05. Moment generating functions and properties

05. Moment generating functions and properties - University...

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University of California, Los Angeles Department of Statistics Statistics 100B Instructor: Nicolas Christou Moment generating functions Definition: M X ( t ) = Ee tX Therefore, If X is discrete M X ( t ) = X x e tX P ( x ) If X is continuous M X ( t ) = Z x e tX f ( x ) dx Aside: e x = 1 + x 1! + x 2 2! + x 3 3! + ··· Similarly, e tx = 1 + tx 1! + ( tx ) 2 2! + ( tx ) 3 3! + ··· Let X be a discrete random variable. M X ( t ) = X x e tX P ( x ) = X x " 1 + tx 1! + ( tx ) 2 2! + ( tx ) 3 3! + ··· # P ( x ) or M X ( t ) = X x P ( x ) + t 1! X x xP ( x ) + t 2 2! X x x 2 P ( x ) + t 3 3! X x x 3 P ( x ) + ··· To find the k th moment simply evaluate the k th derivative of the M X ( t ) at t = 0. EX k = [ M X ( t )] k th derivative t =0 For example: First moment: M X ( t ) 0 = X x xP ( x ) + 2 t 2! X x x 2 P ( x ) + ··· We see that M X (0) 0 = x xP ( x ) = E ( X ). 1
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Similarly, Second moment M X ( t ) 00 = X x x 2 P ( x ) + 6 t 3! X
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This note was uploaded on 01/14/2011 for the course STATS 100B 100B taught by Professor Christou during the Winter '10 term at UCLA.

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05. Moment generating functions and properties - University...

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