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20071ee131A_1_131a4sol

20071ee131A_1_131a4sol - EE 131A Homework#4 Solution Winter...

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EE 131A Homework #4 Winter 2007 Solution K. Yao 1. X is a random variable with a binomial distribution with parameter n and p = 0 . 9 a. Prob. the system will declare an airplane = P (1 X 2) = 1 - P ( X = 0) = 1 - 2 0 (0 . 9) 0 (0 . 1) 2 = 0 . 99 b. Prob. the system will declare an airplane = P (1 X 4) = 1 - P ( X = 0) = 1 - 4 0 (0 . 9) 0 (0 . 1) 4 = 0 . 9999 3. Prob. the system will declare an airplane = P (1 X n ) = 1 - P ( X = 0) = 1 - n 0 (0 . 9) 0 (0 . 1) n = 0 . 999 (0 . 1) n = 0 . 001 ⇐⇒ n = ln 0 . 001 ln 0 . 1 = 3 2. a. P ( X 6) = 0 . 250 . b. P ( X 12) = 1 - P ( 11) = 1 - 0 . 943 = 0 . 057 . c. P ( X = 8) = P ( X 8) - P ( X 7) = 0 . 596 - 0 . 416 = 0 . 180 . 3. (a) 40! 30!5!3!2! ( . 6) 30 ( . 3) 5 ( . 07) 3 ( . 03) 2 (b) P (30 good, 4 fair, 6 others) = 40! 30!4!6! ( . 6) 30 ( . 3) 4 ( . 1) 6 (c) ( . 97) 40 4. S Y = { 1 , 2 , 3 , 4 } ; P [ Y = 1] = P [ { a } ] = 1 2 ; P [ Y = 2] = P [ { b } ] = 1 4 ; P [ Y = 3] = P [ { c } ] = 1 8 ; P [ Y = 4] = P [ { d, e } ] = 2 16 = 1 8 . 5. a. ( 1 2 ) 16 b. ( 16 7 ) ( 1 2 ) 16 c. P (15H) + P (16H) = ( 16 15 ) ( 1 2 ) 16 + ( 1 2 ) 16 6. a. F ( u ) > 1. b. F ( u ) is decreasing for some u . c. F ( u ) is not continuous from the right. 7. a. F (0 - ) = lim x 0 - F ( x ) = lim x 0 1 2 e x = 1 2 . Therefore, P ( X = 0) = F (0) - F (0 - ) = (1 - 1 4 e - 0 ) - 1 2 = 1 4 . b. P (0 < X 2) = F (2) - F (0) = (1 - 1 4 e - 2 ) - (1 - 1 4 e - 0 ) = 1 4 (1 - e - 2 ). c. P (0 X 2) = F (2) - F (0) + P ( X = 0) = 1 4 (1 - e - 2 ) + 1 4 = 1 2 - 1 4 e - 2 . d. P ( - 3 < X < 0) = F (0) - F ( - 3) - P ( X = 0) = 3 4 - ( 1 2 e - 3 ) - 1 4 = 1 2 - 1 2 e -
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