Week 6

Question 1

Idea of Variance Reduction Techniques

Answer 1

• Exploit controllability of random-number generator to recycle previously used random numbers and include some kind of correlation
• Basic relation used:
  Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y)
• If the last term is negative, then the variable could be reduced
• Cov(X,Y) = Σ(Xi - Xbar)(Yi - Ybar) / (n-1)

Question 2

Control Variates Method

Answer 2

• This method works by relating the r.v. of interest X to another r.v. with known mean Y
• E.g. in queueing simulation X could be delay time and Y service time
• Idea is to use knowledge of E(Y) to control X, when X and Y are correlated
• X(c) = X + c(Y-E(Y))
• By construction, E(X(c)) = E(X)

Question 3

Properties of a Control Variable

Answer 3

• known mean

- correlated with X