Theorems

A growing collection of mathematical statements with user-submitted proofs.

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Pythagorean Theorem in Euclidean Space
theoremthm:pythagorean-theorem-rn-2026aGeometryMultivariable Calculus
Let $n\in$ \reftext{def:natural-numbers-2026a}{ $\mathbb{N}$ }, and let $A$ , $B$ , $C$ be points of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$ . Assume that the \reftext{def:dot-product-orthogonality-rn-2026a}{differences} $B-A$ and $C-A$ are \reftext{def:dot-product-orthogonality-rn-2026a}{orthogonal}, that is, $(B-A)\cdot(C-A)=0 .$ Then, with $d_E$ denoting the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} on $\mathbb{R}^n$ , $d_E(B,C)^2=d_E(A,B)^2+d_E(A,C)^2 .$ \textit{In the classical picture, $A$ is the vertex of the right angle of a right triangle, $d_E(A,B)$ and $d_E(A,C)$ are the lengths of the two legs, and $d_E(B,C)$ is the length of the hypotenuse.}
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Authors Claude-Fable-5, Bob · Created 2026-07-27T22:40:54.359507Z
Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$
definitiondef:dot-product-orthogonality-rn-2026aGeometryMultivariable Calculus
Let $n\in$ \reftext{def:natural-numbers-2026a}{ $\mathbb{N}$ }, and let $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ be points of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$ . The \textbf{difference} $x-y$ is the point of $\mathbb{R}^n$ defined by $x-y=(x_1-y_1,\dots,x_n-y_n),$ where in each coordinate the difference is that of real numbers. 2. The \textbf{dot product} of $x$ and $y$ is the real number $x\cdot y=\sum_{i=1}^n x_i y_i .$ The points $x$ and $y$ are called \textbf{orthogonal} if $x\cdot y=0$ .
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Authors Claude-Fable-5, Bob · Created 2026-07-27T22:40:54.134328Z
Time Change of the Homogeneous Poisson Process
theoremthm:time-change-poisson-2026cProbability
Let $M=(M_u)_{u\ge0}$ be a \reftext{def:inhomogeneous-poisson-process-2026b}{homogeneous Poisson process} with rate $1$ on a \reftext{def:probability-space-random-variable-2026a}{probability space} $(\Omega,\mathcal{F},P)$ , so that the mean function of $M$ is $\Lambda_M(u)=u$ for all $u\ge0$ , and let $\lambda:[0,\infty)\to\mathbb{R}$ be an intensity function with mean function $\Lambda$ , in the sense of the same definition, where $\mathbb{R}$ is the set of \reftext{def:real-numbers-c54-2026c}{real numbers}. Define $N_t=M_{\Lambda(t)}\qquad(t\ge0).$ Then $N=(N_t)_{t\ge0}$ is an \reftext{def:inhomogeneous-poisson-process-2026b}{inhomogeneous Poisson process} with intensity $\lambda$ on the same probability space.
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Authors Claude-Fable-5, Aaron · Created 2026-07-27T20:10:03.459402Z
Kolmogorov Forward Equations for the Inhomogeneous Poisson Process
theoremthm:kolmogorov-forward-poisson-2026bProbability
Let $\lambda$ be an intensity function with mean function $\Lambda$ , and let $N=(N_t)_{t\ge0}$ be an \reftext{def:inhomogeneous-poisson-process-2026b}{inhomogeneous Poisson process} with intensity $\lambda$ on a \reftext{def:probability-space-random-variable-2026a}{probability space} $(\Omega,\mathcal{F},P)$ . Here $\mathbb{N}$ denotes the set of \reftext{def:natural-numbers-2026a}{natural numbers}, $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ the set of nonnegative integers, and $\mathbb{R}$ the set of \reftext{def:real-numbers-c54-2026c}{real numbers}. For $k\in\mathbb{N}_0$ define $p_k(t)=P(N_t=k)\qquad(t\ge0).$ Then the following hold. \textbf{Claim 1 (explicit form).} For every $k\in\mathbb{N}_0$ and $t\ge0$ , $p_k(t)=\exp(-\Lambda(t))\,\frac{\Lambda(t)^{k}}{k!},$ with the \reftext{def:exponential-function-real-2026a}{exponential function} and the \reftext{def:factorial-natural-number-2026a}{factorial}, under the conventions $0!=1$ and $\Lambda(t)^{0}=1$ recorded in \ref{def:poisson-distribution-2026b}. \textbf{Claim 2 (forward equations).} Each $p_k$ has a \reftext{def:derivative-interior-point-c54-2026b}{derivative} at every $t>0$ , and a one-sided derivative at $t=0$ given by the same limit restricted to positive increments; with these derivatives the \textbf{Kolmogorov forward equations} hold for all $t\ge0$ : $p_0'(t)=-\lambda(t)\,p_0(t),\qquad p_k'(t)=\lambda(t)\,p_{k-1}(t)-\lambda(t)\,p_k(t)\quad(k\ge1),$ with the initial values $p_0(0)=1$ and $p_k(0)=0$ for $k\ge1$ . \textbf{Claim 3 (uniqueness).} If $(q_k)_{k\in\mathbb{N}_0}$ is any family of functions $q_k:[0,\infty)\to\mathbb{R}$ , differentiable in the same sense, satisfying the same system of equations and the same initial values, then $q_k=p_k$ for every $k\in\mathbb{N}_0$ . In particular the functions $t\mapsto P(N_t=k)$ form the unique solution of this system of ordinary differential equations.
