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Monte Carlo Methods in Statistics

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Figure 1: Monte Carlo evaluation (1) of the expecta- expecta-tion E[X 3 /(1+X 2 +X 4 )] as a function of the number of simulation when X ∼ N (µ, 1) using (left) one  sim-ulation run and (right) 100 independent runs for (top) µ = 0 and (bottom) µ = 2.5.
Figure 2: (left) Gibbs sampling approximation to the distribution f (x) ∝ exp( − x 2 /2)/(1 +x 2 +x 4 ) against the true density; (right) range of convergence of the approximation to E f [X 3 ] = 0 against the number of iterations using 100 independent run
Figure 3: (left) Random walk Metropolis–Hastings sampling approximation to the distribution f (x) ∝ exp( − x 2 /2)/(1 +x 2 + x 4 ) against the true density for a scale of 1.2 corresponding to an acceptance rate of 0.5; (right) range of convergence of the a

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