Calculus Applied to Probability and Statistics
by
Stefan Waner and Steven R. Costenoble

Answers to Exercises
for
Section 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

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1. E(X) = 3/2, Var(X) = 3/4, (X) = 3^1/2/2

3. E(X) = 20/3, Var(X) = 50/9, (X) = 50^1/2/3

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

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

9. E(X) = 10, Var(X) = 100, (X) = 10

11. E(X) = 33.3333, Var(X) = 1111.1111, (X) = 33.3333

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

15. E(X) = 1, Var(X) = 1, (X) = 1

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

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

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

23. E(X) = 0.8862, Var(X) = 0.2146, (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(X²) = 3, E(X²) E(X)² = 3/4

53. E(X²) = 50, E(X²) E(X)² = 50/9

55. E(X²) = 1/5, E(X²) E(X)² = 0.059375

57. E(X²) = 0.1883, E(X²) E(X)² = 0.0391

59. E(X²) = 200, E(X²) E(X)² = 100

61. Comparing answers suggests that E(X²) E(X)² = Var(X). Thus, E(X²) = 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) = 2.5556, f(x) = 16.1975x^2.5556(1x)

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

Return to Exercises

Stefan Waner (matszw@hofstra.edu)

Steven R. Costenoble (matsrc@hofstra.edu)

Last Updated: September, 1996 Copyright © 1996 StefanWaner and Steven R. Costenoble