STATSLIDE5 - MOMENT GENERATING FUNCTIONS The natural number...

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Unformatted text preview: MOMENT GENERATING FUNCTIONS The natural number e. The number e = { 2 . 7183 . . . } is defined by e = lim( n ) 1 + 1 n n . The binomial expansion indicates that ( a + b ) n = a n + na n 1 b + n ( n 1) 2! a n 2 b 2 + n ( n 1)( n 2) 3! a n 3 b 2 + . Using this, we get 1 + 1 n n = 1 + n 1 n + n ( n 1) 2! 1 n 2 + n ( n 1)( n 2) 3! 1 n 3 + . Taking limits as n of each term of the expansion gives lim( n ) 1 + 1 n n = 1 0! + 1 1! + 1 2! + 1 3! + = e. 1 The expansion of e x . There is also e x = lim( p ) 1 + 1 p px = lim( n ) 1 + x n n ; n = px. Using the binomial expansion in the same way as before, it can be shown that e x = x 0! + x 1! + x 2 2! + x 3 3! + . Also e xt = 1 + xt + x 2 t 2 2! + x 3 t 3 3! + . 2 The moment generating function. The m.f.g. of the random variable x f ( x ) is a random is defined as M ( x, t ) = E ( e xt ) = 1 + tE ( x ) + t 2 2! E ( x 2 ) + t 3 3! E ( x 3 ) + . Consider the following derivatives in respect of t r /r !, which is the coecient associated with E ( x r ) within t M ( x, t ): d dt t r r ! = rt r 1 r ! = t r 1 ( r 1)! d 2 dt 2 t r r ! = r ( r 1) t r 2 r ! = t r 2 ( r 2)! . . . d r dt r t r r ! = { r ( r 1) 2 . 1 } t r ! = 1 . . . d k dt k t r r ! = 0 for k > r. 3 From this, it follows that d r dt r M ( x, t ) = d r dt r ( X h =0 t h h ! E ( x h ) ) = E ( x r ) + tE ( x r +1 ) + t 2 2! E ( x r +2 ) + t 3 3! E ( x r +3 ) + . To eliminate the terms in higher powers of x , we set t = 0: d r dt r M ( x, t ) t =0 = E ( x r ) ....
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This note was uploaded on 03/02/2012 for the course EC 2019 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.

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STATSLIDE5 - MOMENT GENERATING FUNCTIONS The natural number...

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