Soong T.T. Fundamentals of Probability and Statistics for Engineers
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5.27
5.29
and are independent
CHAPTER 6
6.3
6.5
6.8
6.10
6.12
6.14
6.17
6.26
6.28
6.30
6.32
384
Fundamentals of Probability and Statistics for Engineers
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TLFeBOOK
CHAPTER 7
7.1 0.847
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7.6 (a) 4.566 10
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7.30 (a) 0.056, (b) 0.989
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7.36
CHAPTER 8
8.2 (a) (i) Type-1 asymptotic maximum-value distribution is suggested
(ii) 0.025, 46.92
(d) (i) Gamma is suggested, (ii)
(f) (i) Poisson is suggested, (ii)
(h) (i) Normal is suggested, (ii)
(j) (i) Poisson is suggested, (ii)
(i) (i) Lognormal is suggested, (ii)
CHAPTER 9
9.1 1.75, 27.96
9.5 (a)
(c) 0.091, 0.91
Appendix C: Answers to Selected Problems 385
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9.7
9.11
9.14
9.15
9.20
9.22
9.24
9.26
9.30
9.32
9.34
9.36
9.38
CHAPTER 10
10.1 More likely to be accepted at
10.3 Hypothesis is accepted
10.5 Hypothesis is accepted
10.8 Poisson hypothesis is rejected
10.10 Gamma hypothesis is accepted
10.12 Normal hypothesis is accepted
10.14 Poisson hypothesis is accepted
10.16 Hypothesis is accepted
CHAPTER 11
11.1 (a)
386 Fundamentals of Probability and Statistics for Engineers
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(b)
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11.3
11.7
11.9
11.11
11.13
Appendix C: Answers to Selected Problems 387
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TLFeBOOK
Subject Index
Average value, see M ean
Axioms of probab ility, 13–14
Bayes’s theorem, 24–25
Bernoulli trials, 161
Beta distribution, 221–226, 237
generalized, 225–226
mean, 223, 237
variance, 223, 237
Bias, 265
Binomial distribution, 43, 162, 182–183
characteristic function, 164
mean, 164, 184
Poisson approximation, 182–183
table, 365–366
variance, 164, 184
Boole’s inequality, 30
Brownian motion, 106
Cauchy distribution, 126
Central limit theorem, 199–201
Characteristic function, 98
joint, 108
Chebyshev inequality, 86–87
Chi-squared distribution, 219–221, 236
mean, 221, 236
table, 371
variance, 221, 236
Chi-squared text, 316
Coefficient of excess, 83
Coefficient of skewness, 83
Coefficient of variation, 81
Computer software, 3, 375–377
Confidence interval, 295, 296, 298, 302
Confidence limit, 347
Consistency, 274
Correlation, 88–90
perfect, 90
zero, 90
Correlation coefficient, 88–89
Covariance, 88
matr ix, 93
Crame
´
r–Rao inequality, 267–270
lower bound (CR LB), 269
Cumulant, 101
Cumula tive dist ribution function see
Probability distribution function
D
2
distribution, 327
table, 372
De Morgan’s laws, 11–12
D ensit y function see Probabilit y d en sit y
function
D ist ribution function, see Probabilit y
distribution function
Efficiency, 270
asymptotic, 271
Error, 316
type I, 316
type II, 316
Estimate, 264
Estimator, 265
consistent, 274
efficient, 270,271
sufficient, 275
unbiased minimum-variance, 266
Event, 12
Excel 2000, 3
Expectation, 75–76
conditional, 83–85
mathematical, 75
operator, 75
Exponential distribution, 45, 78, 215–219,
236
mean, 215, 236
variance, 215, 236
Exponential failure law, 218
T . T . Soong 2004 J o h n W iley & So n s, L t d
ISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)
Fundamentals of Probability and Statistics for Engineers
TLFeBOOK
Extreme-value distribution, 226,
type I, 228, 237
type II, 233, 237
type III, 234, 237
F a ilure rate, see H a za rd function
Fisher–Neyman factorization criterion, 275
F requency diagram, 248
F unction of random variables, 119, 137
moments, 134
probability distributions, 120
Gamma distribution, 212–215, 236
mean, 213, 236
variance, 213, 236
G aus–Markov theorem, 345
