Week4Solutions - RANDOM PROCESSES IN COMMUNICATION...

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σ µ µ X X X E X E X 2 2 2 2 = = {( ) } { } E X x f x dx b a x dx b a x b a b a b ab a a b a b { } ( ) ( ) ( ) 2 2 2 3 3 3 2 2 1 1 3 1 3 1 3 = = = = = + + −∞ RANDOM PROCESSES IN COMMUNICATION ASSIGNMENT #4 PROBLEM 2.23 f x b a X ( ) , = 1 a x b < < , elsewhere f x X ( ) = 0 µ µ X X a b a b X E X xf x dx b a xdx b a x b a b a b a = = = = = = + −∞ { } ( ) ( ) ( ) 1 1 2 2 1 2 2 2 2
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[ ] σ µ X X E X b ab a b a b ab a b ab a b ab a b a or a b 2 2 2 2 2 2 3 2 2 2 2 2 2 2 1 3 1 2 3 3 3 1 4 2 4 1 4 1 12 2 1 12 1 12 = = + + + = + + = + = { } ( ) ( ) ( ) ( ) PROBLEM 2.29 f x y X Y , ( , ) , = 1 2 0 x y 0 2 y a) for and f x f x y dy dy y x X X Y x x ( ) ( , ) , = = = = −∞ 1 2 2 2 2 2 2 0 x y 0 2 y , f y f x y dx dx x y Y X Y y y ( ) ( , ) , , = = = = −∞ 1 2 2 2 0 0 0 x y 0 2 y b) f x y f x y f y y y X Y X Y Y / , ( / ) ( , ) ( ) / / = = = 1 2 2 1 for , 0 x y 0 2 y
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for f y x f x y f x x x Y X X Y X | , ( | ) ( , ) ( ) / = = = 1 2 1 2 1
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