1. Suppose X has a mean and standard deviation ; let X Z . Calculate the mean of Z, using the mean/variance of a linear combination of random variables. 2. Suppose X has a mean and standard deviation ; let . Calculate the standard Z 3. 4. 5. 6. X deviation of Z, using the mean/variance of a linear combination of random variables. Give the definition of a consistent estimator. What is the formula for the Law of Large Numbers? In words, what does the Law of Large Numbers say? Explain (or prove) why the Law of Large Numbers is true. (You don't need to use Chebyshev's inequality here, although you can if you wish. Just give the main idea why the law is true.) 7. Is 8. Is 9. Is 10. Is X a consistent estimator of ? Explain/prove your answer. 1 n Xi n13 i1 a consistent estimator of ? Explain/prove your answer. 1 n Xi 13 n i1 1 n Xi 13 n i1 a consistent estimator of ? Explain/prove your answer. a consistent estimator of ? Explain/prove your answer. 11. Is S2 a consistent estimator of 2? Explain/prove your answer. 12. Is a consistent estimator of 2? Explain/prove your answer. 1 n X i X 2 n i 1 13. 14. 15. 16. 17. Give the definition of an estimator's bias (the mathematical formula will do). Give the definition of a biased estimator (the mathematical formula will do). Give the definition of an unbiased estimator (the mathematical formula will do). Describe in words what a biased estimator is. Describe in words what an unbiased estimator is. If an estimator is consistent, is it unbiased? Why or why not? (If not, give an example.) If an estimator is unbiased, is it consistent? Why or why not? (If not, give an example.) [Hint: consider the estimator: ^ = 1 X 1+ 1 X 2+ 1 X 3 +0 X 4+ 0 X 5 ++0 X n .] 3 3 3 18. Show that (i.e., calculate, like we did in class) X is an unbiased estimator of . 19. Calculate the bias of n as an estimator of ; show your work in detail. 1 Xi n13 i1 20. Calculate the bias of 21 as an estimator of , where n=21, =103; show your work in 1 Xi n13 i1 detail. 21. Calculate the bias of 22. Calculate the bias of detail. 23. Calculate the bias of 24. Calculate the bias of detail. 25. Calculate the bias of 26. Calculate the bias of 1 n Xi 13 n i1 1 n Xi 13 n i1 1 n Xi 13 n i1 1 n Xi 13 n i1 as an estimator of ; show your work in detail. as an estimator of , where n=21, =103; show your work in as an estimator of ; show your work in detail. as an estimator of , where n=21, =103; show your work in n 1 X i X 2 n i 1 n 1 X i X 2 n i 1 as an estimator of 2; show your work in detail. as an estimator of 2, where n=21, 2=8.1; show your work in detail. Calculate an unbiased estimate of 2 using the random sample {2, 1, -3, 1}. 27. Calculate an estimate of from the random sample {2, 1, -3, 1} using the estimator . 1 n X i X 2 n i 1 28. Calculate s2 from the data set {3, -1, 4, 5, 0}. 29. Calculate s from the data set {3, -1, 4, 5, 0}. 30. Calculate 2 s x from the data set {3, -1, 4, 5, 0}. 31. Calculate s x from the data set {3, -1, 4, 5, 0}. X = 32. Do the calculations to show that n . 33. If a random sample of 15 Xs each have a distribution with a variance of 6, then what is the variance of ? X 34. If a random sample of 15 Xs each have a distribution with a variance of 6, then what is the standard deviation of ? X 35. If a random sample of 15 Xs each have a distribution with a standard deviation of 6, then what is the variance of ? X 36. If a random sample of 15 Xs each have a distribution with a standard deviation of 6, then what is the standard deviation of ? X 37. In general, if we have a random sample of n Xs from a distribution whose mean and variance is and 2, then what is the mean and variance of ? X 38. In general, if we have a random sample of n Xs from a distribution whose mean and standard deviation is and , then what is the mean and standard deviation of ? X [Comment: The next five questions break down the proof of that S 2 is an unbiased estimator of 2 found in the slides.] As part of a proof that S 2 is an unbiased estimator of 2, show that: n X X i i1 2 n n n Xi X 2 Xi X i1 2 2 i1 i1 As part of a proof that S2 is an unbiased estimator of 2, show that: As part of a proof that S2 is an unbiased estimator of 2, show that: As part of a proof that S2 is an unbiased estimator of 2, show that: n E 2 Xi X 2 2 i1 n 2 E Xi n 2 i1 n 2 E X 2 i1 Use the previous four equations to calculate that S2 is an unbiased estimator of 2. 