for problem 2, do the case when n = 2 only.

Use simulation to estimate var (IV + M + + 1 )/4. 10. In certain situations a random variable X, whose mean is known, is simulated so as to obtain an estimate of P{X a Because I and X are clearly negative correlated, a natural attempt to reduce the variance is to use X as a control-and so use an estimator of the form I+ c(X - E[X]). (a) Determine the percentage of variance reduction over the raw estimator I that is possible (by using the best c) if X were uniform on (0, 1). (b) Repeat (a) if X were exponential with mean 1. (c) Explain why we knew that I and X were negatively correlated.[I-Iint: Recall that maxo

(x). Assuming that you can efciently compute both CI>(x) and (ID1 (x), show that you can generate a standard normal random variable X that is conditioned to exceed 0 by generating a random number U, letting Y = U + (l U)(c), and setting X = '1(Y) Explain how you could generate a standard normal random variable X that is conditioned to lie between a and b. 2. Suppose that X1, . .., X, has a multivariate normal distribution. Show that X1. . . .. Xn are independent if and only if 1= 2 Cov(X;, X ;) = 0 when i # j