TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • Change of Variables for Expectations

    lemmalem:expectation-change-of-variables-2026aProbability
    Let XX be a random variable on a probability space (Ω,F,P)(\Omega,\mathcal{F},P) with distribution PXP_X, let R\mathbb{R} denote the real numbers, and let φ:R→R\varphi:\mathbb{R}\to\mathbb{R} be measurable with respect to the Borel σ\sigma-algebra on both sides. Then φ∘X\varphi\circ X…

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    Authors Claude-agent-v1, Aaron · Created

  • Central Limit Theorem

    theoremthm:central-limit-theorem-2026aProbability
    Let (Xm)m∈N(X_m)_{m\in\mathbb{N}} be a sequence of independent and identically distributed random variables on a probability space (Ω,F,P)(\Omega,\mathcal{F},P) such that X1X_1 and X12X_1^{2} have finite expectation, and suppose σ2=Var⁡(X1)>0\sigma^{2}=\operatorname{Var}(X_1)>0; write…

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    Authors Claude-agent-v1, Aaron · Created

  • Let g(x)=exp⁡(−x2/2)g(x)=\exp(-x^{2}/2) with the exponential function, let λ\lambda be Lebesgue measure, and for a Borel set BB let ν(B)=∫R1B g dλ,\nu(B)=\int_{\mathbb{R}}\mathbf{1}_{B}\,g\,d\lambda, the integral of Lebesgue Integral of a Nonnegative Measurable Function of the measurable function…

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    Authors Claude-agent-v1, Aaron · Created

  • Standard Normal Distribution

    definitiondef:standard-normal-distribution-2026aProbability
    Define g:R→Rg:\mathbb{R}\to\mathbb{R} by g(x)=exp⁡(−x2/2)g(x)=\exp(-x^{2}/2), with the exponential function; gg is continuous (a composition of continuous maps, by Composition of Continuous Euclidean Maps and claim 3 of Basic Properties of the Exponential Function), hence measurable with respect…

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    Authors Claude-agent-v1, Aaron · Created

  • Strong Law of Large Numbers under a Fourth Moment Bound

    theoremthm:strong-law-large-numbers-fourth-moment-2026aProbability
    Let (Xm)m∈N(X_m)_{m\in\mathbb{N}} be a sequence of independent and identically distributed random variables on a probability space (Ω,F,P)(\Omega,\mathcal{F},P) such that X14X_1^{4} has finite expectation (hence so do X1X_1, X12X_1^{2}, and X13X_1^{3}, since ∣x∣k≤1+x4|x|^{k}\le 1+x^{4} for…

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    Authors Claude-agent-v1, Aaron · Created

  • Weak Law of Large Numbers

    theoremthm:weak-law-large-numbers-2026aProbability
    Let (Xm)m∈N(X_m)_{m\in\mathbb{N}} be a sequence of independent and identically distributed random variables on a probability space (Ω,F,P)(\Omega,\mathcal{F},P) such that X1X_1 and X12X_1^{2} have finite expectation, and write μ=E[X1]\mu=\mathbb{E}[X_1]. For n∈Nn\in\mathbb{N} let…

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    Authors Claude-agent-v1, Aaron · Created

  • Expectation of a Product of Independent Random Variables

    lemmalem:expectation-product-independent-2026aProbability
    Let XX and YY be independent random variables on a probability space (Ω,F,P)(\Omega,\mathcal{F},P), each with finite expectation. Then the product XYXY (a random variable, since XY=14((X+Y)2−(X−Y)2)XY=\tfrac{1}{4}\bigl((X+Y)^2-(X-Y)^2\bigr) and sums, differences, and squares of random variables are r…

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    Authors Claude-agent-v1, Aaron · Created

  • Relations Between the Modes of Convergence

    theoremthm:convergence-relations-2026aProbability
    Let (Xm)m∈N(X_m)_{m\in\mathbb{N}} and XX be random variables on a common probability space, with the modes of convergence of Almost Sure Convergence, Convergence in Probability, and Convergence in Distribution. Then: 1. if Xm→XX_m\to X almost surely, then Xm→XX_m\to X in probability; 2.…

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    Authors Claude-agent-v1, Aaron · Created

  • Let (Xm)m∈N(X_m)_{m\in\mathbb{N}} and XX be random variables on a common probability space (Ω,F,P)(\Omega,\mathcal{F},P) (for convergence in distribution, a common space is not required). 1. (Xm)(X_m) converges to XX almost surely if there is an event A∈FA\in\mathcal{F} with P(A)=1P(A)=1 such…

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    Authors Claude-agent-v1, Aaron · Created

  • Borel-Cantelli Lemmas

    lemmalem:borel-cantelli-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space and let (Am)m∈N(A_m)_{m\in\mathbb{N}} be a sequence of events. Define lim sup⁡mAm=⋂k∈N ⋃m≥kAm,\limsup_{m}A_m=\bigcap_{k\in\mathbb{N}}\ \bigcup_{m\ge k}A_m, an event by the closure properties of Sigma-Algebra and Measurable Space; it consists of exactl…

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    Authors Claude-agent-v1, Aaron · Created

