53. a.
Exact: .212, .577, .573; Approximate: .211, .567, .596
b.
Exact: .885, .575, .017; Approximate: .885, .579, .012
c.
Exact: .002, .029, .617; Approximate: .003, .033, .599
55. a.
.9409
b.
.9943
57. b.
Normal,
59. a.
1
b.
1
c.
.982
d.
.129
61. a.
.480, .667, .147
b.
.050, 0
63. a.
short plan 1 better, whereas long
plan 2 better
b. 65. a.
.238
b.
.238
c.
.313
d.
.653
e.
.653
f.
.713
67. a.
.424
b.
.567,
c.
60
d.
66
69. a. b.
Exponential with
c.
Exponential with parameter
73. a.
.826, .826, .0636
b.
.664
c.
172.727
77. a.
123.97, 117.373
b.
.5517
c.
.1587
79. a.
9.164, .385
b.
.8790
c.
.4247, skewness
d.
No, since P
81. a.
149.157, 223.595
b.
.9573
c.
.0414
d.
148.41
e.
9.57
f.
125.90
83. 85. b.
87.
Yes, since the pattern in the plot is quite linear.
89.
Yes
91.
Yes
93.
Plot lnx vs. z percentile. The pattern is straight, so a lognormal population distribution is plausible.
95.
The pattern in the plot is quite linear; it is very plausible that strength is normally distributed.
97.
There is substantial curvature in the plot. is a scale param- eter as is for the normal family.
s l
b a 1 b
[⌫a 1 b ⌫
m 1 b][⌫a 1 b 1 m ⌫
b], a 5 b
X , 17,000 5 .9332 nl
l 5 .05
¨ A
i
m | , 24
1l 5 15 1 E[h
1
X] 5 150, E[h
2
X] 5 138.51 1l 5 10 1 E[h
1
X] 5 100, E[h
2
X] 5 112.53 1
1 m 5
239, s
2
5 12.96
99. a.
for
b.
.259, .5, .241
c.
6, 43.2, 7.2
d.
.518
e.
3.75
101. a.
for and for
b.
.917
c.
1.213
103. a.
.9162
b.
.9549
c.
1.3374
105. a.
.3859
b.
.0663
c.
72.97, 119.03
107. b.
for for ,
and for
c.
No.
d. 109. a.
.368, .828, .460
b.
352.53
c. d.
e.
, mode
111. a. b.
No
c. d.
e. 113. b.
c. d.
e.
f. 115. a.
Lognormal
b.
1
c.
2.72, .0185
119. a.
Exponential with
c.
Gamma with parameters and
121. a.
1365
3
b.
1365
2
c.
.000002145
123. b.
Let be a sequence of observations from a
Unif[0, 1] distribution a sequence of random numbers. Then with
, the ’s are observa- tions from an exponential distribution with
.
125. 127. a.
710, 84.423, .684
b.
.376 g
EX EgX l 5
10 x
i
x
i
5 2.1ln1 2 u
i
u
1
, u
2
, u
3
, c
cb a
l 5 1
CV , 1
1, CV . 1 V
X 5 2pl
2 1
1 21 2 pl
2 2
2 m
2
p l
1
1 1 2 pl
2
for x 0 p
1 2 exp 2l
1
x 1 1 2 p1 2 exp 2l
2
x n 2
2 a 2 1b
m 5
150, m | 5 182.99
m 5 201.95
a 1b
exp [2exp 2x 2 ab] exp 2x 2 ab
Y | Bin10, 5
27 F
0 , .5 1 m | . 0
x . 2
5 1
2 1 x 2
x , 2 1, 5 4x 2 x
3
39 1 11
27 F
x 5 0 1 x
7 3
5 7
4 2
3 4
x 0 x , 1
f x 5 x
2
0 y 12 F
y 5 1
48 y
2
2 y
3
18
1. a.
.20
b.
.42
c.
At least one hose is in use at each pump; .70.
d. p
X
x .16, .34, .50 for x 0, 1, 2, respectively; p
Y
y .24, .38, .38 for y 0, 1, 2, respec- tively; .50
e.
No; p0, 0 ⬆ p
X
0 ⭈ p
Y
3. a.
.15
b.
.40
c.
.22
d.
.17, .46
5. a.
.054
b.
.00018
7. a.
.030
b.
.120
c.
.300
d.
.380
e.
Yes
9. a.
3380,000
b.
.3024
c.
.3593
d.
10Kx
2
⫹ .05 for 20 x 30
e.
No
11. a. ⭈ m
1 x
⭈ m
2 y
xy
b. ⭈
[1 ⫹ m
1
⫹ m
2
]
c. ⭈
m
1
⫹ m
2 m
m; Poisson m
1
⫹ m
2
13. a. e
⫺x ⫺ y
for x 0, y 0
b.
.400
c.
.594
d.
.330
15. a. F
y 1 ⫺ e
⫺ly
⫹ 1 ⫺ e
⫺ly 2
⫺ 1 ⫺ e
⫺ly 3
for y 0
b.
23l
17. a.
.25
b.
.318
c.
.637
d.
for ⫺R x R; no f
X
x 5 22R
2
2 x
2
pR
2
e
2 m
1
1 m
2
e
2m
1
2 m
2
e
2m
1
2 m
2
Chapter 5
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