Hw5sol - ∞ e-λx dx = 1 λ 4. For α > 1, E [ Y ] =...

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ISyE 2027B Probability with Applications Fall 2010 Homework 5 Solutions 1. Let X be the IQ score and X N (100 , 15 2 ). Let Z denote the random variable that has a standard Normal distribution, Z N (0 , 1 2 ). (a) P ( X > 130) = P ( X - μ σ > 130 - 100 15 ) = P ( Z > 2) = 0 . 0228 (b) P (90 < X < 110) = P ( 90 - 100 15 < X - μ σ < 110 - 100 15 ) = P ( - 2 3 < Z < 2 3 ) = P ( Z < 2 3 ) - P ( Z < - 2 3 ) = Φ( 2 3 ) - Φ( - 2 3 ) = Φ( 2 3 ) - (1 - Φ( 2 3 )) = 2 × Φ( 2 3 ) - 1 0 . 4950 Note: 1. We use the fact that P ( Z < a ) = P ( Z a ) since Z is a continuous random variable. 2. If you approximate Φ( 2 3 ) with Φ(0 . 67), the answer will be 0.4972. Both are correct. 2. Let X represent the travel time, so X N (40 , 7 2 ). And we want to find x such that P ( X x ) = 0 . 95. P ( X - μ σ < x - μ σ ) = P ( Z < x - 40 7 ) = 0 . 95 By the normal distribution table, we get x - 40 7 = 1 . 645, thus x = 51 . 515 mins. That is, you should leave home at least before 9:08AM to guarantee to be on time with probability 0.95. 3. You need to use partial integral. E [ X ] = 0 xλe - λx dx = 0 xe - λx d ( λx ) = 0 - xd ( e - λx ) = - xe - λx | 0 - 0 e - λx d ( - x ) =
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Unformatted text preview: ∞ e-λx dx = 1 λ 4. For α > 1, E [ Y ] = ∫ ∞ 1 y α y α +1 dy = ∫ ∞ 1 αy-α dy =-α α-1 y-( α-1) | ∞ 1 = α α-1 5. (a) Since LHS = k n ! ( n-k )! k ! = n ! ( n-k )!( k-1)! 1 ISyE 2027B Probability with Applications Fall 2010 RHS = n ( n-1)! ( n-1-( k-1))!( k-1)! = n ! ( n-k )!( k-1)! So LHS = RHS. Done. (b) E [ Z ] = n ∑ k =0 k ( n k ) p k (1-p ) n-k = n ∑ k =1 k ( n k ) p k (1-p ) n-k = n ∑ k =1 n ( n-1 k-1 ) pp k-1 (1-p ) n-k by (a) = np n ∑ k =1 ( n-1 k-1 ) p k-1 (1-p ) n-k = np n-1 ∑ j =0 ( n-1 j ) p j (1-p ) ( n-1)-j let j = k-1 = np ( p + (1-p )) n-1 by the binomial theorem* = np You can also see that as the sum of a binomial probability mass function. 2...
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This note was uploaded on 02/21/2011 for the course ISYE 2027 taught by Professor Zahrn during the Fall '08 term at Georgia Tech.

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Hw5sol - ∞ e-λx dx = 1 λ 4. For α > 1, E [ Y ] =...

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