12 1. We wish to estimate = = [²₂e²x²¹/2²dx = [H(x)ƒ (x)dx via t Monte Carlo simulation using two different approaches:

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12 1 We Wish To Estimate E X 2 Dx H X F X Dx Via T Monte Carlo Simulation Using Two Different Approaches 1 (312.44 KiB) Viewed 9 times

12 1. We wish to estimate = = [²₂e²x²¹/2²dx = [H(x)ƒ (x)dx via t Monte Carlo simulation using two different approaches: (1) defining H(x) = 4e-²/² and f the pdf of the U[-2,2] distribution and (2) defining H(x)=√271₁-25x52) and of the pdf of the N(0,1) distribution. (16 points) a) For both cases estimate via the estimator below, î=N-¹℃H(X,). Use a sample size of N = 100 b) For both cases estimate the relative error K of using N = 100. c) Give a 95% confidence interval for t for both cases using N = 100. d) From part b), assess how large N should be such that the relative width of the confidence interval is less than 0.001, and carry out the simulation with this N. Compare the result with the true value of l. 2. Prove that the structure function of the bridge system given in Figure 1 is given by H(x)=1-(1-x₁x₁ )(1 − X₂ X5 ) (1 — Xq X3 X5 )(1 — X₂ X3 X4 ). (16 points) X₁ X₁ X₂ X₁ Figure 1 Bridge network. 3. Consider the M/M/1 queue. Let X, be the number of customers in the system at timet 20. Run a computer simulation of the process {X₁,t≥0} with λ = 1 and μ = 2, starting with an empty system. Let X denote the steady-state number of people in the system. Find point estimates and confidence intervals for l = E[X] using the batch means and regenerative methods as follows. (16 points) a) For the batch means method run the system for a simulation time of 10,000, discard the observations in the interval [0,100], and use N = 30 batches. b) For the regenerative method, run the system for the same amount of simulation time (10,000) and take as regeneration points the times when an arriving customer finds the system empty. c) For both methods find the requisite simulation time that ensures a relative width of the confidence interval not exceeding 5%. 4. Simulate the two-state Markov chain {X} in question 3, starting in state 1, with transition matrix P = P₁1 P12 P21 P22) and cost matrix c=)=(2, 2)-(23) C (c₁) Each transition from i to jincurs a cost of C. Obtain a confidence interval for the long-run average cost using P₁₁ =1/3 and P22 = 3/4, with 1000 regeneration cycles. (18 points) 5. Consider the integral l= = f* H(x)dx= (b − a)E[H(X)], with X~U(a,b). Let X₁, X be a random sample from U(a,b). Consider the estimators ē = — Σ* H(X) and ("") = 2 Σ, {H(X;)+ H(b+a−X;)}. Prove that if 1 2N H(x) is monotonic in x, then Var(a)) ≤Var(l). In other words, using antithetic random variables is more accurate than using CMC. (16 points) 6. Estimate the expected length of the shortest path for the bridge network in Question 5. That is, estimate l=E[H(X)] where H(X) = min{X₁ + X₁, X₁ + X3 + X5,₂ X₂ + X3 + X49X₂ + Xs}. Use both the CMC estimator, î = 1/-ΣX,H(X) N and the antithetic estimator, 1 7(a) = N/2 {{H(X₂)+H(X(@))} N where X₁ = F-¹(U) and X(ª) = F¯¹ (1–U₂). For both cases, take a sample size of N = 100,000. Assume that the lengths of the links X₁,..., X, are exponentially distributed, with means 1,1,0.5,2, and 1.5, respectively. Compare the results. (18 points) X3