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# 10_21 - STAT 410 Examples for D Fall 2011 Theorem M X t M X...

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STAT 410 Examples for 10/21/2011 Fall 2011 Theorem M X n ( t ) M X ( t ) for | t | < h X X D n 1. Let X n be χ 2 ( n ) . Let Z n = ( ) n n n 2 X - . Find the limiting distribution of Z n . M Z n ( t ) = 2 n t e - M X n ( t / n 2 ) = 2 2 2 2 1 1 n n t n t e - - = 2 2 2 2 n n t n t e e n t - - , t < 2 n . n t e 2 = + + + n n t n t o 1 2 1 2 . M Z n ( t ) = 2 2 1 1 n o n n t - + - , t < 2 n . As n , M Z n ( t ) 2 2 exp t = M Z ( t ) , where Z has Standard Normal N ( 0, 1 ) distribution. Z Z D n , Z ~ N ( 0, 1 ) .

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2. a) 4.3.11 Z n ~ Poisson ( n ) Y n = ( ) n n n Z - M Z n ( t ) = e n ( e t – 1 ) . M Y n ( t ) = ( ) - n n t n Z e E = - n t n t n Z e e E = - n t e n n t Z M = - + - 1 exp n t e n n t .
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10_21 - STAT 410 Examples for D Fall 2011 Theorem M X t M X...

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