Random Vectors: Notes

Probability

The Analysis of Data, volume 1

0
- Front Matter
- 0.1: Contents
- 0.2: Preface
1
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
4
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
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

$ \def\P{\mathsf{\sf P}} \def\E{\mathsf{\sf E}} \def\Var{\mathsf{\sf Var}} \def\Cov{\mathsf{\sf Cov}} \def\std{\mathsf{\sf std}} \def\Cor{\mathsf{\sf Cor}} \def\R{\mathbb{R}} \def\c{\,|\,} \def\bb{\boldsymbol} \def\diag{\mathsf{\sf diag}} \def\defeq{\stackrel{\tiny\text{def}}{=}} $

4.10. Notes

More information on random vectors is available in nearly any probability textbook. Elementary exposition that avoids measure theory is available in most undergraduate probability textbooks, for example (Weiss, 2005) or (Dasgupta, 2010). A rigorous measure-theoretic description is available in (Breiman, 1992), (Billingsley, 1995), (Ash and Doleans-Dade, 1999), or (Resnick, 1999). Convolutions are usually described in detail in signal processing books, for example (Oppenheim et. al, 1996).