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1 Probability Spaces
1.1 Limits and continuity in R
1.2 Set theory
1.3 Fields and sigma–fields
1.4 Probability spaces
1.5 Random variables
1.6 Problems
2 Counting techniques
2.1 Counting techniques
2.2 Multiplication rule
2.3 Permutations
2.4 Combinations
2.5 Further techniques
2.6 Problems
3 Conditional probability, Bayes’ theorem
3.1 Conditional probability
3.2 Independent events
3.3 Bayes’ theorem
3.4 Problems
4 Existence of Probability Measures
4.1 Probability determining classes
4.2 Regularity of probability measures
4.3 Caratheodory extension theorem
4.4 Lebesgue–Stiltjes measures
4.5 Kolmogorov consistency theorem
4.6 Existence of a non–measurable set
4.7 Problems
5 Lebesgue Integral
5.1 Lebesgue integral
5.2 Limits theorems for the Lebesgue integral
5.3 Further properties of Lebesgue integrals
5.4 Lp spaces and convexity
5.5 Product measures
5.6 Differentiation and integration
5.7 Radon–Nikodym theorem
5.8 Change of variables
5.9 Problems
6 Basic properties of random variables
6.1 Discrete random variables
6.2 Continuous random variables
6.3 Mixed random variables
6.4 Mode, median, quartiles, percentiles
6.5 Independence
6.6 Conditional probability
6.7 Problems
7 Expectation
7.1 Expectation
7.2 Expectation and independence
7.3 Variance
7.4 Covariance
7.5 Moments
7.6 Characteristic functions
7.7 Conditional expectation
7.8 Problems
8 Common discrete distributions
8.1 Binomial distribution
8.2 Geometric distribution
8.3 Negative binomial
8.4 Negative binomial distribution
8.5 Multinomial distribution
8.6 Hypergeometric distribution
8.7 Poisson distribution
8.8 Problems
9 Common continuous distributions
9.1 Uniform distribution
9.2 Exponential distribution
9.3 Gamma distribution
9.4 Normal distribution
9.5 Multivariate normal distribution
9.6 Cauchy distribution
9.7 Problems
10 Transformation of random variables
10.1 One dimensional transformations
10.2 Multidimensional transformations
10.3 Chi–square distribution
10.4 t-distribution
10.5 F distribution
10.6 Beta distribution
10.7 Maximums and minimums
10.8 Order statistics
10.9 Problems
11 Convergence of r.v.’s
11.1 Convergence almost surely
11.2 Lemmas of Borel–Cantelli
11.3 Convergence in probability
11.4 Strong law of large numbers
11.5 Convergence in distribution
11.6 Problems
12 The central limit theorem via ch.f.’s
12.1 Convergence in distribution via ch.f.’s
12.2 Central limit theorem
12.3 Generalizations of the CLT
12.4 Problems
13 Conditional Expectation
13.1 Conditional expectation
13.2 Conditional expectation given a r.v.
13.3 Regular conditional probability
13.4 The conditional expectation as a projection
13.5 Problems
14 Exploratory Analysis of Data
14.1 Statistics
14.2 Numerical descriptions of data
14.3 Parametric statistics
14.4 Tables
14.5 Graphical descriptions of data
14.6 Problems
15 Point Estimators
15.1 Introduction
15.2 Method of the moments estimators
15.3 Maximum likelihood estimators
15.4 Basic properties of estimators
15.5 Mean square error
15.6 Point estimators from a normal random sample
15.7 Problems
16 Further Properties of Point Estimators
16.1 Uniform minimum variance unbiased estimators
16.2 Efficiency of estimators
16.3 Asymptotics of estimators
16.4 Equivariant estimators
16.5 Robustness
16.6 Problems
17 Data Reduction
17.1 Likelihood principle
17.2 Sufficient statistics
17.3 Factorization theorem
17.4 Minimal sufficient statistics
17.5 Ancillary and complete statistics
17.6 Rao-Blackwell’s theorem
17.7 Problems
18 Confidence Intervals
18.1 Confidence intervals
18.2 Confidence intervals via pivotal quantities
18.3 Pivoting a continuous cdf
18.4 Shortest confidence intervals
18.5 Asymptotic confidence intervals
18.6 Problems
19 Hypothesis Testing
19.1 Testing statistical hypothesis
19.2 Common tests
19.3 Most powerful tests
19.4 Likelihood ratio tests
19.5 Efficiency of tests
19.6 Using tests to find CI’s
19.7 Problems
20 Statistical Decision Theory
20.1 Loss functions
20.2 Point estimation
20.3 Bayesian confidence intervals
20.4 Zero-one losses
20.5 Problems
A Appendix
A.1 Relations
A.2 Topology
A.2.1 Metric spaces
A.2.2 Topological spaces
A.3 Real analysis
A.3.1 Vector spaces
A.3.2 Banach spaces
A.3.3 Hilbert spaces
A.3.4 Spaces of functions
A.4 Problems
B Tables
B.1 Mean, variance and ch.f. of some common distributions
B.2 Tables for the normal distribution
Indexes
References
Subject Index
Author Index