solution7 - EL630 Homework7 Solutions 1 ∑ ∑ ∑ ∞ =...

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Unformatted text preview: EL630: Homework7 Solutions 1) ∑ ∑ ∑ ∞ =- ∞ = +- ∞ =- = + = + = + 1 1 )! ( 1 )! 1 ( 1 ! 1 1 1 1 l l k k k k l e k e e k k X E λ λ λ λ λ λ λ λ ( 29 ( 29 λ λ λ λ λ λ λ λ-- ∞ =-- =- = - = ∑ e e e l e l l 1 1 1 1 1 )! ( 1 . 2) [ ] [ ] [ ] 2 1 2 / 1 4 1 2 / 1 4 1 2 / 4 1 4 1 ) ( 1 1 2 1 1 = +-- + = + = + =-- ∫ x x xy y dy xy x f X By symmetry, 2 1 ) ( = y f Y also. X and Y are not independent because the product of their marginal densities does not equal the joint density. 3) (a) [ ] [ ] [ ] ) 6 / 1 ( 2 / 4 3 ) 6 / ( 2 / 4 3 ) 2 / ( 4 3 ) ( 1 3 2 1 2 + = + = + = ∫ y x y x dx x xy y f Y . (b) [ ] [ ] ∫ ∫ ∫ ∫ + + = < < < = < < 2 2 / 1 2 1 2 / 1 2 ) 2 / ( 4 3 ) 2 / ( 4 3 ) 2 / 1 ( ) 2 / 1 , 1 ( ) 2 / 1 1 ( dy dx x xy dy dx x xy X P X Y P X Y P 4) (a) [ ] [ ] [ ] [ ] 4 / 2 / 1 4 1 4 / 2 / 1 4 1 4 / 2 / 4 1 ) ( 1 4 1 ) ( 3 3 1 1 4 2 3 1 1 2 2 x x x x xy y x y dy y x xy x f X- +--- + =- + =- + =-- ∫...
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solution7 - EL630 Homework7 Solutions 1 ∑ ∑ ∑ ∞ =...

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