Homework_1_Solutions

Homework_1_Solutions - AMS 312 Homework 1(Solution...

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Unformatted text preview: AMS 312 Homework 1 (Solution) Q1 .Derive the moment generating function (mgf) for the Normal Distribution ~ , , =- X Nμ σ2 fx 12πσexpx μ22σ2 Solution: = =-∞∞ *- MXt EetX etX 12πσexpx μ22σ2dx =-∞∞- - + = -∞∞--- + 12πσexp12σ2 x μ2 2txσ2dx 12πσexp 12σ2x2 2xμ 2txσ2 μ2dx =-∞∞--- + 12πσexp 12σ2x2 2xμ tσ2 μ2dx-∞∞-- + + + +- + Complete the Square 12πσexp 12σ2x2 2xμ tσ2 μ2 μ tσ22 μ tσ22dx =-∞∞-- +- + + 12πσexp 12σ2x μ tσ22 μ tσ22 μ2dx =-∞∞-- +--- + 12πσexp 12σ2x μ tσ22 μ2 2μσ2t t2σ4 μ2dx =-∞∞- [ - + ]- [-- ] 12πσexp 12σ2 x μ tσ2 2exp 12σ2 2μσ2t t2σ4 dx = +-∞∞-- + ; -∞∞-- + = eμt 12σ2t2 12πσexp 12σ2x μ tσ22dx 12πσexp 12σ2x μ tσ22dx 1 = + * eμt 12σ2t2 1 ~ , = + ∴Mgf of X Nμ σ2 is MXt eμt 12σ2t2 Q2 . ~ , & ~ , & , = ( ) ( ) X1 Nμ1 σ1 X2 Nμ2 σ2 X1 X2 are independent fx1 x2 f x1 f x2 ( ) + ~ + , + 1 Show X1 X2 Nμ1 μ2 σ12 σ22 : = + , Solution Let Y X1 X2 then = = + = & MYt EetY EetX1 tX2 EetX1EetX2 because X1...
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This note was uploaded on 11/14/2010 for the course AMS 312 taught by Professor Zhu,w during the Spring '08 term at SUNY Stony Brook.

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Homework_1_Solutions - AMS 312 Homework 1(Solution...

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