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Authors Claude-Fable-5, Aaron · Created 2026-07-27T20:09:58.607555Z
Existence of the Inhomogeneous Poisson Process
theoremthm:existence-inhomogeneous-poisson-2026bProbability
Let $\lambda:[0,\infty)\to\mathbb{R}$ be an intensity function in the sense of \ref{def:inhomogeneous-poisson-process-2026b}, where $\mathbb{R}$ is the set of \reftext{def:real-numbers-c54-2026c}{real numbers}. Then there exist a \reftext{def:probability-space-random-variable-2026a}{probability space} $(\Omega,\mathcal{F},P)$ and an \reftext{def:inhomogeneous-poisson-process-2026b}{inhomogeneous Poisson process} $N=(N_t)_{t\ge0}$ with intensity $\lambda$ on it. In particular, for every real $\theta\ge0$ there is a homogeneous Poisson process with rate $\theta$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-27T20:09:42.716294Z
Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process
definitiondef:inhomogeneous-poisson-process-2026bProbability
Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space} and $\mathbb{R}$ the set of \reftext{def:real-numbers-c54-2026c}{real numbers}. A \textbf{stochastic process} on $[0,\infty)$ is a family $X=(X_t)_{t\ge0}$ of random variables on $(\Omega,\mathcal{F},P)$ indexed by the nonnegative real numbers. The process $X$ has \textbf{independent increments} if for all real $0\le t_0<t_1<\dots<t_r$ the random variables $X_{t_1}-X_{t_0},\dots,X_{t_r}-X_{t_{r-1}}$ are \reftext{def:independence-events-rvs-2026a}{independent}. (Differences of random variables are random variables: $\{X_t-X_s>u\}=\bigcup_{q\in\mathbb{Q}}(\{X_t>q\}\cap\{X_s<q-u\})$ , using \reftext{thm:density-q-rudin-b}{density of the rationals} and the generator criterion of \ref{def:measurable-function-2026a}.) An \textbf{intensity function} is a nonnegative function $\lambda:[0,\infty)\to\mathbb{R}$ that is \reftext{def:continuity-closed-interval-c54-2026b}{continuous} on every closed interval $[0,T]$ . Its \textbf{mean function} is $\Lambda(t)=\int_0^t\lambda(s)\,ds\qquad(t\ge0),$ the Riemann integral, which exists by \ref{lem:continuous-implies-riemann-integrable-c54-2026b}, with $\Lambda(0)=0$ . By additivity on adjacent intervals (\ref{lem:riemann-integral-additivity-adjacent-intervals-c54-2026a}), $\Lambda(t)-\Lambda(s)=\int_s^t\lambda(u)\,du$ for $0\le s\le t$ , and this is nonnegative because every lower sum of a nonnegative function is nonnegative (\reftext{def:upper-lower-sums-partition-c54-2026a}{upper and lower sums}); hence $\Lambda$ is nondecreasing. A stochastic process $N=(N_t)_{t\ge0}$ on $(\Omega,\mathcal{F},P)$ is an \textbf{inhomogeneous Poisson process with intensity $\lambda$ } if: \textbf{1.} $N_0=0$ ; \textbf{2.} $N$ has independent increments; \textbf{3.} for all $0\le s<t$ , the increment $N_t-N_s$ has the \reftext{def:poisson-distribution-2026b}{Poisson distribution} with parameter $\Lambda(t)-\Lambda(s)$ , in the sense of \ref{def:distribution-cdf-random-variable-2026a}. If $\lambda$ is constant with value $\theta\ge0$ , then $\Lambda(t)=\theta t$ (the Riemann integral of a constant, directly from the \reftext{def:upper-lower-sums-partition-c54-2026a}{upper and lower sums}), and $N$ is called a \textbf{homogeneous Poisson process with rate $\theta$ }.
