Let X1, X2, . . . , Xn be a random sample from N(0, 2), where n is odd. Let Y and Z be the

Let X1, X2, . . . , Xn be a random sample from N(0, σ2), where n is odd. Let Y and Z be the mean and median of the sample. Argue that Y and Z − Y are independent so that the variance of Z is Var(Y) + Var(Z− Y). We know that Var(Y) = σ2/n, so that we could estimate the Var(Z − Y) by Monte Carlo. This might be more efficient than estimating Var(Z) directly since Var(Z − Y) ≤ Var(Z). This scheme is often called the Monte Carlo Swindle.