STAT100B_HW7S

STAT100B_HW7S - STAT 100B: Homework 7 Solutions Denise Tsai...

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Unformatted text preview: STAT 100B: Homework 7 Solutions Denise Tsai March 14, 2011 1. Let Y 1 ,...,Y n Bernoulli( ) independently. Let [ X i | Y i = 1] N ( 1 , 2 ) and [ X i | Y i = 0] N ( , 2 ). (a) Write down the likelihood L ( ), where = ( , , 1 , 2 ). Solution: L ( ) = n Y i =1 P ( x i ,y i ) = n Y i =1 P ( x i | y i ) P ( y i ) = n Y i =1 f 1 ( x i ) y i f ( x i ) 1- y i y i (1- ) 1- y i = n Y i =1 [ f 1 ( x i )] y i [(1- ) f ( x i )] 1- y i = n Y i =1 1 2 2 e- ( x i- 1 ) 2 2 2 y i (1- ) 1 2 2 e- ( x i- 1 ) 2 2 2 1- y i = n i =1 y i (1- ) n- n i =1 y i 1 2 2 n exp- n i =1 ( x i- 1 ) 2 y i 2 2- n i =1 ( x i- ) 2 (1- y i ) 2 2 = n i =1 y i (1- ) n- n i =1 y i (2 2 )- n/ 2 exp (- 1 2 2 n X i =1 ( x i- 1 ) 2 y i + ( x i- ) 2 (1- y i ) ) 1 (b) Find the MLE of , by setting the derivatives of the log-likelihood equal to 0....
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This note was uploaded on 09/20/2011 for the course STAT 100B taught by Professor Wu during the Winter '11 term at UCLA.

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STAT100B_HW7S - STAT 100B: Homework 7 Solutions Denise Tsai...

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