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Authors Claude-Fable-5, Aaron · Created 2026-07-27T20:09:39.443141Z
Poisson Distribution
definitiondef:poisson-distribution-2026bProbability
Let $\mathbb{N}$ be the set of \reftext{def:natural-numbers-2026a}{natural numbers} and write $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ for the set of \textbf{nonnegative integers}; let $\mathbb{R}$ be the set of \reftext{def:real-numbers-c54-2026c}{real numbers} and $\mathcal{B}(\mathbb{R})$ the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra}. We use the \reftext{def:factorial-natural-number-2026a}{factorial} $k!$ for $k\in\mathbb{N}$ , extended by the convention $0!=1$ (the cited definition covers only $k\ge1$ ), together with the convention $\mu^{0}=1$ . Fix a real number $\mu\ge0$ . The \textbf{Poisson distribution} with parameter $\mu$ is the function $P_\mu:\mathcal{B}(\mathbb{R})\to[0,1],\qquad P_\mu(B)=\exp(-\mu)\sum_{k\in\mathbb{N}_0,\ k\in B}\frac{\mu^{k}}{k!},$ with the \reftext{def:exponential-function-real-2026a}{exponential function}. The terms are nonnegative, so the sum over the countable index set $\mathbb{N}_0\cap B$ is well-defined independently of ordering as the supremum of its finite partial sums; for the full index set $\mathbb{N}_0$ this unordered sum agrees with the limit of the partial sums of the series $\sum_{k=0}^{\infty}\mu^{k}/k!$ , since those partial sums are nondecreasing and every finite subset of $\mathbb{N}_0$ is contained in an initial segment. $P_\mu$ is a probability \reftext{def:measure-measure-space-2026a}{measure} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ . Countable additivity: if $B_1,B_2,\dots$ are pairwise disjoint Borel sets with union $B$ , then every finite subset of $\mathbb{N}_0\cap B$ meets only finitely many of the $B_j$ , so the supremum of the finite partial sums over $\mathbb{N}_0\cap B$ equals the sum over $j$ of the suprema over the blocks $\mathbb{N}_0\cap B_j$ . Total mass: the sum $\sum_{k\in\mathbb{N}_0}\mu^{k}/k!$ is exactly the defining series $\sum_{k=0}^{\infty}\mu^{k}/k!$ of $\exp(\mu)$ from \ref{def:exponential-function-real-2026a}, so $P_\mu(\mathbb{R})=\exp(-\mu)\exp(\mu)=\exp(0)=1$ by \ref{thm:exponential-properties-2026a}. In particular, for $\mu=0$ only the $k=0$ term is nonzero, so $P_0(B)=1$ if $0\in B$ and $P_0(B)=0$ otherwise; we call $P_0$ the \textbf{unit mass at $0$ }. A random variable has the \textbf{Poisson distribution with parameter $\mu$ } if its \reftext{def:distribution-cdf-random-variable-2026a}{distribution} equals $P_\mu$ ; such a variable lies in $\mathbb{N}_0$ with probability $1$ , since $P_\mu(\mathbb{R}\setminus\mathbb{N}_0)=0$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-27T20:09:27.724392Z
Agreement of the Riemann and Lebesgue Integrals for Continuous Functions on a Closed Interval
lemmalem:riemann-lebesgue-integral-agree-2026aAnalysis
Let $a<b$ be \reftext{def:real-numbers-c54-2026c}{real numbers} and let $h:[a,b]\to\mathbb{R}$ be \reftext{def:continuity-closed-interval-c54-2026b}{continuous on the closed interval} $[a,b]$ . Define the zero extension $\tilde{h}:\mathbb{R}\to\mathbb{R}$ by $\tilde{h}(x)=h(x)$ for $x\in[a,b]$ and $\tilde{h}(x)=0$ otherwise. Then the following hold. \textbf{Claim 1.} $h$ is \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integrable} on $[a,b]$ , by \ref{lem:continuous-implies-riemann-integrable-c54-2026b}. \textbf{Claim 2.} $\tilde{h}$ is \reftext{def:measurable-function-2026a}{measurable} with respect to the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra} and \reftext{def:lebesgue-integral-integrable-2026a}{integrable} with respect to \reftext{thm:lebesgue-measure-real-line-2026a}{Lebesgue measure} $\lambda$ . \textbf{Claim 3.} The two integrals agree: $\int_{\mathbb{R}}\tilde{h}\,d\lambda=\int_{a}^{b}h(x)\,dx,$ where the right-hand side is the Riemann integral of claim 1.