39. [Comment: The next five questions break down the proof that S 2 is an unbiased estimator of 2 that we did in class.] Prove that: Y E[Y ] Y . (This is the Lemma from class.) 2 2 2 2 1 n SS X Xi n i1 . i1 40. Prove that: n 2 i 41. Use the Lemma above to prove that: n E Xi2 n 2 n 2 i1 n 2 E Xi n 2 n2 2 i1 42. Use the Lemma above to prove that: 43. Use the previous three equations to prove that: E[SS] = (n-1) 2. 44. Your colleague at work was gathering some data for you, and you noticed that she accidentally estimated the standard deviation as the square root of the average of the sum of the squared deviations from the sample mean (i.e., she "divided by n"). Her estimate was 270 for the 15 observations she collected. You wish to use an unbiased estimate of the sample variance. What is it (show your work)? What does \"MSE\" stand for? What is the relationship between the MSE, variance, and bias of an estimator? 45. Give the definition of the mean squared error of an estimator. 46. Show that ^ )=Var [ ^ ] + ( Bias ( ^ )) MSE ( 2 . 47. The MSE of an estimator looks a lot like a formula for the variance, although it is not. Explain why the MSE is not the variance - don't simply show that the two formulas differ. Explain what the MSE characterizes, and how this differs from the variance of the estimator in question. 48. You are using an estimator to try to figure out some population parameter . Your estimator is known to have a variance of 6 and a bias of -3. What is the mean squared error of ? 49. Give the formula for S2. Why do we use this estimator? Write sentence or two about how consistency and bias are similar, and how they are different. Write a couple sentences about how consistency and MSE are similar, and how they are different. Write a couple sentences about how bias and MSE are similar, and how they are different. Comment: It's natural to think that since S 2 is an unbiased estimator of the variance, S must be an unbiased estimator of the standard deviation. However, this is not true. The next couple questions guide you through a proof, using some concepts you may have encountered in other economics courses. (Cf. the worked examples for an explicit analysis of a simple case.) 50. By definition, a function f ( x ) is strictly convex (in a given interval) iff f ( p x 1+ (1 p ) x 2 ) < pf ( x 1) + ( 1 p ) f ( x 2 ) , for all x 1 , x 2 , and all 0 p 1 . Notice that f is strictly convex at x if its sec- ond derivative at x '' is positive: f ( x ) >0 . Show that for every positive value v >0, the function f ( v )= v 51. If f ( v )= v is strictly convex. is strictly convex, then what do we know about the square root function v ? 52. Jensen's Inequality is a theorem that states that for any strictly convex function f , f ( E [ X ] ) < E [f ( X ) ] . Use this inequality and the previous questions to prove that E [S] . 53. Is the bias of S as an estimator of positive, negative, or 0? The Normal Distribution and the Central Limit Theorem 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. For Z~N(0,1), use the table above to estimate Pr[Z 1.8]. For Z~N(0,1), use the table above to estimate Pr[Z > 2.2]. For Z~N(0,1), use the table above to estimate Pr[-.2 < Z < .8]. For X ~ N(25, 182), use the table above to estimate Pr[X 31]. For X ~ N(25, 182), use the table above to estimate Pr[X > 22]. For X ~ N(25, 182), use the table above to estimate Pr[22 < X 30]. For Z~N(0,1), what is the z such that Pr[Z z] = .93? For Z~N(0,1), what is the z such that Pr[Z > z] = .94? For Z~N(0,1), what is the z such that Pr[Z > z] = .03? For Z~N(0,1), what is the z such that Pr[Z z] = .04? For Z~N(0,1), what is the z such that Pr[-z Z < z] = .94? For X~N(4,72), what is the x such that Pr[X x] = .93? For X~N(4,72), what is the x such that Pr[X > x] = .94? For X~N(4,72), what is the x such that Pr[X > x] = .03? For X~N(4,72), what is the x such that Pr[X x] = .04? 