Introduction to Probability

Contents......Page 4
1 Sample Space and 
Probability......Page 7
1.1 SETS......Page 9
1.2 PROBABILISTIC MODELS......Page 12
1.3 CONDITIONAL PROBABILITY......Page 22
1.4 TOTAL PROBABILITY THEOREM AND BAYES’ RULE......Page 31
1.5 INDEPENDENCE......Page 37
1.6 COUNTING......Page 47
1.7 SUMMARY AND DISCUSSION......Page 54
2 Discrete Random Variables......Page 56
2.1 BASIC CONCEPTS......Page 57
2.2 PROBABILITY MASS FUNCTIONS......Page 59
2.3 FUNCTIONS OF RANDOM VARIABLES......Page 64
2.4 EXPECTATION, MEAN, AND VARIANCE......Page 66
2.5 JOINT PMFS OF MULTIPLE RANDOM VARIABLES......Page 77
2.6 CONDITIONING......Page 81
2.7 INDEPENDENCE......Page 91
2.8 SUMMARY AND DISCUSSION......Page 97
3 General Random Variables......Page 100
3.1 CONTINUOUS RANDOM VARIABLES AND PDFS......Page 101
3.2 CUMULATIVE DISTRIBUTION FUNCTIONS......Page 110
3.3 NORMAL RANDOM VARIABLES......Page 115
3.4 CONDITIONING ON AN EVENT......Page 119
3.5 MULTIPLE CONTINUOUS RANDOM VARIABLES......Page 125
3.6 DERIVED DISTRIBUTIONS......Page 139
3.7 SUMMARY AND DISCUSSION......Page 151
4 Further Topics on Random Variables and Expectations......Page 152
4.1 TRANSFORMS......Page 153
4.2 SUMS OF INDEPENDENT RANDOM VARIABLES
— CONVOLUTIONS......Page 164
4.3 CONDITIONAL EXPECTATION AS A RANDOM VARIABLE......Page 168
4.4 SUM OF A RANDOM NUMBER OF INDEPENDENT RANDOM
VARIABLES......Page 176
4.5 COVARIANCE AND CORRELATION......Page 180
4.6 LEAST SQUARES ESTIMATION......Page 183
4.7 THE BIVARIATE NORMAL DISTRIBUTION......Page 190
5 Stochastic Processes......Page 194
5.1 THE BERNOULLI PROCESS......Page 196
5.2 THE POISSON PROCESS......Page 208
6 Markov Chains......Page 224
6.1 DISCRETE-TIME MARKOV CHAINS......Page 225
6.2 CLASSIFICATION OF STATES......Page 232
6.3 STEADY-STATE BEHAVIOR......Page 236
6.4 ABSORPTION PROBABILITIES AND EXPECTED TIME
TO ABSORPTION......Page 248
6.5 MORE GENERAL MARKOV CHAINS......Page 256
7 Limit Theorems......Page 268
7.1 SOME USEFUL INEQUALITIES......Page 270
7.2 THE WEAK LAW OF LARGE NUMBERS......Page 272
7.3 CONVERGENCE IN PROBABILITY......Page 274
7.4 THE CENTRAL LIMIT THEOREM......Page 276
7.5 THE STRONG LAW OF LARGE NUMBERS......Page 283
Introduction to Probability: Problem Solutions......Page 286
Errata......Page 370