lect21 - MOMENT GENERATING FUNCTIONS JOINT MGF Outline...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MOMENT GENERATING FUNCTIONS JOINT MGF Outline MOMENT GENERATING FUNCTIONS The Uniqueness Theorem and the Multiplication Rule JOINT MGF 1 / 15 Xinghua Zheng Lect 21: Moment Generating Functions MOMENT GENERATING FUNCTIONS JOINT MGF The Uniqueness Theorem and the Multiplication Rule MOMENT GENERATING FUNCTIONS • Let X be a random variable. For any number t , set M ( t ) := E ( e tX ) =        X x e tx f ( x ) , if X is discrete with pmf f , Z ∞-∞ e tx f ( x ) dx , if X is continuous with density f , • Formal interchanges of “ E ” and “ d / dt ” gives M ( t ) = d dt E ( e tX ) = E d dt e tX = E ( Xe tX ) , M 00 ( t ) = d dt E ( Xe tX ) = E d dt Xe tX = E ( X 2 e tX ) , M ( k ) ( t ) = d dt M ( k- 1 ) ( t ) = E d dt X k- 1 e tX = E ( X k e tX ) , for each positive integer k . • In particular, E ( X ) = , E ( X 2 ) = , and E ( X k ) = for each k . • Because M ( t ) generates the moments of X in the above manner, it is called the moment generating function (MGF) of X . 2 / 15 Xinghua Zheng Lect 21: Moment Generating Functions MOMENT GENERATING FUNCTIONS JOINT MGF The Uniqueness Theorem and the Multiplication Rule SOME DISCRETE EXAMPLES • Suppose X ∼ Poisson ( λ ) . Then M ( t ) = E ( e tX ) = ∑ ∞ k = e tk · e- λ λ k k ! = = exp ( λ ( e t- 1 )) . Taking derivatives of M ( t ) gives M ( t ) = exp ( λ ( e t- 1 )) λ e t ; M 00 ( t ) = exp ( λ ( e t- 1 )) λ 2 e 2 t + exp ( λ ( e t- 1 )) λ e t hence E ( X ) = M ( ) = ; E ( X 2 ) = M 00 ( ) = ; Var ( X ) = . • Suppose X ∼ Binomial ( n , p ) . Then M ( t ) = E ( e tX ) = ∑ n k = e tk · ( n k ) p k q n- k = = ( pe t + q ) n . Taking derivatives of M ( t ) gives M ( t ) = n ( pe t + q ) n- 1 pe t ; M 00 ( t ) = n ( n- 1 )( pe t + q ) n- 2 p 2 e 2 t + n ( pe t + q ) n- 1 pe t hence E ( X ) = M ( ) = ; E ( X 2 ) = M 00 ( ) = ; Var ( X ) = . 3 / 15 Xinghua Zheng Lect 21: Moment Generating Functions MOMENT GENERATING FUNCTIONS JOINT MGF The Uniqueness Theorem and the Multiplication Rule GAMMA RANDOM VARIABLES • Suppose X ∼ Gamma ( r ,λ ) . Then for t < λ , M ( t ) = E ( e tX ) = Z ∞ e tx · λ r e- λ x x r- 1 Γ( r ) dx = λ r ( λ- t ) r Z ∞ ( λ- t ) r e- ( λ- t ) x x r- 1 Γ( r ) dx = λ λ- t r • Taking derivatives of M ( t ) gives M ( t ) = r λ r ( λ- t )- r- 1 ; M 00 ( t ) = r ( r + 1 ) λ r ( λ- t )- r- 2 hence...
View Full Document

This note was uploaded on 11/27/2011 for the course ISOM 3540 taught by Professor Zheu during the Spring '11 term at HKUST.

Page1 / 15

lect21 - MOMENT GENERATING FUNCTIONS JOINT MGF Outline...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online