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 A. 2.2. Let F be the cdf of the Weibull distribution with shape param.pdf
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A. 2.2. Let F be the cdf of the Weibull distribution with shape param.pdf

  1. A. 2.2. Let F be the cdf of the Weibull distribution with shape parameter and scale parameter c, i.e. F(x)=1e(x/c)3,x0. Compute the quantile functionand recall first how to obtain a sample of this distribution from a uniform random variable. For p(0,1), set q=c(log(p))1/. We want to estimate P(X>q) for X with cdf F for p small (e.g. p=.005 and p=.001 ). For one value of >1 and one <1, write the crude Monte Carlo algorithm and estimate this probability by the crude Monte Carlo method. Estimate the variance and relative efficiency and give confidence intervals. - For >2, show that we can reduce the variance by applying importance sampling with instrumental distribution (,1/c). Write the algorithm, apply it and compare with the previous results. - For <1, explain why the Gamma distribution is no longer suitable. Apply importance sampling with the log-normal distribution.
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