hw5-sol-spring2008

# hw5-sol-spring2008 - Homework 5 solution(Spring 2008...

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Unformatted text preview: Homework 5 solution (Spring, 2008) 4.2.11(d). (a) F X ( x ) = F ( x, ∞ ) = ,x ≤ , x, ≤ x ≤ . 5 , 1 / 2 + x- . 5 , . 5 ≤ x ≤ 1 , 1 ,x ≥ 1 , → F X ( x ) = ,x ≤ , x, ≤ x ≤ 1 , 1 ,x ≥ 1 , Note: lim n →∞ e- y/ 2 = 0 F Y ( y ) = F ( ∞ ,y ) = 1 / 2(1- e- y/ 2 ) + 1 / 2(1- e- y/ 4 ) ,y ≥ , ,y ≤ , 4.3.5 1. (a) P ( X ≤ 1 ,Y ≤ 1) = 0 . 1 + 0 . 04 + 0 . 08 + 0 . 2 = 0 . 42 (b) A. Pmf of X X = x 1 2 P ( X = x ) 0.16 0.34 0.5 B. Pmf of Y Y = y 1 2 P ( Y = y ) 0.24 0.38 0.38 (c) E ( X ) = 0 × . 16 + 1 × . 34 + 2 × . 5 = 1 . 34 V ar ( X ) = E ( X 2 )- E ( X ) 2 = 2 . 34- 1 . 34 2 = 0 . 544 E ( Y ) = 0 × . 24 + 1 × . 38 + 2 × . 38 = 1 . 14 V ar ( Y ) = E ( Y 2 )- E ( Y ) 2 = 0 × . 24 + 1 × . 38 + 4 × . 38- 1 . 14 2 = 0 . 6004 4. (a) Dependent since conditional distribution changes as given variable changes. (b) p ( x,y ) = p Y | X ( y | x ) · p ( x ) y 1 x 0.3726 0.1674 1 0.1445 0.0255 2 0.2436 0.0464 (c) P ( X = 1 ,Y = 1) = 0 . 0255 6 = P ( X = 1) · P ( Y = 1) = 0 . 17 × . 2393 = 0 . 040681 ⇒ X and Y are not independent....
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hw5-sol-spring2008 - Homework 5 solution(Spring 2008...

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