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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 chisquare distri bution with degrees of freedom n1. 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...
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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.
 Spring '11
 Qiu

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