Chemical Engineering Hand Written_Notes_Part_94

Chemical Engineering Hand Written_Notes_Part_94 - 190 5....

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190 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES where p ( ϕ 1 , ..., ϕ m )= p ( ϕ ) represent multivariate probability density function of ϕ . Also, let Σ represents population covariance, which is de f ned as Σ = E h ( ϕ ϕ )( ϕ ϕ ) T i (6.3) = E h ( ϕ 1 ϕ 1 )( ϕ 1 ϕ 1 ) T i .... E h ( ϕ 1 ϕ 1 )( ϕ m ϕ m ) T i .... .... .... E h ( ϕ m ϕ m )( ϕ 1 ϕ 1 ) T i .... E h ( ϕ m ϕ m )( ϕ m ϕ m ) T i In fact, operator E ( . ) can be used for computing ensemble (population) expec- tation of any arbitrary function of ϕ , say g ( ϕ ) , i.e., (6.4) E [ g ( ϕ )] = Z −∞ .... Z −∞ g ( ϕ ) p ( ϕ ) d ϕ 1 ...d ϕ m Mean and covariance are speci f c forms function g ( ϕ ) . Given a vector of random variables (RVs) ϕ , in general, we may not have exact knowledge of p ( ϕ 1 ,..., ϕ m ) , ϕ and Σ . However, we can generate estimates of ϕ and Σ if we can conduct experiments and collect data for
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_94 - 190 5....

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