第九章 Monte car积分 Hit-or-Miss Method ◇ Sample mean Method Variance Reduction Technique & Variance Reduction using Rejection Technique % Importance Sampling Method
第九章 Monte Carlo积分 ❖Hit-or-Miss Method ❖Sample Mean Method ❖Variance Reduction Technique ❖Variance Reduction using Rejection Technique ❖Importance Sampling Method
Sample Mean method Start Set N: large inte S1=0 Loc (b-a)un+a N times Vnp S + + Estimate mean H=S//N SM Estimate variance V=SyN-H'2 =(b-a)×0.6745 End
Sample Mean Method Start Set N : large integer s1 = 0, s2 = 0 xn = (b-a) un + a yn = (xn ) s1 = s1 + yn , s2 = s2 + yn 2 Estimate mean m’=s1 /N Estimate variance V’ = s2 /N – m’ 2 End Loop N times N V I b a SM error ' = ( − )0.6745
Sample Mean method g(x)dx Write this as gx dx=(D-al/g(.I a (b-aEg(X where X-unif(a, b)
Sample Mean Method I g(x)dx b a = Write this as: dx b- a 1 I g(x)dx (b- a) g(x) b a b a = = =(b- a)Eg(X) where X~unif(a,b)
Sample Mean method I=(b-a)Eg(x)] where X-unif(a, b) So, we will estimate i by estimating Elg(X)] with Eg(X]=1∑ n where X, X,,.,Xn is a random sample from the uniform(a, b)distribution
Sample Mean Method I =(b- a)Eg(X) where X~unif(a,b) So, we will estimate I by estimating E[g(X)] with = = n i 1 ) ˆ g(Xi n 1 E[g(X)] where X1 , X2 , …, Xn is a random sample from the uniform(a,b) distribution