Assign_2_Solutions

Assign_2_Solutions - STAT 330 Assignment 2 Solutions 1 Take...

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1 STAT 330 Assignment 2 Solutions 1. Take expectation of both sides of the identity given in the hint = = = = 0 1 0 1 0 1 0 1 ) ( )] ( [ )] ( [ ) ( ] [ dx x F x n dx x I E x n dx x I x E n dx x I x n E X E n X n X n X n n Taking the expectation inside the integral sign is justified because all the random variables , 0 ), ( < < x x I X are nonnegative. 2. The moment-generating function for Y is 2 0 ) 1 ( 2 1 0 ) 1 ( 2 1 0 2 1 0 2 1 | | 2 1 1 1 ) 1 ( 2 1 ) 1 ( 2 1 ) ( ) ( t t t dy e dy e dy e e dy e e dy e e e E t M t y y t y ty y ty y ty tY Y = + + = + = + = = = + . Using this, 0 ) 1 ( 2 0 2 2 ) ( ) ( = = = = t t t t Y t M Y E = 0.
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2 3. [ ] [ ] 2 1 0 2 1 ) ( ) ( ) ( 2 1 ) 0 ( ) ( 2 1 4 1 1 ) 2 ( 4 1 1 2 1 ) ( 0 ) 0 ( ) ( 4 1 1 2 1 2 1 4 1 1 ) 1 ( ) ( . 2 | | , 4 1 1 1 1 2 , 2 1 2 ) ( ) ( 2 2 2 3 2 2 2 2 2 2 2 2 2 2 / 2 / 2 / 2 2 / 2 / ) 1 ( = = = = = + = = = =
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