This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MOMENT GENERATING FUNCTIONS The natural number e. Consider the irrational number e = { 2 . 7183 . . . } . This is defined by e = lim( n ) 1 + 1 n n . Consider the series expansion of the expression that is being taken to the limit. 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 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! + . The moment generating function. Now imagine that x f ( x ) is a random variable, and let us define M ( x, t ) = E ( e xt ) = 1 + tE ( x ) + t 2 2! E ( x 2 ) + t 3 3! E ( x 3 ) + . This is the moment generating function or m.g.f. of x . Each of its terms contains one of the moments of the random variable x taken about the origin. The problem is to extract the relevant moment from this expansion. The method is a follows. To find the r th moment E ( x r ), differentiate M ( x, t ) = E ( e xt ) 1 r times in respect of t . Then, set t to zero and the moment drops out. To demon strate this consider the following sequence of derivatives in respect of the coecient associated with E ( x r ) within the series expansion of M ( x, t ): d dt t r r ! = rt r 1 r ! d 2 dt 2 t r r ! = r ( r 1) t r 2 r ! . . . 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. Thus 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!...
View Full
Document
 Spring '12
 D.S.G.Pollock

Click to edit the document details