Alphabeta MathConditional Expectation
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Linear Algebra
Measure Theory
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Radon Nikodym Theorem and Lebesgue Decomposition
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Conditional expectation given a sigma algebra
Theorem
Conditional expectation exists by radon nikodym
Lemma
Conditional expectation is unique almost surely
Definition
Conditional expectation as an ae class
Theorem
Basic algebra and order properties of conditional expectation
Theorem
Taking out what is known
Theorem
Tower property of conditional expectation
Lemma
Conditioning a known variable and an independent variable
Definition
Conditional expectation for nonnegative variables
Theorem
Conditional monotone convergence
Theorem
Conditional fatou and dominated convergence
Lemma
Convex functions have countable supporting line representations
Theorem
Conditional jensen inequality
Lemma
Absolute real powers are Borel measurable and convex
Corollary
Conditional lp contraction
Theorem
Conditional expectation is the l2 orthogonal projection
Definition
Conditional variance
Lemma
Conditional variance is well-defined and has the second-moment formula
Corollary
Conditional variance decomposition
Theorem
Uniform integrability of conditional expectations of one variable
Remark
Conditional expectation is a class not a canonical pointwise function
Corollary