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Showing 151–180 of 1709 reference entries

Mathematics & logicTotal derivative in finite dimensions

Differentiability means a linear approximation with smaller-order error.

DefinitionMathematics & logicDouble integrals on simple regions

A continuous double integral can be evaluated by slicing.

Mathematical statementMathematics & logicGradient, divergence and curl

These coordinate derivatives have different input and output types.

DefinitionMathematics & logicStokes' circulation theorem

Circulation around a boundary equals oriented curl flux.

Mathematical statementMathematics & logicDivergence theorem

Outward boundary flux equals integrated divergence.

Mathematical statementMathematics & logicLocal existence and uniqueness for autonomous ODEs

Local Lipschitz continuity ensures one local trajectory.

Mathematical statementMathematics & logicConstant-coefficient linear ODE systems

The matrix exponential solves a finite linear system.

Mathematical statementMathematics & logicLaplace, heat and wave equation problems

A PDE becomes a solution problem only with a domain and data.

ModelMathematics & logicRandom variables and probability distributions

A distribution assigns probabilities to a random variable's values.

DefinitionMathematics & logicNormal distribution

A normal model is specified by location and positive scale.

DefinitionMathematics & logicBinomial and Poisson count models

Different count mechanisms require different distributions.

DefinitionMathematics & logicWeak law of large numbers

A sample mean concentrates near its mean under stated assumptions.

Mathematical statementMathematics & logicClassical iid central limit theorem

Centered sample averages have an asymptotic normal law.

Mathematical statementMathematics & logicConditional masses and densities

A conditional distribution renormalizes a joint law.

DefinitionMathematics & logicLikelihood and Bayesian updating

Likelihood is a function of a parameter given observed data.

MethodMathematics & logicNull tests, significance and p-values

A p-value is calibrated under a specified null model.

MethodMathematics & logicSampling, randomization and confounding

Selection and treatment assignment answer different questions.

MethodMathematics & logicPermutation inference under exchangeability

Distribution-free procedures still require assumptions.

MethodInformationFinite-alphabet mutual information

Mutual information measures statistical dependence in bits.

DefinitionInformationNear-lossless source coding threshold

Entropy bounds asymptotic compression for an iid source.

Mathematical statementInformationDiscrete memoryless channel capacity

Capacity is the maximal asymptotically reliable rate.

Mathematical statementInformationHamming distance and error correction

Separated codewords permit bounded symbol-error recovery.

Mathematical statementInformationInternet protocol layers

Protocol layers divide communication responsibilities.

DefinitionInformationTCP reliable byte-stream transport

TCP orders bytes and retransmits detected losses.

MethodInformationEncryption, hashes and message authentication

Cryptographic primitives serve distinct security properties.

DefinitionInformationPublic-key digital signatures

A signing key and a verification key have different roles.

DefinitionInformationComputable reductions and undecidability

Reduction direction determines which impossibility transfers.

Mathematical statementInformationDecision classes P and NP

Polynomial solving and polynomial verification differ.

DefinitionInformationNP-hardness and NP-completeness

NP-completeness combines verification and reduction hardness.

DefinitionInformationBounded-error randomized computation

Random choices permit a stated probability of error.

Definition
Mathematical statementSource checked

Weak law of large numbers

Mathematics & logic

A sample mean concentrates near its mean under stated assumptions.

Key knowledge

For iid real variables with mean μ and finite variance, the sample mean X̄ₙ converges in probability to μ: for every ε>0, P(|X̄ₙ−μ|>ε)→0.

Conditions and scope

  • Independence and identical distribution are imposed here as sufficient conditions.
  • This does not say each observation approaches μ or that finite samples are error-free.
Sources · 1

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Probability 1, chapter 9Clare Wallace / Durham UniversityCourse notesmaths.dur.ac.ukRead source

Evidence and review

Formal weak-law and classical CLT statements; overbroad historical/aside claims excluded.

2026-10-10 · Finite-variance iid weak law; broader aside claims excluded.

Related reference entries

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Scientific relationships · 3 / 3

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  1. Uses definition
    Relationship sources · 1

    Linked statements

    • For iid real variables with mean μ and finite variance, the sample mean X̄ₙ converges in probability to μ: for every ε>0, P(|X̄ₙ−μ|>ε)→0.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • An estimator sequence Tₙ is consistent for θ when Pθ(|Tₙ−θ|>ε)→0 for every ε>0. Asymptotic normality instead specifies a normalized error's limiting distribution; neither property follows merely from an estimator being a likelihood maximizer.
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • For iid finite-alphabet X₁,…,Xₙ, −n⁻¹log₂p(X₁,…,Xₙ) converges in probability to H(X₁). With high probability the sequence lies in a typical set of about 2ⁿᴴ probability-scale possibilities, which supports near-lossless block compression.
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math-law-large-numbers
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university-reference.1
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University reference