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Lectures on Probability.pdf
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概率不等式,包含所有概率研究中所存在的反例。Marco Taboga - Lectures on Probability Theory and Mathematical Statistics-CreateSpace Independent Publishing Platform (2012)
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Lectures on Probability Theory
and Mathematical Statistics
Second Edition
Marco Taboga
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ii
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Contents
I Mathematical tools 1
1 Set theory 3
1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Set membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Set inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.7 De Morgan’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.8 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Permutations 9
2.1 Permutations without repetition . . . . . . . . . . . . . . . . . . . . 9
2.1.1 De…nition of permutation without repetition . . . . . . . . . 9
2.1.2 Number of permutations without repetition . . . . . . . . . . 10
2.2 Permutations with repetition . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 De…nition of permutation with repetition . . . . . . . . . . . 11
2.2.2 Number of permutations with repetition . . . . . . . . . . . . 11
2.3 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 k-permutations 15
3.1 k-permutations without repetition . . . . . . . . . . . . . . . . . . . 15
3.1.1 De…nition of k-permutation without repetition . . . . . . . . 15
3.1.2 Number of k-permutations without repetition . . . . . . . . . 16
3.2 k-permutations with repetition . . . . . . . . . . . . . . . . . . . . . 17
3.2.1 De…nition of k-permutation with repetition . . . . . . . . . . 17
3.2.2 Number of k-permutations with repetition . . . . . . . . . . . 18
3.3 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Combinations 21
4.1 Combinations without repe tition . . . . . . . . . . . . . . . . . . . . 21
4.1.1 De…nition of combination without repetition . . . . . . . . . 21
4.1.2 Number of combinations without repetition . . . . . . . . . . 22
4.2 Combinations with repe tition . . . . . . . . . . . . . . . . . . . . . . 22
4.2.1 De…nition of combination with repetition . . . . . . . . . . . 23
4.2.2 Number of combinations with repetition . . . . . . . . . . . . 23
4.3 More details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.1 Binomial coe¢ cients and binomial expansions . . . . . . . . . 25
4.3.2 Recursive formula for binomial coe¢ cients . . . . . . . . . . . 25
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iv CONTENTS
4.4 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Partitions into groups 27
5.1 De…nition of partition into groups . . . . . . . . . . . . . . . . . . . 27
5.2 Number of partitions into groups . . . . . . . . . . . . . . . . . . . . 28
5.3 More details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3.1 Multinomial expansions . . . . . . . . . . . . . . . . . . . . . 29
5.4 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6 Sequences and limits 31
6.1 De…nition of sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2 Countable and uncountable sets . . . . . . . . . . . . . . . . . . . . . 32
6.3 Limit of a sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3.1 The limit of a sequence of real numbers . . . . . . . . . . . . 32
6.3.2 The limit of a sequence in general . . . . . . . . . . . . . . . 34
7 Review of di¤erentiation rules 39
7.1 Derivative of a constant function . . . . . . . . . . . . . . . . . . . . 39
7.2 Derivative of a power function . . . . . . . . . . . . . . . . . . . . . . 39
7.3 Derivative of a logarithmic function . . . . . . . . . . . . . . . . . . . 40
7.4 Derivative of an exponential function . . . . . . . . . . . . . . . . . . 40
7.5 Derivative of a linear combination . . . . . . . . . . . . . . . . . . . 41
7.6 Derivative of a product of functions . . . . . . . . . . . . . . . . . . . 41
7.7 Derivative of a composition of functions . . . . . . . . . . . . . . . . 42
7.8 Derivatives of trigonometric functions . . . . . . . . . . . . . . . . . 43
7.9 Derivative of an inverse function . . . . . . . . . . . . . . . . . . . . 43
8 Review of integration rules 45
8.1 Inde…nite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.1.1 Inde…nite integral of a constant function . . . . . . . . . . . . 46
8.1.2 Inde…nite integral of a power func tion . . . . . . . . . . . . . 46
8.1.3 Inde…nite integral of a logarithmic function . . . . . . . . . . 47
8.1.4 Inde…nite integral of an exponential function . . . . . . . . . 47
8.1.5 Inde…nite integral of a linear combination of functions . . . . 47
8.1.6 Inde…nite integrals of trigonometric functions . . . . . . . . . 48
8.2 De…nite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8.2.1 Fundamental theorem of calculus . . . . . . . . . . . . . . . . 48
8.2.2 De…nite integral of a linear combination of functions . . . . . 49
8.2.3 Change of variable . . . . . . . . . . . . . . . . . . . . . . . . 50
8.2.4 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . 51
8.2.5 Exchanging the b ou nds of integration . . . . . . . . . . . . . 51
8.2.6 Subdividing the integral . . . . . . . . . . . . . . . . . . . . . 51
8.2.7 Leibniz integral rule . . . . . . . . . . . . . . . . . . . . . . . 52
8.3 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
9 Special functions 55
9.1 Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.1.1 De…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.1.2 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.1.3 Relation to the factorial function . . . . . . . . . . . . . . . . 56
9.1.4 Values of the Gamma function . . . . . . . . . . . . . . . . . 57
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CONTENTS v
9.1.5 Lower incomplete Gamma function . . . . . . . . . . . . . . . 58
9.2 Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9.2.1 De…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9.2.2 Integral representations . . . . . . . . . . . . . . . . . . . . . 59
9.3 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
II Fundamentals of probability 67
10 Probability 69
10.1 Sample space, sample points and events . . . . . . . . . . . . . . . . 69
10.2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
10.3 Properties of probability . . . . . . . . . . . . . . . . . . . . . . . . . 71
10.3.1 Probability of the empty set . . . . . . . . . . . . . . . . . . . 71
10.3.2 Additivity and sigma-additivity . . . . . . . . . . . . . . . . . 72
10.3.3 Probability of the complement . . . . . . . . . . . . . . . . . 72
10.3.4 Probability of a union . . . . . . . . . . . . . . . . . . . . . . 73
10.3.5 Monotonicity of probability . . . . . . . . . . . . . . . . . . . 73
10.4 Interpretations of probability . . . . . . . . . . . . . . . . . . . . . . 74
10.4.1 Classical interpretation of probability . . . . . . . . . . . . . 74
10.4.2 Frequentist interpretation of probability . . . . . . . . . . . . 74
10.4.3 Subjectivist interpretation of probability . . . . . . . . . . . . 74
10.5 More rigorous de…nitions . . . . . . . . . . . . . . . . . . . . . . . . . 75
10.5.1 A more rigorous de…nition of event . . . . . . . . . . . . . . . 75
10.5.2 A more rigorous de…nition of probability . . . . . . . . . . . . 76
10.6 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
11 Zero-probability events 79
11.1 De…nition and discuss ion . . . . . . . . . . . . . . . . . . . . . . . . . 79
11.2 Almost sure and almost surely . . . . . . . . . . . . . . . . . . . . . 80
11.3 Almost sure events . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
11.4 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
12 Conditional probability 85
12.1 Intro du ction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
12.2 The case of equally likely sample points . . . . . . . . . . . . . . . . 85
12.3 A more general approach . . . . . . . . . . . . . . . . . . . . . . . . 87
12.4 Tackling division by zero . . . . . . . . . . . . . . . . . . . . . . . . . 90
12.5 More details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
12.5.1 The law of total probability . . . . . . . . . . . . . . . . . . . 90
12.6 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
13 Bayes’rule 95
13.1 Statement of Bayes’rule . . . . . . . . . . . . . . . . . . . . . . . . . 95
13.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
13.3 Solved exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
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