022607 - Lecture 10 Recall Segueing into discussing"Monte...

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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
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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>...
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  • Spring '07
  • Algorithms, Probability theory, Stanislaw Ulam

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