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Test 4 Solutions (Fall 2010)

# Test 4 Solutions (Fall 2010) - 1 NAME ISyE 3770 Solutions...

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Unformatted text preview: 1 NAME ISyE 3770 Solutions Test 4 Fall 2010 1. Let X be the outcome of a 4-sided die toss. Find E [ X 2 + 1]. Solution: E [ X 2 + 1] = E [ X 2 ] + 1 = 1 4 (1 2 + 2 2 + 3 2 + 4 2 ) + 1 = 8 . 5 . 2. Suppose that the lifetime of a transistor is exponential with a mean of 100,000 hours. Further suppose that the transistor has already survived 300,000 hours. Find the probability that it will fail in the next 100,000 hours. Solution: P ( X 400 | X 300) = P ( X 100) = 1- e- x = 1- e- 1 = 0 . 6321. 3. Suppose that X and Y are the scores that a student will receive on the verbal and math portions of the SAT test. Further suppose that X and Y are both Nor (500 , 100 2 ) and that Cov ( X,Y ) = 50 2 . Find the probability that the total score, X + Y , will exceed 1200. (You can assume that X + Y is normal.) Solution: X + Y N (1000 , 2 ), where 2 = Var ( X ) + Var ( Y ) + 2 Cov ( X,Y ) = 10000 + 10000 + 2(2500) = 25000 . Therefore, X + Y N (1000 , 25000). This implies that P ( X + Y &amp;gt; 1200) = P Z &amp;gt; 1200- 1000 25000 = P ( Z &amp;gt; 1 . 265) = 0 . 103 . 4. If X 1 ,...,X 100 are i.i.d. from some distribution with mean 0 and variance 100, find the probability that the sample mean X is between- 1 and 1. Solution: Note that X Nor (0 , 1). Thus, P (- 1 X 1) 2(1)- 1 = 2(0 . 8413)- 1 = 0 . 6826 . 2 5. What theorem says that a properly standardized sample mean can be approxi- mated by a normal random variable as the sample size becomes large? Solution: Central Limit Theorem 6. Find the normal quantile value - 1 (0 . 95)....
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Test 4 Solutions (Fall 2010) - 1 NAME ISyE 3770 Solutions...

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