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Authors Claude-Fable-5, Aaron · Created 2026-07-27T19:26:07.074925Z
Moments and Stability of the Standard Normal Distribution
lemmalem:gaussian-stability-2026aProbability
Let $\mathbb{R}$ be the set of \reftext{def:real-numbers-c54-2026c}{real numbers} and $\mathbb{N}$ the set of \reftext{def:natural-numbers-2026a}{natural numbers}. \textbf{Claim 1.} Let $Z$ be a \reftext{def:standard-normal-distribution-2026a}{standard normal} random variable on a \reftext{def:probability-space-random-variable-2026a}{probability space}. Then $Z$ , $Z^{2}$ , and $|Z|^{3}$ are \reftext{def:lebesgue-integral-integrable-2026a}{integrable}, and the \reftext{def:expectation-variance-2026a}{expectation and variance} satisfy $\mathbb{E}[Z]=0,\qquad \mathbb{E}[Z^{2}]=\operatorname{Var}(Z)=1.$ \textbf{Claim 2.} If $Z_1$ and $Z_2$ are \reftext{def:independence-events-rvs-2026a}{independent} standard normal random variables on a common probability space and $a,b$ are positive real numbers with $a^{2}+b^{2}=1$ , then $aZ_1+bZ_2$ is a standard normal random variable. \textbf{Claim 3.} If $n\in\mathbb{N}$ with $n\ge 1$ and $Z_1,\dots,Z_n$ are independent standard normal random variables on a common probability space, then $(Z_1+\cdots+Z_n)/\sqrt{n}$ is a standard normal random variable, where $\sqrt{n}$ denotes the positive \reftext{thm:nonnegative-real-has-unique-square-root-2026a}{square root} of $n$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-23T04:24:17.483047Z
Existence of Independent Sequences with Prescribed Distributions
theoremthm:existence-independent-sequence-2026aProbability
Let $(\nu_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{sequence} of probability \reftext{def:measure-measure-space-2026a}{measures} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ , where $\mathbb{R}$ is the set of \reftext{def:real-numbers-c54-2026c}{real numbers}, $\mathcal{B}(\mathbb{R})$ is the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra}, and $\mathbb{N}$ is the set of \reftext{def:natural-numbers-2026a}{natural numbers}. Then there exist a \reftext{def:probability-space-random-variable-2026a}{probability space} $(\Omega,\mathcal{F},P)$ and a sequence $(X_m)_{m\in\mathbb{N}}$ of random variables on it that is \reftext{def:independence-events-rvs-2026a}{independent} and such that $X_m$ has \reftext{def:distribution-cdf-random-variable-2026a}{distribution} $\nu_m$ for every $m$ . One may take $\Omega=(0,1)$ , $\mathcal{F}=\{B\in\mathcal{B}(\mathbb{R}):B\subseteq(0,1)\}$ , and $P$ the restriction to $\mathcal{F}$ of \reftext{thm:lebesgue-measure-real-line-2026a}{Lebesgue measure}, as in \ref{thm:existence-iid-sequence-2026a}; taking all $\nu_m$ equal to a fixed probability measure $\nu$ recovers the statement of that theorem.
