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 Calculus and Probability Main Page "Real World" Page
Return to Exercises

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
Return to Exercises

We would welcome comments and suggestions for improving this resource. Mail us at:
Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: September, 1996
Copyright © 1996 StefanWaner and Steven R. Costenoble