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solution3 - Homework 3 solution 1 Problem 1 We can just use...

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Homework 3 solution 1 Problem 1 We can just use the algorithm described in HW2: Step1: Generate x uniform(0,1) distribution. Step2: Generate u from uniform(0,1). Step3: Accept x if u < f ( x )1 { 0 <x< 1 } c , and reject x otherwise. R code: x<-NULL num=0 while(num<100) { y<-runif(1) u<-runif(1) if(u<=dweibull(y,5,3)/dweibull(1,5,3)) { x<-c(x,y) num<-num+1 } } mean(x) sd(x)/10 In my run, the estimate for the posterior mean is 0.8367417, and the standard error is 0.01437222. 1

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2 Problem 2 a) Naive Monte Carlo method: Step1: Draw 1000 samples x i from uniform(0,1). Step2: Estimate E [ sin ( x 2 )] by the sample mean of sin ( x 2 i ) and estimate the standard error by sd ( sin ( x 2 i )) / 1000. R code: x=runif(1000) m_naive=mean(sin(x^2)) sd_naive=sd(sin(x^2))/sqrt(1000) m_naive sd_naive In my run, the estimate for E [ sin ( x 2 )] is 0.3092447, and the standard error is 0.0083495. b) Importance sampling method: Step1: Draw 1000 samples x i from g ( x ). Step2: Calculate the weights as
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solution3 - Homework 3 solution 1 Problem 1 We can just use...

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