Probability: An Introduction (Oxford Science Publications)

Front Cover......Page 1
Back Cover......Page 2
Title Page......Page 6
Copyright......Page 7
Preface......Page 8
Contents......Page 10
A. Basic Probability......Page 14
1.2 Outcomes and events......Page 16
1.3 Probabilities......Page 18
1.4 Probability spaces......Page 20
1.5 Discrete sample spaces......Page 22
1.6 Conditional probabilities......Page 24
1.7 Independent events......Page 25
1.8 The partition theorem......Page 27
1.9 Probability measures are continuous......Page 28
1.10 Worked problems......Page 29
1.11 Problems......Page 31
2.1 Probability mass functions......Page 35
2.2 Examples......Page 37
2.3 Functions of discrete random variables......Page 41
2.4 Expectation......Page 42
2.5 Conditional expectation and the partition theorem......Page 45
2.6 Problems......Page 47
3.1 Bivariate discrete distributions......Page 49
3.2 Expectation in the multivariate case......Page 50
3.3 Independence of discrete random variables......Page 51
3.4 Sums of random variables......Page 55
3.5 Problems......Page 56
4.1 Generating functions......Page 58
4.2 Integer-valued random variables......Page 59
4.3 Moments......Page 62
4.4 Sums of independent random variables......Page 64
4.5 Problems......Page 66
5.1 Distribution functions......Page 69
5.2 Examples of distribution functions......Page 72
5.3 Continuous random variables......Page 73
5.4 Some common density functions......Page 76
5.5 Functions of random variables......Page 78
5.6 Expectations of continuous random variables......Page 80
5.7 Problems......Page 83
B. Further Probability......Page 86
6.1 Random vectors and independence......Page 88
6.2 Joint density functions......Page 90
6.3 Marginal density functions and independence......Page 93
6.4 Sums of continuous random variables......Page 96
6.5 Changes of variables......Page 98
6.6 Conditional density functions......Page 101
6.7 Expectations of continuous random variables......Page 103
6.8 Conditional expectation and the bivariate normal
distribution......Page 105
6.9 Problems......Page 109
7.1 A general note......Page 114
7.2 Moments......Page 117
7.3 Variance and covariance......Page 119
7.4 Moment generating functions......Page 124
7.5 Characteristic functions......Page 128
7.6 Problems......Page 133
8.1 The law of averages......Page 137
8.2 Chebyshev\'s inequality and the weak law......Page 139
8.3 The central limit theorem......Page 142
8.4 Convergence in distribution, and characteristic functions......Page 145
8.5 Problems......Page 149
C. Random Processes......Page 154
9.2 A model for population growth......Page 156
9.3 The generating-function method......Page 158
9.4 An example......Page 160
9.5 The probability of extinction......Page 162
9.6 Problems......Page 165
10.1 One-dimensional random walks......Page 166
10.2 Transition probabilities......Page 167
10.3 Recurrence and transience in random walks......Page 169
10.4 The Gambler\'s Ruin problem......Page 172
10.5 Problems......Page 177
11.1 Life at a telephone switchboard......Page 181
11.2 Poisson processes......Page 184
11.3 Inter-arrival times and the exponential distribution......Page 187
11.4 Population growth, and the simple birth process......Page 190
11.5 Birth and death processes......Page 195
11.6 A simple queueing model......Page 198
11.7 Problems......Page 203
Appendix
Difference equations......Page 208
Answers to exercises......Page 211
Remarks on the problems......Page 214
Reading list......Page 220
Index......Page 221