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Authors Claude-Fable-5, Aaron · Created 2026-07-23T04:24:11.066288Z
Joint Distribution, Expectations, and Block Independence for Independent Random Variables
theoremthm:independent-block-functions-2026aProbability
Throughout, $r$ is a \reftext{def:natural-numbers-2026a}{natural number} with $r\ge 1$ , $\mathbb{R}^r$ is \reftext{def:euclidean-space-rn-2026a}{Euclidean space}, $\mathbb{R}$ denotes the \reftext{def:real-numbers-c54-2026c}{real numbers}, and $\mathcal{B}(\mathbb{R})$ the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra}. Define the \textbf{ $r$ -fold product Borel $\sigma$ -algebra} $\mathcal{B}_r$ on $\mathbb{R}^r$ iteratively: $\mathcal{B}_1=\mathcal{B}(\mathbb{R})$ and, for $2\le k\le r$ , $\mathcal{B}_k=\mathcal{B}_{k-1}\otimes\mathcal{B}(\mathbb{R})$ , the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$ -algebra} on $\mathbb{R}^{k}$ , identifying the \reftext{def:cartesian-product-sets-2026a}{Cartesian product} $\mathbb{R}^{k-1}\times\mathbb{R}$ with $\mathbb{R}^{k}$ via $((x_1,\dots,x_{k-1}),x_k)\mapsto(x_1,\dots,x_k)$ . Given probability \reftext{def:measure-measure-space-2026a}{measures} $\nu_1,\dots,\nu_r$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ , define the probability measure $\nu_1\otimes\cdots\otimes\nu_r$ on $(\mathbb{R}^r,\mathcal{B}_r)$ iteratively by \ref{thm:product-measure-2026a} (probability measures are $\sigma$ -finite). Call a function $\varphi:\mathbb{R}^r\to\mathbb{R}$ \textbf{jointly Borel} if it is \reftext{def:measurable-function-2026a}{measurable} from $(\mathbb{R}^r,\mathcal{B}_r)$ to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ . Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space} and let $V_1,\dots,V_r$ be \reftext{def:independence-events-rvs-2026a}{independent} random variables on it with \reftext{def:distribution-cdf-random-variable-2026a}{distributions} $\nu_1,\dots,\nu_r$ . Then the following hold. \textbf{Claim 1.} The map $V:\Omega\to\mathbb{R}^r$ given by $V(\omega)=(V_1(\omega),\dots,V_r(\omega))$ is measurable from $(\Omega,\mathcal{F})$ to $(\mathbb{R}^r,\mathcal{B}_r)$ , and its distribution $P_V:\mathcal{B}_r\to[0,1]$ , $P_V(C)=P(V\in C)$ , is a probability measure equal to $\nu_1\otimes\cdots\otimes\nu_r$ . \textbf{Claim 2.} If $\varphi:\mathbb{R}^r\to\mathbb{R}$ is jointly Borel and either nonnegative or bounded, then $\varphi(V_1,\dots,V_r)=\varphi\circ V$ is a random variable and its \reftext{def:expectation-variance-2026a}{expectation} is defined (as an element of $[0,\infty]$ in the nonnegative case, and as a real number in the bounded case, where $\varphi\circ V$ is \reftext{def:lebesgue-integral-integrable-2026a}{integrable}), with $\mathbb{E}[\varphi(V_1,\dots,V_r)]=\int_{\mathbb{R}^r}\varphi\,d(\nu_1\otimes\cdots\otimes\nu_r).$ In particular this expectation depends only on $\varphi$ and the distributions $\nu_1,\dots,\nu_r$ . \textbf{Claim 3 (block independence).} Let $I=\{i_1<\dots<i_p\}$ and $J=\{j_1<\dots<j_q\}$ be disjoint nonempty subsets of $\{1,\dots,r\}$ , and let $\varphi:\mathbb{R}^p\to\mathbb{R}$ and $\psi:\mathbb{R}^q\to\mathbb{R}$ be jointly Borel. Then $\varphi(V_{i_1},\dots,V_{i_p})$ and $\psi(V_{j_1},\dots,V_{j_q})$ are independent random variables. \textbf{Claim 4.} Each coordinate projection $(x_1,\dots,x_r)\mapsto x_i$ and the addition map $(x_1,\dots,x_r)\mapsto x_1+\cdots+x_r$ are jointly Borel; moreover, if $\varphi:\mathbb{R}^r\to\mathbb{R}$ is jointly Borel and $t:\mathbb{R}\to\mathbb{R}$ is Borel measurable, then $t\circ\varphi$ is jointly Borel.
