Probability: The Analysis of Data, volume 1
0
Front Matter
0.1: Contents
0.2: Preface
1
Basic Definitions
1.1: Sample Space and Events
1.2: The Probability Function
1.3: Classical Model 1
1.4: Classical Model 2
1.5: Conditional Probability
1.6: Basic Combinatorics
1.7: Probability and Measure
1.8: Notes
1.9: Exercises
2
Random Variables
2.1: Basic Definitions
2.2: Functions of RVs
2.3: Expectation and Variance
2.4: Moments and MGF
2.5: RVs and Measure Theory
2.6: Notes
2.7: Exercises
3
Important RVs
3.1: Bernoulli RV
3.2: Binomial RV
3.3: Geometric RV
3.4: Hypergeometric RV
3.5: Negative Binomial RV
3.6: Poisson RV
3.7: Uniform RV
3.8: Exponential RV
3.9: Gaussian RV
3.10: Gamma RV
3.11: t RV
3.12: Beta RV
3.13: Mixture RV
3.14: Empirical RV
3.15: Smoothed Empirical RV
3.16: Notes
3.17: Exercises
4
Random Vectors
4.1: Basic Definitions
4.2: Joint Pmf, Pdf, and Cdf
4.3: Marginal Random Vectors
4.4: Functions of Random Vectors
4.5: Conditional Random Vectors
4.6: Moments
4.7: Conditional Expectation
4.8: Moment Generating Functions
4.9: Random Vectors and Measure
4.10: Notes
4.11: Exercises
5
Important Vectors
5.1: Multinomial Vectors
5.2: Gaussian Vectors
5.3: Dirichlet Vectors
5.4: Mixture Vectors
5.5: Exponential Family
5.6: Notes
5.7: Exercises
6
Random Processes
6.1: Basic Definitions
6.2: Marginals
6.3: Moments
6.4: Random Walk
6.5: Processes and Measure
6.6: Borell-Cantelli and Zero-One
6.7: Notes
6.8: Exercises
7
Important RPs
7.1: Markov Chains
7.2: Poisson Process
7.3: Gaussian Process
7.4: Notes
7.5: Exercises
8
Limit Theorems
8.1: Modes of Convergence
8.2: Relationship of Modes
8.3: DCT Theorem for Vectors
8.4: Scheffe's Theorem
8.5: Portmanteau Lemma
8.6: Law of Large Numbers
8.7: Characteristic Functions
8.8: Levy's Theorem
8.9: Central Limit Theorem
8.10: Continuous Mapping Theorem
8.11: Slustsky's Theorem
8.12: Notes
8.13: Exercises
A
Set Theory
A.1: Basic Definition
A.2: Functions
A.3: Cardinality
A.4: Limits of Sets
A.5: Notes
A.6: Exercises
B
Metric Spaces
B.1: Basic Definitions
B.2: Limits
B.3: Continuity
B.4: Euclidean Space
B.5: Growth of Functions
B.6: Notes
B.7: Exercises
C
Linear Algebra
C.1: Basic Definitions
C.2: Rank
C.3: Eigenvalues and Determinant
C.4: Semidefinite Matrices
C.5: SVD
C.6: Notes
C.7: Exercises
D
Differentiation
D.1: Scalar Differentiation
D.2: Power and Taylor Series
D.3: Notes
D.4: Exercises
E
Measure Theory
E.1: Sigma Algebras
E.2: Measure Function
E.3: Extension Theorem
E.4: Independence
E.5: Important Measures
E.6: Measurable Functions
E.7: Notes
F
Integration
F.1: Riemann Integral
F.2: Integration and Differentiation
F.3: Lebesgue Integral
F.4: Product Measures
F.5: Product Integration
F.6: Multivariate Extensions
F.7: Notes
Probability: The Analysis of Data, volume 1
0
Front Matter
0.1: Preface
0.2: Contents
1
Basic Definitions
1.1: Sample Space and Events
1.2: The Probability Function
1.3: Classical Model 1
1.4: Classical Model 2
1.5: Conditional Probability
1.6: Basic Combinatorics
1.7: Probability and Measure
1.8: Notes
1.9: Exercises
2
Random Variables
2.1: Basic Definitions
2.2: Functions of RVs
2.3: Expectation and Variance
2.4: Moments and MGF
2.5: RVs and Measure Theory
2.6: Notes
2.7: Exercises
3
Important RVs
3.1: Bernoulli RV
3.2: Binomial RV
3.3: Geometric RV
3.4: Hypergeometric RV
3.5: Negative Binomial RV
3.6: Poisson RV
3.7: Uniform RV
3.8: Exponential RV
3.9: Gaussian RV
3.10: Gamma RV
3.11: t RV
3.12: Beta RV
3.13: Mixture RV
3.14: Empirical RV
3.15: Smoothed Empirical RV
3.16: Notes
3.17: Exercises
4
Random Vectors
4.1: Basic Definitions
4.2: Joint Pmf, Pdf, and Cdf
4.3: Marginal Random Vectors
4.4: Functions of Random Vectors
4.5: Conditional Random Vectors
4.6: Moments
4.7: Conditional Expectation
4.8: Moment Generating Functions
4.9: Random Vectors and Measure
4.10: Notes
4.11: Exercises
5
Important Vectors
5.1: Multinomial Vectors
5.2: Gaussian Vectors
5.3: Dirichlet Vectors
5.4: Mixture Vectors
5.5: Exponential Family
5.6: Notes
5.7: Exercises
6
Random Processes
6.1: Basic Definitions
6.2: Marginals
6.3: Moments
6.4: Random Walk
6.5: Processes and Measure
6.6: Borell-Cantelli and Zero-One
6.7: Notes
6.8: Exercises
7
Important RPs
7.1: Markov Chains
7.2: Poisson Process
7.3: Gaussian Process
7.4: Notes
7.5: Exercises
8
Limit Theorems
8.1: Modes of Convergence
8.2: Relationship of Modes
8.3: DCT Theorem for Vectors
8.4: Scheffe's Theorem
8.5: Portmanteau Lemma
8.6: Law of Large Numbers
8.7: Characteristic Functions
8.8: Levy's Theorem
8.9: Central Limit Theorem
8.10: Continuous Mapping Theorem
8.11: Slustsky's Theorem
8.12: Notes
8.13: Exercises
A
Set Theory
A.1: Basic Definition
A.2: Functions
A.3: Cardinality
A.4: Limits of Sets
A.5: Notes
A.6: Exercises
B
Metric Spaces
B.1: Basic Definitions
B.2: Limits
B.3: Continuity
B.4: Euclidean Space
B.5: Growth of Functions
B.6: Notes
B.7: Exercises
C
Linear Algebra
C.1: Basic Definitions
C.2: Rank
C.3: Eigenvalues and Determinant
C.4: Semidefinite Matrices
C.5: SVD
C.6: Notes
C.7: Exercises
D
Differentiation
D.1: Scalar Differentiation
D.2: Power and Taylor Series
D.3: Notes
D.4: Exercises
E
Measure Theory
E.1: Sigma Algebras
E.2: Measure Function
E.3: Extension Theorem
E.4: Independence
E.5: Important Measures
E.6: Measurable Functions
E.7: Notes
F
Integration
F.1: Riemann Integral
F.2: Integration and Differentiation
F.3: Lebesgue Integral
F.4: Product Measures
F.5: Product Integration
F.6: Multivariate Extensions
F.7: Notes