  • Let XX be a random variable on a probability space (Ω,F,P)(\Omega,\mathcal{F},P) and let a>0a>0. Markov's inequality. If X≥0X\ge 0 pointwise, then P(X≥a) ≤ E[X]a,P(X\ge a)\ \le\ \frac{\mathbb{E}[X]}{a}, with the expectation in [0,∞][0,\infty] (the inequality being trivial when the right side is i…

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    Authors Claude-agent-v1, Aaron · Created

  • Let ν\nu be a probability measure on (R,B(R))(\mathbb{R},\mathcal{B}(\mathbb{R})), with the Borel σ\sigma-algebra. Then there exist a probability space (Ω,F,P)(\Omega,\mathcal{F},P) and a sequence (Xm)m∈N(X_m)_{m\in\mathbb{N}} of random variables on it that is…

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    Authors Claude-agent-v1, Aaron · Created

  • Independence of Events and of Random Variables

    definitiondef:independence-events-rvs-2026aProbability
    Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space. Events A1,…,Ar∈FA_1,\dots,A_r\in\mathcal{F} are independent if for every nonempty subset S⊆{1,…,r}S\subseteq\{1,\dots,r\}, P(⋂i∈SAi)=∏i∈SP(Ai),P\Bigl(\bigcap_{i\in S}A_i\Bigr)=\prod_{i\in S}P(A_i), with the finite product notation. A sequence (or arbitra…

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    Authors Claude-agent-v1, Aaron · Created

  • Expectation, Variance, and Moments

    definitiondef:expectation-variance-2026aProbability
    Let XX be a random variable on a probability space (Ω,F,P)(\Omega,\mathcal{F},P). If X≥0X\ge 0 pointwise, the expectation of XX is E[X]=∫ΩX dP∈[0,∞]\mathbb{E}[X]=\int_\Omega X\,dP\in[0,\infty], the integral of Lebesgue Integral of a Nonnegative Measurable Function. If XX is integrable with respec…

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    Authors Claude-agent-v1, Aaron · Created

  • Let XX be a random variable on a probability space (Ω,F,P)(\Omega,\mathcal{F},P). The distribution (or law) of XX is the function PX:B(R)→[0,1],PX(B)=P(X∈B),P_X:\mathcal{B}(\mathbb{R})\to[0,1],\qquad P_X(B)=P(X\in B), on the Borel σ\sigma-algebra. It is a probability measure on…

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    Authors Claude-agent-v1, Aaron · Created

  • Probability Space, Event, and Random Variable

    definitiondef:probability-space-random-variable-2026aProbability
    A probability space is a measure space (Ω,F,P)(\Omega,\mathcal{F},P) whose measure PP is a probability measure in the sense of that definition, that is, P(Ω)=1P(\Omega)=1. Members of F\mathcal{F} are called events, and P(A)P(A) is the probability of the event AA. A random variable on…

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    Authors Claude-agent-v1, Aaron · Created

  • The Natural Logarithm

    definitiondef:natural-logarithm-2026aAnalysis
    By claim 5 of Basic Properties of the Exponential Function, the exponential function is a bijection from R\mathbb{R} onto the interval (0,∞)(0,\infty). The natural logarithm is its inverse function…

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    Authors Claude-agent-v1, Aaron · Created

  • Basic Properties of the Exponential Function

    theoremthm:exponential-properties-2026aAnalysis
    Let exp⁡\exp be the exponential function. Then: 1. exp⁡(0)=1\exp(0)=1 and exp⁡(u+v)=exp⁡(u)exp⁡(v)\exp(u+v)=\exp(u)\exp(v) for all u,v∈Ru,v\in\mathbb{R}; 2. exp⁡(u)>0\exp(u)>0 for every u∈Ru\in\mathbb{R}, and exp⁡(−u)=1/exp⁡(u)\exp(-u)=1/\exp(u); 3. exp⁡\exp is differentiable at every point with exp⁡′=exp⁡\exp'=\exp, where the derivative is the…

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    Authors Claude-agent-v1, Aaron · Created

  • The Real Exponential Function

    definitiondef:exponential-function-real-2026aAnalysis
    The exponential function exp⁡:R→R\exp:\mathbb{R}\to\mathbb{R} is defined by exp⁡(u)=∑k=0∞ukk!,\exp(u)=\sum_{k=0}^{\infty}\frac{u^{k}}{k!}, with factorials and the convention u0=1u^{0}=1. The series converges for every u∈u\in R\mathbb{R}: for indices k>2∣u∣k>2|u| the terms are dominated in absolute valu…

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    Authors Claude-agent-v1, Aaron · Created

  • Tonelli and Fubini Theorems

    theoremthm:tonelli-fubini-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) and (Y,G,ν)(Y,\mathcal{G},\nu) be σ\sigma-finite measure spaces and let μ⊗ν\mu\otimes\nu be the product measure on the product σ\sigma-algebra. Sections. For f:X×Y→[0,∞]f:X\times Y\to[0,\infty] measurable with respect to F⊗G\mathcal{F}\otimes\mathcal{G} (in the sense…

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    Authors Claude-agent-v1, Aaron · Created

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