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Authors Claude-Fable-5, Aaron · Created 2026-07-23T04:24:04.478830Z
Smooth Test Function Criterion for Convergence in Distribution
theoremthm:smooth-test-convergence-distribution-2026aAnalysisProbability
Let $(X_m)_{m\in\mathbb{N}}$ and $X$ be \reftext{def:probability-space-random-variable-2026a}{random variables}, not necessarily on a common probability space, and let $\mathbb{R}$ denote the \reftext{def:real-numbers-c54-2026c}{real numbers}. Call a function $f:\mathbb{R}\to\mathbb{R}$ an \textbf{admissible test function} if $f$ is bounded, $f$ is a \reftext{def:ck-map-euclidean-open-set-2026b}{ $C^3$ map} on $\mathbb{R}=\mathbb{R}^1$ , and its first, second, and third derivatives are bounded. Then the following hold. For every admissible test function $f$ and every random variable $Y$ , the composition $f\circ Y$ is a random variable with finite \reftext{def:expectation-variance-2026a}{expectation}; in particular $\mathbb{E}[f(X_m)]$ and $\mathbb{E}[f(X)]$ are defined real numbers. If $\mathbb{E}[f(X_m)]\longrightarrow\mathbb{E}[f(X)]\qquad(m\to\infty)$ for every admissible test function $f$ , then $X_m\to X$ in distribution, in the sense of \ref{def:convergence-modes-2026a}.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T23:11:55.606597Z
Taylor Expansion with Third-Order Remainder Bound
lemmalem:taylor-third-order-remainder-2026aAnalysis
Let $\mathbb{R}$ denote the \reftext{def:real-numbers-c54-2026c}{real numbers} and let $f:\mathbb{R}\to\mathbb{R}$ be a \reftext{def:ck-map-euclidean-open-set-2026b}{ $C^3$ map} on $\mathbb{R}=\mathbb{R}^1$ , and suppose its third derivative is bounded: there is $M_3\ge 0$ with $|f'''(x)|\le M_3$ for all $x\in\mathbb{R}$ , where $f'$ , $f''$ , $f'''$ denote the iterated one-dimensional \reftext{def:derivative-interior-point-c54-2026b}{derivatives}. Then for all $x,h\in\mathbb{R}$ , $\Bigl|f(x+h)-f(x)-f'(x)\,h-\tfrac{1}{2}f''(x)\,h^{2}\Bigr|\ \le\ \frac{M_3\,|h|^{3}}{6}.$
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T23:11:49.681071Z
Change of Variables for Expectations
lemmalem:expectation-change-of-variables-2026aProbability
Let $X$ be a \reftext{def:probability-space-random-variable-2026a}{random variable} on a probability space $(\Omega,\mathcal{F},P)$ with \reftext{def:distribution-cdf-random-variable-2026a}{distribution} $P_X$ , let $\mathbb{R}$ denote the \reftext{def:real-numbers-c54-2026c}{real numbers}, and let $\varphi:\mathbb{R}\to\mathbb{R}$ be \reftext{def:measurable-function-2026a}{measurable} with respect to the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra} on both sides. Then $\varphi\circ X$ is a random variable (preimages compose), and, with the \reftext{def:expectation-variance-2026a}{expectation} notation $\mathbb{E}$ : if $\varphi\ge 0$ , then, with integrals as in \ref{def:lebesgue-integral-nonnegative-2026a}, $\mathbb{E}[\varphi(X)]=\int_\Omega \varphi\circ X\,dP=\int_{\mathbb{R}}\varphi\,dP_X\qquad\text{in }[0,\infty];$ in general, $\varphi\circ X$ is \reftext{def:lebesgue-integral-integrable-2026a}{integrable} with respect to $P$ if and only if $\varphi$ is integrable with respect to $P_X$ , and in that case the displayed identity holds in $\mathbb{R}$ . In particular, \reftext{def:expectation-variance-2026a}{expectations, moments, and variances} of random variables depend only on their distributions, and identically distributed random variables share them.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T23:11:44.979348Z
Central Limit Theorem
theoremthm:central-limit-theorem-2026aProbability
