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15. Notes

An incredible variety of special distributions have been studied over the years, and new ones are constantly being added to the literature. To truly deserve the adjective special, a distribution should have a certain level of mathematical elegance and economy, and should arise in interesting and diverse applications.

Books

The definitive works on special distributions are the books by Johnson and Kotz and their co-authors:

• Univariate Discrete Distributions, second edition, by Norman L. Johnson, Samuel Kotz, and Andrienne W. Kemp, published by John Wiley & Sons (1992).
• Continuous Univariate Distributions, Volume 1, second edition, by Norman L. Johnson, Samuel Kotz, and N. Balakrishnan, published by John Wiley & Sons (1994)
• Continuous Univariate Distributions, Volume 2, second edition, by Norman L. Johnson, Samuel Kotz, and N. Balakrishnan, published by John Wiley & Sons (1995)
• Discrete Multivariate Distributions, by Norman L. Johnson, Samuel Kotz, and N. Balakrishnan, published by John Wiley & Sons (1997)
• Continuous Multivariate Distributions: Models and Applications, second edition, by Samuel Kotz, N. Balakrishnan, and Normal L. Johnson, published by John Wiley & Sons (2000).

Web Sites

• Compendium of Common Probability Distributions. The compendium is an extensive list of common probability distributions and their properties, including continuous, discrete, and mixed distributions.

Selected Answers for Section 1

1.6. f(x) = (1 / b) exp[-(x - a) / b] for x > a.

1.7. f(x) = 1 / {b [1 + (x - a) / b]²} for x in R.

Selected Answers for Section 2

2.22. Let X denote the volume of beer in liters

1. P(X > 0.48) = 0.9772
2. x_0.95 = 0.51645

2.23. Let X denote the radius of the rod and Y the radius of the hole

P(Y - X < 0) = 0.0028.

2.24. Let X denote the combined weight of the 5 peaces, in ounces.

P(X > 45) = 0.0127.

Selected Answers for Section 3

3.8. P(X >3) = 17 exp(-3) / 2 ~ 0.4232.

3.9.

1. Q₁ = 0.287, Q₂ = 0.693, Q₃ = 1.396, Q₃ - Q₁ = 1.109.
2. Q₁ = 0.961, Q₂ = 1.678, Q₃ = 2.692, Q₃ - Q₁ = 1.731.
3. Q₁ = 1.727, Q₂ = 2.674, Q₃ = 3.920, Q₃ - Q₁ = 2.193.

3.14. Let X denote the petal length in centimeters.

1. E(X) = 4.
2. sd(X) = 2

3.21. Let X denote the lifetime in hours.

1. P(X > 300) = 13 exp(-3) ~ 0.6472.
2. E(X) = 400
3. sd(X) = 200

3.26.

1. P(18 < Y < 25) ~ 0.4095.
2. y₈₀ ~ 25.325.

Selected Answers for Section 4

4.5.

1. Q₁ = 0.102, Q₂ = 0.455, Q₃ = 1.323, Q₃ - Q₁ = 1.221.
2. Q₁ = 0.575, Q₂ = 1.386, Q₃ = 2.773, Q₃ - Q₁ = 2.198.
3. Q₁ = 2.675, Q₂ = 4.351, Q₃ = 6.626, Q₃ - Q₁ = 3.951.
4. Q₁ = 6.737, Q₂ = 9.342, Q₃ = 12.549, Q₃ - Q₁ = 5.812.

4.14. Let Z denote the distance from the missile to the target.

P(Z < 20) = 1 - exp(-2) ~ 0.8647.

4.16.

1. P(15 < X < 20) = 0.3252, P(15 < X < 20) ~ 0.3221
2. x_.075 = 21.605, x_0.75 ~ 22.044

Selected Answers for Section 5

5.5.

1. Q₁ = -1, Q₂ = 0, Q₃ = 1, Q₃ - Q₁ = 2
2. Q₁ = -0.816, Q₂ = 0, Q₃ = 0.816, Q₃ - Q₁ = 1.632
3. Q₁ = -0.727, Q₂ = 0, Q₃ = 0.727, Q₃ - Q₁ = 1.454
4. Q₁ = -0.7, Q₂ =0, Q₃ = 0.7, Q₃ - Q₁ = 1.4.

Selected Answers for Section 6

6.4.

1. Q₁ = 0.528, Q₂ = 1, Q₃ = 1.895, Q₃ - Q₁ = 1.367
2. Q₁ = 0.529, Q₂ = 0.932, Q₃ = 1.585, Q₃ - Q₁ = 1.056
3. Q₁ = 0.631, Q₂ = 1.073, Q₃ = 1.890, Q₃ - Q₁ = 1.259
4. Q₁ = 0.645, Q₂ = 1, Q₃ = 1.551, Q₃ - Q₁ = 0.906.

Selected Answers for Section 9

9.13.

1. Q₁ = 0.25, Q₂ = 0.5, Q₃ = 0.75, Q₃ - Q₁ = 0.5.
2. Q₁ = 0.091, Q₂ = 0.206, Q₃ = 0.370, Q₃ - Q₁ = 0.279
3. Q₁ = 0.630, Q₂ = 0.794, Q₃ = 0.909, Q₃ - Q₁ = 0.279
4. Q₁ = 0.194, Q₂ = 0.314, Q₃ = 0.454, Q₃ - Q₁ = 0.260.
5. Q₁ = 0.546, Q₂ = 0.686, Q₃ = 0.806, Q₃ - Q₁ = 0.260.
6. Q₁ = 0.379, Q₂ = 0.5, Q₃ = 0.621, Q₃ - Q₁ = 0.242.

Selected Answers for Section 10

10.7. Q₁ = 0.5364 Q₂ = 0.8326, Q₃ = 1.1774, Q₃ - Q₁ = 0.6411.

10.24.

1. P(T > 1500) = 0.1966
2. E(T) = 940.656, sd(T) = 787.237
3. h(t) = 0.000301 t^0.2.

Selected Answers for Section 11

11.3. P(X > 4) = 1 - 49 / 6² ~ 0.1725.

11.7.

1. E(X) = 1.1106
2. sd(X) = 0.5351

Selected Answers for Section 12

12.6. Q₁ = 1.1006, Q₂ = 1.2599, Q₃ = 1.5874, Q₃ - Q₁ = 0.4868

12.7. Q₁ = 1.1547, Q₂ = 1.4142, Q₃ = 2, Q₃ - Q₁ = 0.8453

12.16. Let X denote income.

1. P(2000 < X < 4000) = 0.1637 so the proportion is 16.37%
2. Q₂ = 1259.92
3. Q₁ = 1100.64, Q₃ = 1587.40, Q₃ - Q₁ = 486.76
4. E(X) = 1500
5. sd(X) = 866.03
6. F^-1(0.9) = 2154.43

Selected Answers for Section 13

13.2. P(-1 < X < 2) = 0.6119.

13.7. Q₁ = -1.0986, Q₂ = 0, Q₃ = 1.0986, Q₃ - Q₁ = 2.1972

13.8. F^-1(0.1) = -2.1972, F^-1(0.9) = 2.1972.

Selected Answers for Section 14

14.6. P(X > 20) = 0.1497

14.7. Q₁ = 0.5097, Q₂ = 1, Q₃ = 1.9621, Q₃ - Q₁ = 1.4524

14.11.

1. E(X) = exp(5 / 2) = 12.1825.
2. sd(X) = [exp(6) - exp(5)]^1/2 = 15.9692.

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