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# 07_20 - STAT 410 Examples for D Summer 2009 Theorem 4 M X t...

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STAT 410 Examples for 07/20/2009 Summer 2009 Theorem 4 M X n ( t ) M X ( t ) for | t | < h X X D n Example 16(a) : 4.3.11 Z n ~ Poisson ( n ) Y n = ( 29 n n n Z - M Z n ( t ) = e n ( e t – 1 ) . M Y n ( t ) = ( 29 - 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 . n n e t n t n t e z n t 6 2 1 3 2 + + + = for some z between 0 and n t . Thus, for some z between 0 and n t , M Y n ( t ) = + + + - 6 2 3 2 exp n n e t n t n t n n t z = + 6 2 3 2 exp n e t t z . As n , M Y n ( t ) 2 2 exp t = M Z ( t ), where Z has Standard Normal N ( 0, 1 ) distribution. Z Y D n , Z ~ N ( 0, 1 ).

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Example 16(b) : 4.3.14 X 1 , X 2 , … , X n are i.i.d. Poisson ( 1 ) Y n = ( 29 1 X - n n a) M X 1 ( t ) = e ( e t – 1 ) . M Y n ( t ) = ( 29 - 1 X E n n t e = ( 29 + + + - n t n t n e e 2 1 X ... X X E = n n t n t e 1 X M - = - + - 1 exp n t e n n t . b)
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07_20 - STAT 410 Examples for D Summer 2009 Theorem 4 M X t...

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