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How do we generate the statistics of a function of a random variable? – Why is the method called “Monte Carlo?” How do we use the uniform random number generator to generate other distributions? – Are other distributions directly available in matlab? How do we accelerate the brute force approach? – Probability distributions and moments Web links: Monte Carlo Simulation SOURCE:

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Basic Monte Carlo Given a random variable X and a function h(X): sample X: [x 1,x 2,…,x n ]; Calculate [h(x 1 ),h(x 2 ),…,h(x n )]; use to approximate statistics of h. Example: X is U[0,1]. Use MCS to find mean of X 2 x=rand(10); y=x.^2; %generates 10x10 random matrix mean=sum(y)/10 x = mean= What is the true mean SOURCE: SOURCE:

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Obtaining distributions Histogram: y=randn(100,1); hist(y)

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Cumulative density function Cdfplot(y) [f,x]=ecdf(y);

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Histogram of average x=rand(100); y=sum(x)/100; hist(y)

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Histogram of average x=rand(1000); y=sum(x)/1000; hist(y ) What is the law of large numbers?

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Distribution of x 2 x=rand(10000,1); x2=x.^2; hist(x2,20)

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Other distributions Other distributions available in matlab For example, Weibull distribution r=wblrnd(1,1,1000); hist(r,20)

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Correlated Variables For normal distribution can use Matlab’s mvnrnd R = MVNRND(MU,SIGMA,N) returns a N-by- D matrix R of random vectors chosen from the multivariate normal distribution with 1-by-D mean vector MU, and D-by-D covariance matrix SIGMA.

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Example mu = [2 3]; sigma = [1 1.5; 1.5 3]; r = mvnrnd(mu,sigma,20); plot(r(:,1),r(:,2),'+') What is the correlation coefficient?

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Problems Monte Carlo Use Monte Carlo simulation to estimate the mean and standard deviation of x 2, when X follows a Weibull distribution with a=b=1. Calculate by Monte Carlo simulation and check by integration the correlation coefficient between x and x 2, when x is uniformly distributed in [0,1]

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Latin hypercube sampling X = lhsnorm(mu,SIGMA,n) generates a latin hypercube sample X of size n from the multivariate normal distribution with mean vector mu and covariance matrix SIGMA. X is similar to a random sample from the multivariate normal distribution, but the marginal distribution of each column is adjusted so that its sample marginal distribution is close to its theoretical normal distribution.

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Comparing MCS to LHS mu = [2 2]; sigma = [1 0; 0 3]; r = lhsnorm(mu,sigma,20); sum(r)/20 ans = r = mvnrnd(mu,sigma,20); sum(r)/20 ans =

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Evaluating probabilities of failure Failure is defined in terms of a limit state function that must satisfy g(r)>0, where r is a vector of random variables. Probability of failure is estimated as the ratio of number of negative g’s, m, to total MC sample size, N The accuracy of the estimate is poor unless N is much larger than 1/P f For small P f

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problems probability of failure 1.Derive formula for the standard deviation of estimate of P f 2.If x is uniformly distributed in [0,1], use MCS to estimate the probability that x2>0.95 and estimate the accuracy of your estimate from the formula. 3. Calculate the exact value of the answer to Problem 2 (that is without MCS). Source: Smithsonian Institution Number:

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Separable Monte Carlo Usually limit state function is written in terms of response vs. capacity g=C(r)-R(r)>0 Failure typically corresponds to structures with extremely low capacity or extremely high response but not both Can take advantage of that in separable MC

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Reading assignment Ravishankar, Bharani, Smarslok B.P., Haftka R.T., Sankar B.V. (2010)“Error Estimation and Error Reduction in Separable Monte Carlo Method ” AIAA Journal,Vol 48(11), 2225–2230. Source: Page11.htm

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