# STATSLIDE5 - MOMENT GENERATING FUNCTIONS The natural number...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ) ....
View Full Document

## 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.

### Page1 / 12

STATSLIDE5 - MOMENT GENERATING FUNCTIONS The natural number...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online