quiz03_2_solution

quiz03_2_solution - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Civil and Environmental Engineering 1.017/1.010 Computing and Data Analysis for Environmental Applications/ Uncertainty in Engineering Quiz 2 (Solutions provided at the end of each problem) Tuesday, November 4, 2003 Problem 1 ( 15 points) Sometimes we may want to generate two correlated random variables for stochastic simulation applications. For example, the duration and intensity of rain storm may be highly uncertain but positively correlated. One option for generating correlated variables x and y is to obtain y from: y = ax + b ε where x and are independent random variables with means equal to 0.0 and variances equal to 1.0 and a and b are specified constants. a. (5 points) What are the mean and variance of y ? (expressed as functions of a and b ): Solution: E [ y ] = E [ ax+b ] = aE [ x ] +bE [ ] = 0 Var [ y ] = Var [ ax+b ] = a 2 Var [ x ] + b 2 Var [ ] = a 2 + b 2 b. (10 points) Select a and b so that y has a variance of 2 and the correlation between x and y is 0.5. Show all relevant calculations. Solution: a 2 + b 2 = 2 Correl ( x,y ) = 0.5 = Cov ( x,y ) / ( Std [ x ] Std [ y ]) Cov ( x,y ) = E [( x- x )( y- y )] y- y = a ( x- x ) +b ( - ) so: Cov ( x,y ) = E [ a ( x- x ) 2 +b ( x- x ) ( - )] = aVar [ x ] = a a = .5* 2 = 0.707 ; b = 1.22 Problem 2 ( 25 points) 1 Suppose that the time between failures of a structural component is modeled as an exponentially distributed random variable. You want to use the 10% quantile x 10 [defined by F x ( x 10 ) = 0.10] as an indication of how often the component should be tested. You have the following 10 recorded times between failures (in hrs):
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quiz03_2_solution - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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