Homework_1_Solutions

Homework_1_Solutions - AMS 312 Homework 1 (Solution) Q1...

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Unformatted text preview: AMS 312 Homework 1 (Solution) Q1 .Derive the moment generating function (mgf) for the Normal Distribution ~ , , =- X N 2 fx 12expx 222 Solution: = =- *- MXt EetX etX 12expx 222dx =-- - + = ---- + 12exp122 x 2 2tx2dx 12exp 122x2 2x 2tx2 2dx =---- + 12exp 122x2 2x t2 2dx--- + + + +- + Complete the Square 12exp 122x2 2x t2 2 t22 t22dx =--- +- + + 12exp 122x t22 t22 2dx =--- +--- + 12exp 122x t22 2 22t t24 2dx =-- [ - + ]- [-- ] 12exp 122 x t2 2exp 122 22t t24 dx = +--- + ; --- + = et 122t2 12exp 122x t22dx 12exp 122x t22dx 1 = + * et 122t2 1 ~ , = + Mgf of X N 2 is MXt et 122t2 Q2 . ~ , & ~ , & , = ( ) ( ) X1 N1 1 X2 N2 2 X1 X2 are independent fx1 x2 f x1 f x2 ( ) + ~ + , + 1 Show X1 X2 N1 2 12 22 : = + , Solution Let Y X1 X2 then = = + = & MYt EetY EetX1 tX2 EetX1EetX2 because X1...
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Homework_1_Solutions - AMS 312 Homework 1 (Solution) Q1...

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