Alphabeta MathArticle items · 02 — Expectation of a nonnegative or integrable random variable
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Random elements and real random variables
- The nonnegative Lebesgue integral
- Finite sums and finite products, by recursion
- Riemannian distance is a metric
- The recursion theorem
- The principle of mathematical induction
- Finite sums and finite products of natural numbers, ∑_k<n aₖ and ∏_k<n aₖ in ℕ
- Laws of finite sums and products in ℕ, and ι(∑_k<n aₖ) = ∑_k<n ι(aₖ)
- Multiplication of natural numbers
- Multiplication is associative
- Multiplication is commutative
- Zero and one under multiplication
- Cancellation for multiplication by a nonzero factor
- Addition is cancellative
- Order on the natural numbers
- Every nonzero natural number is a successor
- The von Neumann naturals form a Peano system
- The canonical natural ι(n) = n · 1_F of a field
- Finite, countably infinite, countable, uncountable
- The pigeonhole principle on ℕ
- Equinumerous sets, A ≈ B and A ⪯ B
- Injection, surjection, bijection
- The natural numbers ℕ (von Neumann)
- On ℕ the order is membership: m < n ⟺ m ∈ n
- A function is a relation f with (a,b) ∈ f and (a,c) ∈ f implying b = c; f : A → B, the value f(a), domain and codomain
- The reals form a totally ordered field
- Probability is additive on every finite pairwise-disjoint family of events
- The product g₀ g₁ ⋯ gₙ₋₁ of a finite list in a monoid, by recursion, with the empty product (n = 0) equal to the identity
- Generalised associativity: in a monoid the product of a finite list does not depend on the bracketing, and in a commutative monoid it does not depend on the order of the factors either
- Semigroup and monoid
- The sum ∑_i ∈ S aᵢ over a finite index set, and its product form
- The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Cᵏ maps and multi-index derivative notation in Euclidean space
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
- The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations
- Independent events, pairwise independence, and mutual independence of a finite family
- The sigma-algebra generated by a function
- Independent sigma-algebras and independent events
- Closure properties of measurable functions used by the integral
- Real and imaginary parts, complex conjugation, and modulus
- A smooth vector field is a smooth section of the tangent bundle
- The velocity derivation of a smooth curve
- Nonnegative simple measurable functions
- For a set x ≠ ∅ the collection { z : ∀ s (s ∈ x → z ∈ s) } is a set, and it does not depend on the member of x used to separate it
- The unordered pair {x,y} and the singleton {x} = {x,x}
- There is exactly one set with no elements, written ∅
- Probability measures and probability spaces
- Lebesgue outer measure on ℝⁿ
- Carathéodory measurable sets
- Uniform continuity of a map of metric spaces: one δ serving every point
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Rational powers aʳ of a positive base
- Order on the rationals
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Integer powers aᵐ
- Complete ordered field (least-upper-bound property)
- Ordered field
- Sigma-algebras
- Series in the nonnegative extended real line
- The extended real line ℝ̄ = ℝ ∪ {-∞, +∞}, its order, and the arithmetic that is left undefined
- Characteristic function of a normal law
- A non-negative operator has a unique non-negative square root
- Countably many independent copies of a prescribed law exist
- AC supplies countable selections and prescribed serial paths
- Characteristic functions under affine maps and independent sums
- Uniqueness of a law from its characteristic function
- Expectations factor over finite products of independent random variables
- The Lebesgue integral is linear on L¹(μ)
- Standard normal and normal laws
- Moments, variance, and covariance on a probability space
- The Axiom of Choice
- The exponential is a continuous bijection from ℝ onto (0,∞)
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
DefinitionLibrary prerequisite
Expectation of a nonnegative or integrable random variable
DefinitionLibrary prerequisite
Expectation of a real random variable on a finite probability space
DefinitionLibrary prerequisite
Extended riemannian distance on a disconnected manifold
DefinitionLibrary prerequisite
The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ
DefinitionLibrary prerequisite
The cardinality | A| of a finite set
DefinitionLibrary prerequisite
Finite probability spaces, outcome weights, events, and event probabilities
DefinitionLibrary prerequisite
Real random variables on finite probability spaces and their finite distributions
DefinitionLibrary prerequisite
A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
DefinitionLibrary prerequisite
A finite sum in a commutative monoid indexed by an arbitrary finite set
DefinitionLibrary prerequisite
First-order systems, initial value problems, and solutions on intervals
DefinitionLibrary prerequisite
Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree
DefinitionLibrary prerequisite
Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
DefinitionLibrary prerequisite
The Hessian matrix and critical points of a scalar field
DefinitionLibrary prerequisite
Pairwise and mutual independence of finite-valued random variables
DefinitionLibrary prerequisite
Independent random elements
DefinitionLibrary prerequisite
The indicator random variable of an event
DefinitionLibrary prerequisite
Integrable real and complex functions, and their integrals
DefinitionLibrary prerequisite
Integral curves of a vector field
DefinitionLibrary prerequisite
The integral of a nonnegative simple function
DefinitionLibrary prerequisite
The intersection ⋂ x of a nonempty set, the binary intersection a ∩ b := ⋂{a,b}, and disjointness
DefinitionLibrary prerequisite
Law or distribution of a random element
DefinitionLibrary prerequisite
Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
DefinitionLibrary prerequisite
Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
DefinitionLibrary prerequisite
Maximum and minimum of a set
DefinitionLibrary prerequisite
Measures on sigma-algebras
DefinitionLibrary prerequisite
Multivariate normal law, including singular covariance
DefinitionLibrary prerequisite
The natural logarithm as the inverse of the exponential function
DefinitionLibrary prerequisite