Simulation provides a powerful method for evaluating single and multidimensional integrals. For instance, let f be a function of an r-valued vector ( y1, . . . , yr ), and suppose that we want to estimate the quantity θ, defined by

To accomplish this, note that if U1,U2, . . . ,Ur are independent uniform random variables on (0, 1), then

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0 = f(y1 y2yr) dy dy2,..., dyr Jo Jo