Introduction to Probability Theory

Introduction to Probability Theory is a Gtu book for Gujarat Technological University Computer and IT Branch Students in Engineering First Year by G Shanmugam and S Ravindran.

First Indian Reprint of the 1971 edition by Houghton Mifflin Company. The first of three volumes. The first volume, Introduction to Probability Theory, presents the fundamental ideas of probability theory and also prepares the student both for courses in statistics and for further study in probability theory, including stochastic processes.

Probability spaces; Combinatorial analysis; Discrete random variables; Expectation of discrete random variables; Continuous random variables; Jointly distributed random variables; Expectations and the central limit theorem; Moment generating functions and characteristic functions; Random walks and Poisson processes.

 

Table of contents :

General Preface……Page 5
Preface……Page 7
Table of Contents……Page 9
1 Probability Spaces……Page 12
1.1. Examples of random phenomena……Page 13
1.2. Probability spaces……Page 17
1.3. Properties of probabilities……Page 21
1.4. Conditional probability……Page 25
1.5. Inclependence……Page 29
2.1. ordered samples……Page 38
2.2. Petrmutations……Page 41
2.3. Connbinations ( unordered samples)……Page 42
2.4. Partitions……Page 45
2.5. Union of events·……Page 49
2.6. Matching problems·……Page 51
2.7. Occupancy problems·……Page 54
2.8. Number of empty boxes·……Page 55
3 Discrete Random Variables……Page 60
3.1 . Definitions……Page 61
3.2. Computations with densities……Page 68
3.3. Discrete random vectors……Page 71
3.4. Independent random variables……Page 74
3.5. Infiinite sequences of Bernoulli trials……Page 81
3.6. Sums of independent random variables……Page 83
4 Expectations of Discrete Random Variables……Page 93
4.1. Definition of expectation……Page 95
4.2. Properties of expectation……Page 96
4.3. Moments……Page 103
4.4. Variance of a sum……Page 107
4.5. Correlation coefficient……Page 110
4.6. Chebyshev’s inequality……Page 111
5 Continuous Random Variables……Page 120
5.1. Random variables and their distribution functions……Page 121
5.2. Densities of continuous random variables……Page 126
5.3. Normal, exponential, and gamma densities……Page 135
5.4. Inverse distribution functions·……Page 142
6.1. Properties of bivariate distributions……Page 150
6.2. Distribution of sums and quotients……Page 156
6.3. Conditional densities……Page 163
6.4. Properties of multivariate distributions……Page 168
6.5. Order statistics·……Page 171
6.6. Sampling distributions·……Page 174
6.7. Multidimensional changes of variables·……Page 177
7.1 . Expectations of continuous random variables……Page 184
7.2. A general definition of expectation……Page 185
7.3. Moments of continuous random variables……Page 188
7.4. Conditional expectation……Page 192
7.5. The Central Limit Theorem……Page 194
8.1. Moment generating functions……Page 208
8.2. Characteristic functions……Page 211
8.3. Inversion formulas and the Continuity Theorem……Page 216
8.4. The Weak Law of Large Numbers and the Central Limit Theorem……Page 220
9.1. Random walks……Page 227
9.2. Simple random walks……Page 230
9.3. Construction of a Poisson process……Page 236
9.4. Distance to particles……Page 239
9.5. Waiting times……Page 241
CHAPTER 1……Page 250
CHAPTER 2……Page 251
CHAPTER 3……Page 252
CHAPTER 5……Page 254
CHAPTER 6……Page 256
CHAPTER 7……Page 258
CHAPTER 8……Page 259
CHAPTER 9……Page 260
Table I Values of the standard normal distribution function……Page 263
Table I Values of the standard normal distribution function……Page 264
Index……Page 266