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Unformatted text preview: EAS6939 Homework #6 1. In engineering, the random wind pressure ( Y ) is typically modeled as a quadratic transformation of the random wind speed ( X ). If 2 ~ ( , ) x x X N μ σ and Y = X 2 , find approximation of Y based on the first-order Taylor series expansion about mean of X and equivalent linearization. For equivalent linearization, consider Y L = aX + b , where a and b are optimal parameters. Calculate the mean and variance of the above approximations of Y . Solution: 1) First-order Taylor series expansion: 2 ( ) ( ) 2 x L x x x x x dY Y Y X X dX μ μ μ μ μ = = + − = − Therefore, the approximate mean and variance are 2 2 [ ] 2 [ ] Y L x x x E Y E X μ μ μ μ = = − = 2 2 2 2 (2 ) [ ] 2 Y x x x Var X σ μ μ σ = = 2) Equivalent linearization: The model parameters a and b can be obtained from: 2 , minimize [ ] L a b E Y Y − For a general case, the minimizing conditions become 2 3 2 [ ] [ ] [ ] [ ] [ ] aE X bE X E X aE X b E X + = + = From class, 2 2 2 [ ] x x E X σ μ = + 3 2 3 [ ] 2 x x x E X μ σ μ = + Therefore, parameters a and b are calculated by 2 2 2 x x...
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- Spring '08
- Normal Distribution, Probability distribution, Probability theory, probability density function, Cumulative distribution function