σ 2 /n )) = σ 2 . (4) Problem 2: Suppose we observe X 1 ,...,X n N ( μ 1 2 ), and Y 1 ,...,Y m N ( μ 2 2 ), and X 1 ,...,X n , Y 1 ,...,Y m are all independent of each other. Suppose we want to test whether μ 1 = μ 2 . (1) If we known σ 2 , then what is an appropriate test statistic, and what is the distribution of this test statistic under the null hypothesis? A: The test statistic is

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 note was uploaded on 03/30/2011 for the course STAT 100B taught by Professor Wu during the Winter '11 term at UCLA.

### Page1 / 2

100BHWS3 - STAT 100B HW III solution Problem 1: Suppose X1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online