022607 - 01gxdx For some “complicated” function g Monte...

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

View Full Document Right Arrow Icon
Lecture 10: 02/26/2007 Recall: Segueing into discussing “Monte Carlo stuff.” Theoretical Tool : Strong Law of Large Numbers (SLLN). If x 1 , x 2 , x 3 ,… are iid random variables with finite (conversion) mean / expected value xbar, we have with probability 1, →∞ = = limn 1nk 1nxk x And variance of estimate -> 0 like 1/n. In fact, = ( = - ) varianceestimate E 1ni 1 nxi x 2 = ( ) variance xi n = ( )- E xi2 x2n Where we remember that the variance(xi) is the same for all i. Stanislaw Ulam, at Los Alamos, mid-40’s recognized (along w/ others) how new “computers” might give life to “statistical simulation”—rigorous basis=SLLN; computers would arbitrate actual calculations. Ulam coined name “Monte Carlo” for this collection of methods. First “interesting” problem: compute
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.

Unformatted text preview: 01gxdx For some “complicated” function g. Monte Carlo approach: Make independent draws from a uniform distribution on [0,1] (lots of computer ways to do this.)—let the corresponding random variables ve x 1 ,x 2 , …, then by SLLN: = ( )→ →∞→ 1nk 1ng xi n 01gxdx =expected value of the rv g(x i ) wrt uniform distribution (Nb: {g(x i ): 1<=i<=0} is an iid sequence, (with y i =g(x i )), write common mean ybar= 01gxdx ) Computer’s roles: 1.) Generating random numbers 2.) Computing partial sums, n->infinity Another problem: (actually a special case of integration): compute area or volume of some complicated region D in some “easy” space V. <crash> <see onenote>...
View Full 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