G a ussian dist ribution, see N ormal
distribution
Geometric distribution, 167, 184
mean, 168, 184
variance, 168, 184
Gumbel’s extreme value, 228
distribution, 228
H azard function, 218
H istogram, 248
cumulative, 327
Hypergeometric distribution, 167, 184
mean, 184
variance, 184
H ypothesis t esting, see Test of hypothesis
Independence, 19–20
mutua l, 18
Interarrival time, 215
Jacobian, 149
K olmogorov–Smirnov test, 327
Law of large numbers, 96
Least-square estimator, 354–355
covariance, 356
linear unbiased minimum variance, 344
mean, 355
variance, 355
Likelihood equation, 288
Likelihood function, 288
Linear regression, 335
multiple, 354
other models, 357
simple, 335
variance, 343
Lognormal distribution, 209–212, 236
mean, 211, 236
variance, 211, 236
MacLaurin series, 99
M arkovian property, 27
M arkov’s inequality, 115
M a ss function, see Probabilit y m a s s f u n ct io n
M aximum likelihood estimate, 288
M aximum likelihood estimator, 288–289
consistency, 289
efficiency, 289
invariance property, 290
Mean, 76–77
conditional, 84
Median, 76
M ode, 78
Moment, 76, 78
central, 79
joint, 87
joint central, 87
M oment estimat e, 278
Moment estimator (ME), 278–280
combined, 284
consistency, 279
Moment-generating function, 112, 117
Multinomial distribution, 172, 184
covariance, 173
mean, 173, 184
variance, 173, 184
Mutual exclusiveness, 13
Negative binomial distribution, 169, 184
mean, 171, 184
variance, 171, 184
Normal distribution, 107, 196–199, 236
bivariate, 111
characteristic function, 198
mean, 198, 236
multivariate, 205
standardized, 201
table, 369
variance, 198, 236
N ormal equation, 338
N uisance parameter, 284
Parameter estimation, 259
interval estimation, 294–295
maximum likelihood method, 287
moment method, 278
point estimation, 277
390 Subject Index
TLFeBOOK
Pascal dist ribution, see N egative binomia l
distribution
Poisson distribution, 173–176, 184
mean, 176, 184
table, 367
variance, 176, 184
Population, 259
Probability, 13
assignment, 16, 17
conditional, 20–21
function, 13
measure, 13
Probability density function (pdf), 44–46
conditional, 62–63
joint (jpdf), 49–51
margina l, 57
Probability distribution function (PDF), 39–41
bivariate, 49
conditional, 61
joint (JPDF), 49–51
margina l, 50
mixed-type, 46
Probability mass function (pmf), 41, 43
conditional, 61
joint (jpmf), 51–55
margina l, 52
R andom experiment, 12
R a ndom sa mple, see Sa mple
R andom variable, 37–39
continuous, 38
discrete, 38
function of, 120
sum of, 145
R andom vector
R andom walk, 52
R ange space, 120
Regression coefficient, 336
confidence interval, 347
least-square estimate, 344
test of hypothesis, 316
R elative likelihood, 16–17
Reliability, 60, 218
R esidual, 337
R eturn period, 169
Sample, 259
size, 260
value, 260
Sample mean, 97, 261
mean, 261
variance, 261
Sample moment, 263–264
Sample point, 12
Sample space, 12
Sample variance, 262–263
mean, 262
variance, 262
Schwarz inequality, 92
Set, 8–12
complement o f, 9
countable (enumerable), 8
disjoint, 10
element, 8
empty, 9
finite, 8
infinite, 8
subset of, 8
uncountable (nonenumerable), 8
Set operation, 9–12
difference, 10
intersection (product), 10
union (sum), 9
Significance level, 319
Spreadsheet, 3
Standard deviation, 79–81
Statistic, 260
sufficient, 275
St atistica l independence, see
Independence
Sterling’s formula, 107
St u d en t ’s t - d ist r ib u t io n , 298–299
table, 370
Sum of random variables, 93,
145–146
characteristic function, 104–105
moment, 94
probability distribution, 106, 146
Test of hypothesis, 316
Total probability theorem, 23
Tree diagram, 27–28
U nbiasedness, 265
Uniform distribution, 57, 189, 236
bivariate, 193
mean, 192, 236
variance, 192, 236
U nimodal distribution, 79
Variance, 79, 82
Venn diagram, 9
Weibull distribution, 235
Subject Index
391
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