Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Expectation is linear for every finite family of random variables, without any independence hypothesis
Statement
For a finite family of real random variables on one finite probability space and real scalars , No independence hypothesis is required. For , both sides are .
Facts & Assumptions
Given: A finite probability space, random variables , and real scalars , indexed by a finite set .
Expectation is a finite weighted sum over outcomes (Expectation of a real random variable on a finite probability space).
Finite sums are additive and compatible with real scaling (Laws of finite sums and finite products).
Finite Fubini permits the two finite summations to be interchanged (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule).
Proof
Expanding the left side gives .
Finite Fubini and distributivity turn step 1.1 into .
The calculation uses no independence identity. If , the two sums in step 2.1 are empty and equal .
Depends on
Used by
- The expected number of triangles in G(n,p) is binom n3p³ Example
- False: linearity of expectation requires independence False statement
- A Bernoulli(p) variable has mean p and variance p(1-p); a binomial(n,p) variable has mean np and variance np(1-p) Lemma
- Cauchy-Schwarz for finite random variables: E[XY]² leE[X²]E[Y²] Lemma
- The expected number of cycles of length at most ℓ in G(n,p) Lemma
- Var(X)=E[X²]-E[X]² and Cov(X,Y)=E[XY]-E[X]E[Y] Lemma
- The deletion-alteration method converts an expected defect count into a deterministic lower bound Proposition
- An n-vertex graph of minimum degree δ>1 has a dominating set of size at most n(log(δ+1)+1)/(δ+1) Theorem
- Covariance is symmetric and bilinear in finite linear combinations Theorem
- Every finite graph with m edges has a cut containing at least m/2 edges Theorem
- Every k-uniform hypergraph with fewer than 2ᵏ⁻¹ edges is 2-colourable Theorem
- Every nonempty finite set of n nonzero integers has a sum-free subset of size greater than n/3 Theorem
- Szele's bound: for every n≥1, some n-vertex tournament has at least n!/2ⁿ⁻¹ Hamilton paths Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 55 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Matousek and J. Vondrak, The Probabilistic Method, Lemma 3.1.1 (standard reference, not scraped)
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 6.2 (standard reference, not scraped)