Schaum’s outline of theory and problems of probability, random variables, and random processes

Hwei Hsu9780070306448, 0-07-030644-3

Designed for students in various disciplines of engineering, science, mathematics, management and business, this effective study tool includes hundreds of problems with step-by-step solutions and another 150 problems with hints or delayed answers. The solved problems illustrate and strongly reinforce vital theory and techniques. Topics taught include functions of random variables; expectation and limit theorems; and estimation theory, decision theory and queuing theory.

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Table of contents :
Cover……Page 1
Preface……Page 4
Contents……Page 6
B. Sample Space……Page 9
A. Set Operations……Page 10
B. Venn Diagram……Page 11
C. Identities……Page 12
A. Relative Frequency Definition……Page 13
C. Elementary Properties of Probability……Page 14
A. Definition……Page 15
1.8 Independent Events……Page 16
Sample Space & Events……Page 17
Algebra of Sets……Page 20
Notion & Axioms of Probability……Page 24
Equally Likely Events……Page 28
Conditional Probability……Page 32
Total Probability……Page 36
Independent Events……Page 39
Supplementary Problems……Page 43
B. Events Defined by Random Variables……Page 46
B. Properties of Fx(X)……Page 47
C. Determination of Probabilities from Distribution Function……Page 48
B. Probability Density Functions……Page 49
C. Variance……Page 50
A. Bernoulli Distribution……Page 51
C. Poisson Distribution……Page 52
D. Uniform Distribution……Page 53
E. Exponential Distribution……Page 54
F. Normal (or Gaussian) Distribution……Page 55
Random Variables……Page 56
Distribution Functions……Page 57
Discrete Random Variables & PMF……Page 61
Continuous Random Variables & PDF……Page 64
Mean & Variance……Page 69
Some Special Distributions……Page 75
Conditional Distributions……Page 79
Supplementary Problems……Page 82
A. Definition……Page 87
C. Properties of Fxy(x, y)……Page 88
D. Independent Random Variables……Page 89
C. Marginal PDF……Page 90
D. Properties of fY|X(y | x)……Page 91
3.7 Covariance & Correlation Coefficient……Page 92
3.8 Conditional Means & Conditional Variances……Page 93
A. Definitions……Page 94
C. N-Variate Normal Distribution……Page 96
Bivariate Random Variables & Joint Distribution Functions……Page 97
Discrete Random Variables–Joint PMF……Page 101
Continuous Random Variables–Joint PDF……Page 106
Conditional Distributions……Page 112
Covariance & Correlation Coefficient……Page 115
Conditional Means & Conditional Variances……Page 118
N-Dimensional Random Vectors……Page 119
Special Distributions……Page 121
Supplementary Problems……Page 126
B. Determination of fY(y) from fX(x)……Page 130
B. Two Functions of Two Random Variables……Page 131
B. n Functions of n Random Variables……Page 132
C. Linearity Property of Expectation……Page 133
B. Joint Moment Generating Function……Page 134
B. Joint Characteristic Functions……Page 135
B. Strong Law of Large Numbers……Page 136
Functions of One Random Variable……Page 137
Functions of Two Random Variables……Page 144
Functions of n Random Variables……Page 152
Expectation……Page 155
Moment Generating Functions……Page 158
Characteristic Functions……Page 162
Laws of Large Numbers & Central Limit Theorem……Page 163
Supplementary Problems……Page 166
A. Probabilistic Descriptions……Page 169
5.4 Classification of Random Processes……Page 170
D. Processes with Stationary Independent Increments……Page 171
F. Normal Processes……Page 172
A. Transition Probability Matrix……Page 173
B. Higher-Order Transition Probabilities–Chapman-Kolmogorov Equation……Page 174
D. Classification of States……Page 175
E. Absorption Probabilities……Page 176
A. Definitions……Page 177
C. Poisson Processes……Page 178
Random Processes……Page 180
Characterization of Random Processes……Page 182
Classification of Random Processes……Page 187
Discrete-Parameter Markov Chains……Page 193
Poisson Processes……Page 206
Wiener Processes……Page 212
Supplementary Problems……Page 214
B. Stochastic Derivatives……Page 217
A. Autocorrelation Functions……Page 218
C. Power Spectral Density……Page 219
D. Cross Power Spectral Densities……Page 220
A. Linear Systems……Page 221
B. Response of Continuous-Time Linear System to Random Input……Page 222
C. Response of Discrete-Time Linear System to Random Input……Page 223
B. Fourier Series……Page 224
C. Karhunen-Loeve Expansion……Page 225
B. Discrete-Time Random Processes……Page 226
Continuity, Differentiation, Integration……Page 227
Power Spectral Densities……Page 233
White Noise……Page 237
Response of Linear Systems to Random Inputs……Page 239
Fourier Series & Karhunen-Loeve Expansions……Page 244
Fourier Transform of Random Processes……Page 248
Supplementary Problems……Page 250
B. Efficient Estimators……Page 255
7.5 Bayes’ Estimation……Page 256
7.7 Linear Mean Square Estimation……Page 257
Properties of Point Estimators……Page 258
Maximum-Likelihood Estimation……Page 261
Bayes’ Estimation……Page 263
Mean Square Estimation……Page 266
Linear Mean Square Estimation……Page 268
Supplementary Problems……Page 270
B. Hypothesis Testing & Types of Errors……Page 272
A. Maximum-Likelihood Test……Page 273
C. Neyman-Pearson Test……Page 274
F. Minimax Test……Page 275
Hypothesis Testing……Page 276
Decision Tests……Page 279
Supplementary Problems……Page 287
C. Useful Formulas……Page 289
9.3 Birth-Death Process……Page 290
9.4 M/M/1 Queueing System……Page 291
9.5 M/M/s Queueing System……Page 292
9.7 M/M/s/K Queueing System……Page 293
Solved Problems……Page 294
Supplementary Problems……Page 303
AppA Normal Distribution……Page 305
B.1 Continuous-Time Fourier Transform……Page 307
B.2 Discrete-Time Fourier Transform……Page 308
Index……Page 311
Backcover……Page 320