Let X Unif(1, 2) and let Y = X. 2 a) Use the delta method to...

 

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Let X Unif(1, 2) and let Y = X. 2 a) Use the delta method to approximate the mean and variance of Y. b) Compute the exact values of E(Y) and Var(Y). c) Which approximations are better, the ones in this exercise or the ones in Example 4.12.1? Why? Example 4.12.1. Q. Let X-Unif (0, 1) and let Y= X. Approximate E(Y) and Var(Y). A. E(X) = and Var(X) = Let g(x) = . Then g'(x) = -1, and 9"(z) = -, so g() = = 0.707, g'(4) = = 2/2 = 1.414, and 9"() -2/2=-0.707. This gives the following estimates: E(X) . 1 2 1 = 0.707, Var(X) = 0.042, 11 2 E(X) 2 + = = 0.471. 22 12 2 2 12 1 In this case, we can use the cdf method to compute the mean and variance exactly (see Exercise 4.69). Here we will settle for a simulation: X The approximations in this case are not as good as we might have hoped, in part because the linear approximation to g(x)= is not very good near 0. This same method would work better on an interval other than [0, 1]. (See Exercise 4.70.) 4 Let X Unif(1, 2) and let Y = X. 2 a) Use the delta method to approximate the mean and variance of Y. b) Compute the exact values of E(Y) and Var(Y). c) Which approximations are better, the ones in this exercise or the ones in Example 4.12.1? Why? Example 4.12.1. Q. Let X-Unif (0, 1) and let Y= X. Approximate E(Y) and Var(Y). A. E(X) = and Var(X) = Let g(x) = . Then g'(x) = -1, and 9"(z) = -, so g() = = 0.707, g'(4) = = 2/2 = 1.414, and 9"() -2/2=-0.707. This gives the following estimates: E(X) . 1 2 1 = 0.707, Var(X) = 0.042, 11 2 E(X) 2 + = = 0.471. 22 12 2 2 12 1 In this case, we can use the cdf method to compute the mean and variance exactly (see Exercise 4.69). Here we will settle for a simulation: X The approximations in this case are not as good as we might have hoped, in part because the linear approximation to g(x)= is not very good near 0. This same method would work better on an interval other than [0, 1]. (See Exercise 4.70.) 4