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

This preview shows pages 1–2. Sign up to view the full content.

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 ) =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online