69. For X~N(4,72), what are the x L and x U that are equidistant from the mean, and such that Pr [ x L X xU ] =.94 ? 70. Many test scores are (approximately) normally distributed. Your professor reports that the midterm average was a 78, and that the standard deviation was 8. You got an 88 on the midterm. Approximately what percentile does that put you in? (I.e., what percentage of the students did you score higher than?) 71. Many test scores are (approximately) normally distributed. Your professor reports that the midterm average was a 78, and that the standard deviation was 8. What is the approximate range of scores for the middle 50% of the scores? 72. Many test scores are (approximately) normally distributed. Your professor reports that the midterm average was a 78, and that the standard deviation was 8. What is the approximate range of scores for the top 10% of the scores? 73. Many test scores are (approximately) normally distributed. Your professor reports that the midterm average was a 78, and that the standard deviation was 8. What is the approximate range of scores for the bottom 10% of the scores? 74. There is a stock that, in the past has produced, on average annual return of 3.7%, with a standard deviation of 2.9%. These returns are, you assume, normally distributed. You will invest in this stock for one year. What is the probability that you will lose money (i.e. experience a negative return)? 75. There is a stock that, in the past has produced, on average annual return of 3.7%, with a standard deviation of 2.9%. These returns are, you assume, normally distributed. You will invest $350 in this stock for one year. What is the probability that you will receive a return worth at least $75? 76. Kimmy Schmidt has an investment! Its monthly fluctuation is predicted to be normally distributed! The mean should be $2,000, and it should have about $127 as its standard deviation! Kimmy wants to know what values of her investment will be in the top 5% of this investment! Help her find it! Show your work! A group of Mexicans have taken the New Black, and decided to try to build a large wall of ice at the border to keep out White Walkers, white wraiths, white horsemen of death, and White Zombie (both the heavy metal band and its fans), hereafter collectively known as \"Whities\". (This wall will be the best wall. It's going to be a big beautiful wall, made out of the best ice. Many people are saying this is the best ice in the world. Many, many people have said this.) However, to do this, they need to raise the capital to cover the construction costs. Thus, the ultimate question of interest is, Can the Mexicans afford to build a wall at the border to keep out Whities? To build the wall, the group will need to raise 175 billion Gold Dragon coins (bGD), and through a wide variety of fundraising schemes, their sum total projected funds is assumed to be distributed as X N(159, 82). (All units are in bGDs.) However, Carlos Slim has also indicated interest in this project, and may make his own investment in it, which would be independent from the primary funds indicated by X. (Carlos Slim is the best investor. He's the best, the smartest. His real estate developments are better than anyone else's. They're fantastic, simply wonderful, every one of them. And he knows everything about ice, too. A very, very beautiful man.) However, there are several ways he might do this. In each case, assess the probability that the wall will be built. 77. CS0: He decides not to invest. 78. CS1: He invests 10 bGD. 79. CS2: He invests the revenues from a particular telecom company; those funds are distributed as N(10, 62). 80. CS3: He will cover all remaining costs, provided that the other funding sources (i.e. X) reach at least 150 bGD. 81. CS4: He will invest 20 bGD, provided that the other funding sources reach at least 145 bGD. 82. CS5: He will invest the returns from his telecom expansion into the Isle of Pyke. There is a 30% chance that the Ironborn residents will \"pay the iron price\