Quiz1_Solutions

Quiz1_Solutions - AMS312.01 Quiz 1 Spring 2010 ♠♣♥♦...

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Unformatted text preview: AMS312.01 Quiz 1 Spring 2010 ♠♣♥♦ Name: ________________________________ ID: ____________________ Signature: ________________________ Instruction: This is a close book exam. Anyone who cheats in the exam shall receive a grade of F. Please provide complete solutions for full credit. The exam goes from 12:50-2:10pm. Good luck! 1 (25 points). For a random variable following the normal distribution with mean μ and variance σ 2 , please derive its moment generating function. Solution: Let X be a R.V, its Moment Generation Function (m.g.f.) is defined as a special mathematical expectation with g(X) = ( ) tX E e . That is ( ) ( ) ( ) tX tx X M t E e e f x dx ∞- ∞ = = ∫ Now we derive the m.g.f. of 2 ( , ) X N μ σ : 2 2 ( ) 2 1 ( ) ( ) ( ) 2 x tx tx tx X M t E e e f x dx e e dx μ σ πσ-- ∞ ∞-∞-∞ = = = ∫ ∫ 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 x x x x t x tx e dx e dx μ μ μ σ μ σ σ πσ πσ- +-- + ∞ ∞---∞-∞ = = ∫ ∫ 2 2 2 2 2 2 2 2 2 4 2 2 2 ( ( )) ( ) ( ( )) 2 2 2 2 1 1 2 2 x t t x t t t e dx e e dx μ σ μ σ μ μ σ μ σ σ σ σ σ πσ πσ- +- + +- + + ∞ ∞---∞-∞ = = ∫ ∫ 2 2 2 2 2 2 1 t t t t e e σ σ μ μ + + = ⋅ = 2. Suppose you are on a game show, and you are given a choice of three doors. Behind one door is a car, behind the others, nothing. You would win if you have chosen the door with the car behind it. After you have chosen a door (say, door #1), the host, who knows what’s behind the door, would reveal one of the other two doors, and he would always...
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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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Quiz1_Solutions - AMS312.01 Quiz 1 Spring 2010 ♠♣♥♦...

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