The method of control variates is a technique for reducing the variance of a simulation estimator. Suppose that we wish to estimate θ = E(W). A control variate is another random variable V that is positively correlated with W and whose mean μ we know. Then, for every constant k > 0, E(W ˆ’ kV + kμ) = θ. Also, if k is chosen carefully, Var(W ˆ’ kV + kμ) W(i) = g(X(i))/f(X(i)),
V(i) = h(X(i))/f(X(i)),
Y(i) = W(i) ˆ’ kV(i),
for all i. Our estimator of ˆ« g(x) dx is then

a. Prove that E(Z) = ˆ« g(x) dx.
b. Let Var(W(i)) = σ2W and Var(V(i)) = σ2V . Let ρ be the correlation between W(i) and V(i). Prove that the value of k that makes Var(Z) the smallest is k = σWρ/σV.

Y) + kc. Z=E