Cochran_Sample_Variance

Cochran_Sample_Variance - Let X 1 ,...,X n be a collection...

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

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Let X 1 ,...,X n be a collection of random variables from a normal distribution with unknown mean μ and unknown variance σ 2 . Show that ( n- 1) S 2 σ 2 = ∑ n i =1 ( X i- X ) 2 /σ 2 follows a chi-square distri- bution with degrees of freedom n-1. I am presenting two proofs here. Both proofs need knowledge in linear algebra. Proof method 1: As discussed in the classroom, we can expand ∑ n i =1 ( X i- X ) 2 in the following way: n X i =1 ( X i- X ) 2 = n X i =1 ( X i- μ )- ( X- μ ) 2 = n X i =1 ( X i- μ ) 2- 2( X i- μ )( X- μ ) + ( X- μ ) 2 = n X i =1 ( X i- μ ) 2- 2( X- μ ) n X i =1 ( X i- μ ) + n ( X- μ ) 2 = n X i =1 ( X i- μ ) 2- 2( X- μ )( n X- nμ ) + n ( X- μ ) 2 = n X i =1 ( X i- μ ) 2- 2 n ( X- μ ) 2 + n ( X- μ ) 2 = n X i =1 ( X i- μ ) 2- n ( X- μ ) 2 Hence we have n X i =1 ( X i- X ) 2 σ 2 = n X i =1 ( X i- μ ) 2 σ 2- n ( X- μ ) 2 σ 2 = n X i =1 ( X i- μ ) 2 σ 2- ( X- μ ) 2 σ 2 / √ n Let U i = ( X i- μ ) /σ ∼ N (0 , 1) and U = ( U 1 + U...
View Full Document

This note was uploaded on 01/19/2012 for the course IEN 312 taught by Professor Qiu during the Spring '11 term at University of Miami.

Page1 / 2

Cochran_Sample_Variance - Let X 1 ,...,X n be a collection...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online