Answers to Selected Exercises

1.6. $f x 1 b exp x a b, x a$

1.7. $f x 1 b 1 x a b 2, x$

2.22. Let $X$ denote the volume of beer in liters

$X 0.48 0.9772$
$x 0.95 0.51645$

2.23. Let $X$ denote the radius of the rod and $Y$ the radius of the hole. $Y X 0 0.0028$ .

2.24. Let $X$ denote the combined weight of the 5 peaches, in ounces. $X 45 0.0127$ .

3.8. $X 3 17 2 3 0.432$

3.9.

$Q 1 0.287, Q 2 0.693, Q 3 1.396, Q 3 Q 1 1.109,$
$Q 1 0.961, Q 2 1.678, Q 3 2.692, Q 3 Q 1 1.731,$
$Q 1 1.727, Q 2 2.674, Q 3 3.920, Q 3 Q 1 2.193,$

3.14. Let $X$ denote the petal length in centimeters.

$X 4$
$X 2$

3.21. Let $X$ denote the lifetime in hours.

$X 300 133 0.6472$
$X 400$
$X 200$

3.26.

$18 X 25 0.3860$ , $18 X 25 0.4095$
$y 80 25.038$ , $y 80 25.325$

4.5.

$Q 1 0.102, Q 2 0.455, Q 3 1.323, Q 3 Q 1 1.221,$
$Q 1 0.575, Q 2 1.386, Q 3 2.773, Q 3 Q 1 2.198,$
$Q 1 2.675, Q 2 4.351, Q 3 6.626, Q 3 Q 1 3.951,$
$Q 1 6.737, Q 2 9.342, Q 3 12.549, Q 3 Q 1 5.812,$

4.14. Let $Z$ denote the distance from the missile to the target. $Z 20 12 0.8647$ .

4.16.

$15 X 20 0.3252$ , $15 X 20 0.3221$
$x 0.75 21.605$ , $x 0.75 22.044$

5.6.

$Q 1 1, Q 2 0, Q 3 1, Q 3 Q 1 2,$
$Q 1 0.816, Q 2 0, Q 3 0.816, Q 3 Q 1 1.632,$
$Q 1 0.727, Q 2 0, Q 3 0.727, Q 3 Q 1 1.454,$
$Q 1 0.7, Q 2 0, Q 3 0.7, Q 3 Q 1 1.4,$

5.13.

$0.8 T 1.2 0.650$ , $0.8 T 1.2 0.673$
$x 0.90 1.372$ , $x 0.90 1.281$

6.4.

$Q 1 0.528, Q 2 1, Q 3 1.895, Q 3 Q 1 1.367,$
$Q 1 0.529, Q 2 932, Q 3 1.585, Q 3 Q 1 1.056,$
$Q 1 0.631, Q 2 1.073, Q 3 1.890, Q 3 Q 1 1.259,$
$Q 1 0.645, Q 2 1, Q 3 1.551, Q 3 Q 1 0.906,$

8.14.

$Q 1 0.25, Q 2 0.5, Q 3 0.75, Q 3 Q 1 0.5,$
$Q 1 0.091, Q 2 0.206, Q 3 0.370, Q 3 Q 1 0.279,$
$Q 1 0.630, Q 2 0.794, Q 3 0.9009, Q 3 Q 1 0.279,$
$Q 1 0.194, Q 2 0.314, Q 3 0.454, Q 3 Q 1 0.260,$
$Q 1 0.546, Q 2 0.686, Q 3 0.806, Q 3 Q 1 0.260,$
$Q 1 0.379, Q 2 0.5, Q 3 0.621, Q 3 Q 1 0.242,$

9.7. $Q 1 0.5364, Q 2 0.8326, Q 3 1.1774, Q 3 Q 1 0.6411,$

9.24.

$T 1500 0.1966$
$T 940.656$ , $T 787.237$
$h t 0.0.000301 t 0.2$

10.3. $X 4 14962 0.1725$

10.7.

$X 1.1106$
$X 0.5351$

11.6. $Q 1 1.1006, Q 2 1.2599, Q 3 1.5874, Q 3 Q 1 0.4868$

11.16. Let $X$ denote income.

$2000 X 4000 0.1637$ so the proportion is 16.37%
$Q 2 1259.92$
$Q 1 1100.64, Q 3 1587.40, Q 3 Q 1 486.76$
$X 1500$
$X 866.03$
$F 0.9 2154.43$

12.2. $1 X 2 0.6119$

12.7. $Q 1 1.0986, Q 2 0, Q 3 1.0986, Q 3 Q 1 2.1972$

12.8. $F 0.1 2.1972$ , $F 0.9 2.1972$

13.6. $X 20 0.1497$

13.7. $Q 1 0.5097, Q 2 1, Q 3 1.9621, Q 3 Q 1 1.4524$

13.11.

$X 5 2 12.1825$
$X 65 15.9629$

15.4.

$f y 1 0.2303 y$ for $y 1 10 1$ .
$Y 0.3909$ , $Y 0.0622$ ,

15.6.

$n$	$N 1 n$
1	0.3010
2	0.1761
3	0.1249
4	0.0969
5	0.0792
6	0.0669
7	0.0580
8	0.0512
9	0.0458

$N 1 3.4402$ , $N 1 6.0567$

15.8.

$N 1 n 1 N 2 n 2$
$n 2$ \ $n 1$	1	2	3	4	5	6	7	8	9
0	0.0414	0.0212	0.0142	0.0107	0.0086	0.0072	0.0062	0.0054	0.0048
1	0.0378	0.0202	0.0138	0.0105	0.0084	0.0071	0.0061	0.0053	0.0047
2	0.0348	0.0193	0.0134	0.0102	0.0083	0.0069	0.0060	0.0053	0.0047
3	0.0322	0.0185	0.0130	0.0100	0.0081	0.0068	0.0059	0.0052	0.0046
4	0.0300	0.0177	0.0126	0.0098	0.0080	0.0067	0.0058	0.0051	0.0046
5	0.0280	0.0170	0.0122	0.0092	0.0078	0.0066	0.0058	0.0051	0.0045
6	0.0263	0.0164	0.0119	0.0093	0.0077	0.0065	0.0057	0.0050	0.0045
7	0.0248	0.0158	0.0116	0.0091	0.0076	0.0064	0.0056	0.0050	0.0045
8	0.0235	0.0152	0.0113	0.0090	0.0074	0.0063	0.0055	0.0049	0.0044
9	0.0223	0.0147	0.0110	0.0088	0.0073	0.0062	0.0055	0.0049	0.0044

15.10.

$n$	$N 2 n$
0	0.1197
1	0.1139
2	0.1088
3	0.1043
4	0.1003
5	0.0967
6	0.0934
7	0.0904
8	0.0876
9	0.0850

$N 2 4.1874$ , $N 2 0.8254$

15.12.

0.0008171
0.001377
0.003205

Answers to Selected Exercises

5. Special Distributions

1. Introduction

2. The Normal Distribution

3. The Gamma Distribution

4. The Chi-Square Distribution

5. The Student $t$ Distribution

6. The $F$ distribution

8. The Beta Distribution

9. The Weibull Distribution

10. The Zeta Distribution

11. The Pareto Distribution

12. The Logistic Distribution

13. The Lognormal Distribution

15. Benford's Law