Chemical Engineering Hand Written_Notes_Part_94

Chemical Engineering Hand Written_Notes_Part_94 - 190 5...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 fi 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 fi 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 ϕ in each experiments. Let us assume that we have carried out experiments and collected N data vectors © ϕ (1) , .... , ϕ ( N ) ª
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern