Test 3 Solutions (Fall 2009)

# Test 3 Solutions (Fall 2009) - 1 NAME ISyE 3770 Test 3...

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NAME ISyE 3770 — Test 3 Solutions — Fall 2009 This test is 90 minutes long. You are allowed one cheat sheet. 1. Suppose U 1 ,U 2 ,U 3 are i.i.d. Unif(0,4). Find P ( U 2 < U 3 < U 1 ). Solution: 1 6 (one of 6 permutations.) 2. Suppose that the lifetime of a light bulb is exponential with a mean of 1000 hours. Further suppose that the bulb has already survived 2000 hours. Find the probability that it will fail sometime in the next 1000 hours. Solution: By memoryless, P ( X < 3000 | X > 2000) = P ( X < 1000) = 1 - e - λt = 1 - e - 1 = 0 . 63 3. Suppose that X and Y are both Nor(1,9) random variables with Cov ( X,Y ) = 3. Find Var ( X - Y ). Solution: Var ( X - Y ) = Var ( X ) + Var ( Y ) - 2 Cov ( X,Y ) = 9 + 9 - 6 = 12 4. Suppose that the number of typographical errors in a book is Poisson with rate 0.1/page. What’s the probability that there will be exactly 2 errors in a chapter of 30 pages? Solution: Let X be the number of errors in a chapter of 30 pages. Then X Pois(30 * 0 . 1) = Pois(3) P ( X = 2) = e - 3 3 2 2! = 0 . 224 5. TRUE or FALSE? If X 1 ,X 2 ,...,X 100 are i.i.d. Exp(3), then the sample mean ¯ X is approximately normal. Solution: TRUE. (By CLT) 6. Suppose Z is a standard normal random variable. Find z such P ( - z Z z ) = 0 . 95. Solution:

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## This note was uploaded on 08/27/2011 for the course ISYE 3770 taught by Professor Goldsman during the Spring '07 term at Georgia Tech.

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Test 3 Solutions (Fall 2009) - 1 NAME ISyE 3770 Test 3...

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