assi_soln - 1 Solutions of Assignment#1 STAT 330 Due in...

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Unformatted text preview: 1 Solutions of Assignment #1 - STAT 330 Due in class: Thursday May 19 Important Note: You need to print out this page as the cover page for your assignment. LAST NAME: FIRST NAME: ID. NO.: QUESTION 1. /12 QUESTION 2. /8 QUESTION 3. /3 QUESTION 4. /2 QUESTION 5. /9 QUESTION 6. /6 TOTAL: /40 2 1. Solution: (a) (i) ∞ X x =0 f ( x ) = 1 ⇒ ∞ X x =0 k (0 . 3) x = k · 1 1- . 3 = 1 ∴ k = 0 . 7 (ii) F ( x ) = P ( X ≤ x ) = X r ≤ x f ( r ) = x X r =0 . 7(0 . 3) r = 0 . 7 · 1- . 3 x +1 1- . 3 = 1- . 3 x +1 , x = 0 , 1 , 2 ,... which gives a step function: F ( x ) = 1- . 3 n +1 , n ≤ x < n + 1 , otherwise where n = 0 , 1 , 2 ,... (iii) E ( X ) = ∞ X x =0 xf ( x ) = ∞ X x =0 x · . 7 · (0 . 3) x = ∞ X x =1 . 21 · x (0 . 3) x- 1 = 0 . 21 ∞ X x =1 x (0 . 3) x- 1 = 0 . 21( 1 1- . 3 ) 2 = 3 7 where we used the technique: ∞ X n =1 ny n- 1 = ∞ X n =1 ( y n ) = ( ∞ X n =1 y n ) = ( y 1- y ) = 1 (1- y ) 2 , for | y | < 1 3 (b) (i) Z ∞ kx 2 e- λx dx = 1 , let y = λx, then Z ∞ k ( y λ ) 2 e- y 1 λ dy = k λ 3 Z ∞ y 3- 1 e- y dy = k λ 3 Γ(3) = 2 k λ 3 ⇒ k = λ 3 2 (ii) When x > 0: F ( x ) = Z x f ( t ) dt = Z x λ 3 2 t 2 e- λt dt = Z λx 1 2 y 2 e- y dy, let y = λt = Z λx 1 2 y 2 d (- e- y ) =- 1 2 y 2 e- y λx + Z λx e- y · ydy =- 1 2 ( λx ) 2 e- λx + y (- e- y ) λx + Z λx e- y dy = 1- e- λx 1 + λx + 1 2 ( λx ) 2 When x ≤ 0: F ( x ) = 0, therefore, F ( x ) = 1- e- λx (1 + λx + 1 2 ( λx ) 2 ) , x > , x ≤ 4 (iii) E ( X ) = Z ∞-∞ xf ( x ) dx = Z ∞ x · λ 3 2 x 2 e- λx dx = λ 3 2 Z ∞ x 3 e- λx dx = 1 2 λ Z ∞ y 4- 1 e- y dy, let y = λx = 1 2 λ Γ(4) = 3! 2 λ = 3 λ (c) (i) Z ∞ k (1 + x )- ( θ +1) dx = 1 ∴- k θ (1 + x )- θ ∞ = 1 k θ = 1 ⇒ k = θ (ii) when x > 0: F ( x ) = Z x θ (1 + t )- ( θ +1) dt = θ [- (1 + t )- θ θ | x ] = 1- (1 + x )- θ when x ≤ 0: F ( x ) = 0 Therefore, F ( x ) = 1- (1 + x )- θ , x > , x ≤ (iii) E ( X ) = Z ∞ x · θ (1 + x )- ( θ +1) dx = Z ∞ 1 ( y- 1) θy- ( θ +1) dy, let y = 1 + x = Z ∞ 1 θy- θ dy- Z ∞ 1 θy- ( θ +1) dy = θ · 1- θ + 1 y- θ +1 ∞ 1- θ ·...
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This note was uploaded on 07/13/2011 for the course STAT 330 taught by Professor Paulasmith during the Spring '08 term at Waterloo.

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assi_soln - 1 Solutions of Assignment#1 STAT 330 Due in...

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