Solution to HW 10 - Solutions to EE351K homework assignment...

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Solutions to EE351K homework assignment 10 1. a. 0 0 0 2 1 1 ( ) ( ) | ( ) | | 1 1 (1 ) Y s s s d d E Y M s ds ds s s = = = = = = = - - 2 2 2 0 0 0 2 2 3 1 2 ( ) ( ) | ( ) | | 2 1 (1 ) Y s s s d d E Y M s ds ds s s = = = = = = = - - 3 3 3 0 0 0 3 3 4 1 6 ( ) ( ) | ( ) | | 6 1 (1 ) Y s s s d d E Y M s ds ds s s = = = = = = = - - b. Since X and Y are independent, and W=X+Y, then 5 1 ( ) ( ) ( ) (1 ) W X Y M s M s M s s = = - 2 2 2 0 0 0 2 2 5 7 1 30 ( ) ( ) | ( ) | | 30 (1 ) (1 ) W s s s d d E W M s ds ds s s = = = = = = = - - 2. Consider Kn as the sum of n random variables, each of which is a Bernoulli trial with mean 0.2 and standard deviation 0.4. Hence Kn has mean 0.2n and standard deviation 0.4 n a. 100 ( ) 20 E K = b. 100 4 K σ = c. 100 100 20 18 20 ( 18) ( ) 1 ( 0.5) (0.5) 0.6915 4 4 K P K P - - = = - Φ - = Φ = d. 100 100 20 (16 24) ( 1 1) (1) ( 1) 2 (1) 1 0.6826 4 K P K P - = - ≤ ≈ Φ - Φ - = Φ - = 3. a. Each bird follows a uniform distribution to land on a spot in the 100 span. Suppose they are independent on each other. Denote the first two birds as X and Y, and we are trying to compute E(|X-Y|). 100 100 , 0 0 100 100100 0 0 0 100 100 2 2 100 0 4 4 0 0 3 100 3 2 0 4 4 1 1 (| |) | | ( , ) | | 100 100 1 1 1 1 ( ) ( ) 100 100 100 100 1 1 1 1 [( ) | ] [( ) | ] 10 2 10 2 1 1 1 1 | (5000 50 10 6 10 6 X Y x x x x E X Y x y f x y dydx x y dydx x y dydx y x dydx xy y dx y xy dx x x x x - = - = - = - + - = - + - = + + - ∫∫
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