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Dr. Hackney STA Solutions pg 34

Dr. Hackney STA Solutions pg 34 - Second Edition 3-7 3.22 a...

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Second Edition 3-7 3.22 a. E( X ( X - 1)) = x =0 x ( x - 1) e - λ λ x x ! = e - λ λ 2 x =2 λ x - 2 ( x - 2)! (let y = x - 2) = e - λ λ 2 y =0 λ y y ! = e - λ λ 2 e λ = λ 2 E X 2 = λ 2 + E X = λ 2 + λ Var X = E X 2 - (E X ) 2 = λ 2 + λ - λ 2 = λ. b. E( X ( X - 1)) = x =0 x ( x - 1) r + x - 1 x pr (1 - p ) x = x =2 r ( r + 1) r + x - 1 x - 2 pr (1 - p ) x = r ( r + 1) (1 - p ) 2 p 2 y =0 r + 2 + y - 1 y pr + 2(1 - p ) y = r ( r - 1) (1 - p ) 2 p 2 , where in the second equality we substituted y = x - 2, and in the third equality we use the fact that we are summing over a negative binomial( r + 2 , p ) pmf. Thus, Var X = E X ( X - 1) + E X - (E X ) 2 = r ( r + 1) (1 - p ) 2 p 2 + r (1 - p ) p - r 2 (1 - p ) 2 p 2 = r (1 - p ) p 2 . c. E X 2 = 0 x 2 1 Γ( α ) β α x α - 1 e - x/β dx = 1 Γ( α ) β α 0 x α +1 e - x/β dx = 1 Γ( α ) β α Γ( α + 2) β α +2 = α ( α + 1) β 2 . Var X = E X 2 - (E X ) 2 = α ( α + 1) β 2 - α 2 β 2 = αβ 2 . d. (Use 3.3.18) E X = Γ( α +1)Γ(
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