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Monte Carlo integration is a method to numerically estimate integral like I(g) = Sa g(x)dx. The idea is to sample X1, X2, . This problem introduces a variation on the Monte Carlo integration technique. Let h(x) be a density function on [a, b]. Generate X1,..., Xn independently from h(x) and estimate I(g) by Î(g) = 1 Xn independent Unif(a, b) and estimate I(g) as E1 9(X;). ... g(X;) i=1 h(X;)* n (a) Show that E[Ï(g)] = I(g).

Monte Carlo integration is a method to numerically estimate integral like I(g) = Sa g(x)dx. The idea is to sample X1, X2, . This problem introduces a variation on the Monte Carlo integration technique. Let h(x) be a density function on [a, b]. Generate X1,..., Xn independently from h(x) and estimate I(g) by Î(g) = 1 Xn independent Unif(a, b) and estimate I(g) as E1 9(X;). ... g(X;) i=1 h(X;)* n (a) Show that E[Ï(g)] = I(g).

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Monte Carlo integration is a method to numerically estimate integral like I(g) = S% g(x)dx. The idea is to sample X1, X2, ..., Xn independent Unif(a, b) and estimate I(g) as E19(X;). This problem introduces a variation on the Monte Carlo integration technique. Let h(x) be a density function on [a, b]. Generate X1,..., Xn independently from h(x) and estimate I(g) by n i=1 (a) Show that E[Î (9)] = I(g).
Transcribed Image Text:Monte Carlo integration is a method to numerically estimate integral like I(g) = S% g(x)dx. The idea is to sample X1, X2, ..., Xn independent Unif(a, b) and estimate I(g) as E19(X;). This problem introduces a variation on the Monte Carlo integration technique. Let h(x) be a density function on [a, b]. Generate X1,..., Xn independently from h(x) and estimate I(g) by n i=1 (a) Show that E[Î (9)] = I(g).
(b) Find an expression for Var(I(g)). Give an example for which it is finite and an example for which it is infinite. Note that if it is finite, the law of large numbers implies that Í(g) → I(g) as n → 0.
Transcribed Image Text:(b) Find an expression for Var(I(g)). Give an example for which it is finite and an example for which it is infinite. Note that if it is finite, the law of large numbers implies that Í(g) → I(g) as n → 0.
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