## Answers to ExercisesforSection 3: Mean, Median, Variance and Standard Deviation

 2. Probability Density Functions: Uniform, Exponential, Normal, and Beta 3. Mean, Median, Variance and Standard Deviation 4. You're the Expert Creating a Family Trust Calculus and Probability Main Page "Real World" Page

1. \$E(X) = 3/2,  Var(X) = 3/4,  σ(X) = 3^{1/2}/2\$

3. \$E(X) = 20/3,  Var(X) = 50/9,  σ(X) = 50^{1/2}/3\$

5. \$E(X) = 3/8,  Var(X) = 0.05975,  σ(X) = 0.2437\$

7. \$E(X) = 0.3863,  Var(X) = 0.03909,  σ(X) = 0.1977\$

9. \$E(X) = 10,  Var(X) = 100,  σ(X) = 10\$

11. \$E(X) = -33.3333,  Var(X) = 1111.1111,  σ(X) = 33.3333\$

13. \$E(X) = 3/2,  Var(X) = 3/4,  σ(X) = /2\$

15. \$E(X) = 1,  Var(X) = 1,  σ(X) = 1\$

17. \$E(X) = 0.4286,  Var(X) = 0.05442,  σ(X) = 0.2333\$

19. \$E(X) = 0.6774,  Var(X) = 0.0304,  σ(X) = 0.1742\$

21. \$E(X) = 0.4413,  Var(X) = 0.07852,  σ(X) = 0.2802\$

23. \$E(X) = 0.8862,  Var(X) = 0.2146,  σ(X) = 0.4633\$

25. \$2\$

27. \$0.2310\$

29. \$- 23.1049\$

30. \$34.6574\$

31. \$0.2929\$

33. \$0.25\$

35-40. Proofs

41. \$0.35\$

43. \$0.61\$

45. \$0.65\$

47. \$0.41\$

49. \$0.83\$

51. \$E(X^2) = 3,  E(X^2) - E(X)^2 = 3/4\$

53. \$E(X^2) = 50,  E(X^2) - E(X)^2 = 50/9\$

55. \$E(X^2) = 1/5,  E(X^2) - E(X)^2 = 0.059375\$

57. \$E(X^2) = 0.1883,  E(X^2) - E(X)^2 = 0.0391\$

59. \$E(X^2) = 200,  E(X^2) - E(X)^2 = 100\$

61. Comparing answers suggests that \$E(X^2) - E(X)^2 = Var(X).\$  Thus, \$E(X^2) = E(X)^2 + Var(X).\$

63. \$\$25,000\$

65. \$34\$ months

67. \$8,267\$ years

69. \$100,000,000\$ years

71. \$5.3009\$

73. \$100.4988\$

75 (a) \$β = 2.5556,  f(x) = 16.1975x^{2.5556}(1-x)\$

75 (b) \$M(X) = 0.66,\$  a little smaller than the mean. This indicates that more students scored below the mean than above it.

77.

79. Missing words: variance, median.

81. Values of \$X\$ are more likely to be close to the mode than anywhere else. Thus an interval about the mode determines the most popular values of \$X.\$

 2. Probability Density Functions: Uniform, Exponential, Normal, and Beta 3. Mean, Median, Variance and Standard Deviation 4. You're the Expert Creating a Family Trust Calculus and Probability Main Page "Real World" Page