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Midterm2sol

Midterm2sol - NAME MAE 108 Probability and Statistical...

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NAME: MAE 108 – Probability and Statistical Methods for Engineers - Winter 2010 Midterm # 2 - February 22, 2010 50 minutes, open book, open notes, calculator allowed, no cell phones. Write your answers directly on the exam. 40 points total. (1) (10 points) Consider a random variable X . The PDF for X is given by f X ( x ) = ce - 2 x , 0 < x < and zero otherwise. (a) Find the value of c Solution: R 0 f X ( x ) dx = 1 so c = 2. (b) Calculate P ( X > 2). Solution: P ( X > 2) = R 2 2 e - 2 x dx = [ - e - 2 x ] 2 = e - 4 1 . 83% (c) Calculate the mean E ( X ) Solution: E ( X ) = R 0 2 xe - 2 x dx = 1 2 R 0 ue - u du = 1 2 by integration by parts (d) Calculate the standard deviation σ X . Solution: E ( X 2 ) = R 0 2 x 2 e - 2 x dx = 1 4 R 0 u 2 e - u du = 1 2 by integration by parts so σ X = p E ( X 2 ) - E ( X ) 2 = q 1 2 - 1 4 = 1 2

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(2) (10 points) The SAT mathematics test scores across the population of high school seniors follow a normal distribution with mean 500 and standard deviation 100. If five seniors are randomly chosen, find the probability that: (a) All of them scored below 600.
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Midterm2sol - NAME MAE 108 Probability and Statistical...

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