| Preface |
|
v | |
| Acknowledgments |
|
vii | |
Week
1. Probability |
|
1 | (13) |
|
|
|
1 | (3) |
|
1.2 The laws of set theory |
|
|
4 | (2) |
|
1.3 Conditional probability |
|
|
6 | (1) |
|
|
|
7 | (2) |
|
|
|
9 | (1) |
|
1.6 Combinatorial formulas |
|
|
9 | (4) |
|
|
|
13 | (1) |
Week
2. Random Variables |
|
14 | (18) |
|
2.1 Random variables and their distributions |
|
|
14 | (2) |
|
2.2 Some quantitative characteristics: Median and quantiles |
|
|
16 | (1) |
|
2.3 Independent random variables |
|
|
16 | (1) |
|
2.4 Discrete random variables |
|
|
17 | (1) |
|
2.5 Examples of discrete distributions |
|
|
18 | (4) |
|
2.6 Continuous distributions |
|
|
22 | (2) |
|
2.7 Examples of continuous random variables |
|
|
24 | (6) |
|
|
|
30 | (2) |
Week
3. Joint Distributions |
|
32 | (13) |
|
3.1 The joint distribution of two random variables |
|
|
32 | (1) |
|
3.2 Discrete random vectors |
|
|
33 | (1) |
|
3.3 Review of double integrals |
|
|
34 | (2) |
|
3.4 Continuous random vectors |
|
|
36 | (1) |
|
3.5 Uniform distribution for vectors |
|
|
37 | (1) |
|
3.6 Several continuous random variables |
|
|
38 | (1) |
|
3.7 Bivariate normal density |
|
|
38 | (1) |
|
3.8 The distribution of independent random variables |
|
|
39 | (1) |
|
|
|
40 | (1) |
|
3.10 Conditional frequencies |
|
|
40 | (1) |
|
3.11 Conditional densities |
|
|
41 | (1) |
|
3.12 Bayes' formula for the densities |
|
|
42 | (2) |
|
|
|
44 | (1) |
Week
4. Transformations of the Distributions |
|
45 | (16) |
|
4.1 The density for the invertible functions |
|
|
45 | (3) |
|
4.2 Sums of random variables |
|
|
48 | (2) |
|
4.3 Ratio distribution/quotients |
|
|
50 | (3) |
|
4.4 Transformation of the joint density |
|
|
53 | (3) |
|
|
|
56 | (1) |
|
4.6 Distributions for maximums and minimums |
|
|
56 | (2) |
|
|
|
58 | (2) |
|
|
|
60 | (1) |
Week
5. Expectation of Random Variables |
|
61 | (11) |
|
5.1 Expectation of a discrete random variable |
|
|
61 | (2) |
|
5.2 Expectation of a continuous random variable |
|
|
63 | (1) |
|
5.3 The case of the distributions with heavy tails (fat tails) |
|
|
63 | (1) |
|
5.4 Expectations of functions of random variables |
|
|
64 | (2) |
|
5.5 Joint probability distributions |
|
|
66 | (1) |
|
5.6 Expectation of linear combination of random variables |
|
|
66 | (2) |
|
5.7 Expectation of a product |
|
|
68 | (1) |
|
5.8 Probability of an event |
|
|
69 | (1) |
|
|
|
70 | (1) |
|
|
|
71 | (1) |
Week
6. Variance and Covariance |
|
72 | (13) |
|
6.1 Variance and standard deviation |
|
|
72 | (2) |
|
6.2 Existence of variance and 2nd moment |
|
|
74 | (1) |
|
6.3 Variance of linear transformations of random variables |
|
|
75 | (1) |
|
6.4 Chebyshev's inequality |
|
|
75 | (3) |
|
6.5 Example: Binomial distribution |
|
|
78 | (1) |
|
|
|
79 | (1) |
|
6.7 Covariance and linear transformations |
|
|
79 | (1) |
|
|
|
80 | (2) |
|
6.9 Example: Investment portfolios |
|
|
82 | (1) |
|
6.10 Example: Linearly related random variables |
|
|
83 | (1) |
|
|
|
84 | (1) |
Week
7. Conditional Expectations |
|
85 | (9) |
|
7.1 Conditional expectations given an observed value |
|
|
85 | (2) |
|
7.2 Conditional expectation as an expectation |
|
|
87 | (1) |
|
|
|
87 | (1) |
|
7.4 Conditional expectation as a random variable |
|
|
88 | (3) |
|
7.5 Expectation as the best estimate |
|
|
91 | (2) |
|
|
|
93 | (1) |
Week
8. Moment Generating Functions |
|
94 | (11) |
|
8.1 Definition of moment generating function |
|
|
94 | (1) |
|
8.2 m.g.f. for a sum of independent random variables |
|
|
95 | (2) |
|
8.3 m.g.f. and linear transformation of the random variable |
|
|
97 | (1) |
|
8.4 m.g.f. and the moments |
|
|
98 | (2) |
|
8.5 Calculation of the moments using m.g.f. |
|
|
100 | (1) |
|
8.6 On the polynomial and exponential m.g.f. |
|
|
101 | (3) |
|
|
|
104 | (1) |
Week
9. Analysis of Some Important Distributions |
|
105 | (16) |
|
9.1 Normal (Gaussian) distribution |
|
|
105 | (2) |
|
9.2 Bivariate normal density |
|
|
107 | (4) |
|
9.3 Chi-squared distribution |
|
|
111 | (2) |
|
|
|
113 | (1) |
|
|
|
114 | (3) |
|
9.6 Some other distributions |
|
|
117 | (2) |
|
|
|
119 | (2) |
Week
10. Limit Theorems |
|
121 | (13) |
|
10.1 Classical limits for the sequences of real numbers |
|
|
121 | (1) |
|
10.2 Types of limits for random variables |
|
|
121 | (1) |
|
10.3 The Law of Large Numbers |
|
|
122 | (1) |
|
10.4 Convergence of random variables in distribution |
|
|
123 | (2) |
|
10.5 The Central Limit Theorem |
|
|
125 | (7) |
|
|
|
132 | (2) |
Week
11. Statistical Inference: Point Estimation |
|
134 | (14) |
|
11.1 Maximum Likelihood Estimation |
|
|
134 | (3) |
|
11.2 The method of moments |
|
|
137 | (2) |
|
|
|
139 | (1) |
|
11.4 Properties of the sample mean and sample variance |
|
|
139 | (1) |
|
11.5 Properties of Gaussian samples |
|
|
140 | (4) |
|
11.6 Some useful distributions for Gaussian samples |
|
|
144 | (3) |
|
|
|
147 | (1) |
Week
12. Statistical Inference: Interval Estimation |
|
148 | (12) |
|
12.1 Critical values for the distributions |
|
|
148 | (1) |
|
12.2 Interval estimation: Confidence interval |
|
|
149 | (4) |
|
|
|
153 | (4) |
|
12.4 Confidence intervals and hypothesis testing |
|
|
157 | (1) |
|
|
|
158 | (2) |
| Appendix 1: Solutions for the Problems for Weeks 1-12 |
|
160 | (27) |
| Appendix 2: Sample Problems for Final Exams |
|
187 | (8) |
| Appendix 3: Some Bonus Challenging Problems |
|
195 | (3) |
| Appendix 4: Statistical Tables |
|
198 | (7) |
| Bibliography |
|
205 | (2) |
| Index |
|
207 | (2) |
| Legend of Notations and Abbreviations |
|
209 | |