Let $(X_m)_{m\in\mathbb{N}}$ be a \reftext{def:independence-events-rvs-2026a}{sequence of independent and identically distributed random variables} on a probability space $(\Omega,\mathcal{F},P)$ such that $X_1$ and $X_1^{2}$ have finite \reftext{def:expectation-variance-2026a}{expectation}, and suppose $\sigma^{2}=\operatorname{Var}(X_1)>0$ ; write $\mu=\mathbb{E}[X_1]$ , $\sigma$ for the positive square root of $\sigma^{2}$ (\ref{thm:nonnegative-real-has-unique-square-root-2026a}), and $S_n=X_1+\cdots+X_n$ . Then $\frac{S_n-n\mu}{\sigma\sqrt{n}}\ \longrightarrow\ Z\qquad(n\to\infty)\quad\text{in distribution},$ where $Z$ is a \reftext{def:standard-normal-distribution-2026a}{standard normal} random variable; explicitly, in the sense of \ref{def:convergence-modes-2026a}, $P\Bigl(\frac{S_n-n\mu}{\sigma\sqrt{n}}\le t\Bigr)\ \longrightarrow\ \Phi(t)\qquad\text{for every }t\in\mathbb{R},$ the convergence holding at every $t$ because $\Phi$ is continuous everywhere by claim 3 of \ref{thm:gaussian-integral-2026a}.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T05:52:44.151458Z
The Gaussian Weight Defines a Probability Distribution
theoremthm:gaussian-integral-2026aAnalysisProbability
Let $g(x)=\exp(-x^{2}/2)$ with the \reftext{def:exponential-function-real-2026a}{exponential function}, let $\lambda$ be \reftext{thm:lebesgue-measure-real-line-2026a}{Lebesgue measure}, and for a \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel set} $B$ let $\nu(B)=\int_{\mathbb{R}}\mathbf{1}_{B}\,g\,d\lambda,$ the integral of \ref{def:lebesgue-integral-nonnegative-2026a} of the \reftext{def:measurable-function-2026a}{measurable} function $\mathbf{1}_B\,g$ . Then: $\nu$ is a \reftext{def:measure-measure-space-2026a}{measure} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (countable additivity follows from \ref{thm:monotone-convergence-2026a} applied to the partial sums of indicators); the total mass $c=\nu(\mathbb{R})=\int_{\mathbb{R}}g\,d\lambda$ is a finite positive real number; in particular the normalization $N=\nu/c$ of \ref{def:standard-normal-distribution-2026a} is a probability measure; the function $t\mapsto\nu\bigl((-\infty,t]\bigr)$ is \reftext{def:continuous-at-point-c54-2026b}{continuous} at every point of $\mathbb{R}$ (single points have $\nu$ -mass $0$ ), so the cumulative distribution function $\Phi$ of the standard normal distribution is continuous on all of $\mathbb{R}$ ; with $\pi$ defined as the value $(\lambda\otimes\lambda)(D)$ of the \reftext{thm:product-measure-2026a}{product measure} on the closed unit disk $D=\{(x,y)\in\mathbb{R}^2:x^{2}+y^{2}\le 1\}$ (a Borel subset of the plane for the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$ -algebra}), the total mass satisfies $c^{2}=2\pi,$ by \ref{thm:tonelli-fubini-2026a} applied to $(x,y)\mapsto g(x)g(y)$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T05:52:30.586666Z
Standard Normal Distribution
definitiondef:standard-normal-distribution-2026aProbability
Define $g:\mathbb{R}\to\mathbb{R}$ by $g(x)=\exp(-x^{2}/2)$ , with the \reftext{def:exponential-function-real-2026a}{exponential function}; $g$ is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous} (a composition of continuous maps, by \ref{thm:composition-continuous-euclidean-2026a} and claim 3 of \ref{thm:exponential-properties-2026a}), hence \reftext{def:measurable-function-2026a}{measurable} with respect to the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra} by the generator criterion there (preimages of open sets are open). By claim 2 of \ref{thm:gaussian-integral-2026a}, the quantity $c=\int_{\mathbb{R}}g\,d\lambda$ (the integral of \ref{def:lebesgue-integral-nonnegative-2026a} with respect to \reftext{thm:lebesgue-measure-real-line-2026a}{Lebesgue measure} $\lambda$ ) is a finite positive real number. The \textbf{standard normal distribution} is the function $N:\mathcal{B}(\mathbb{R})\to[0,1],\qquad N(B)=\frac{1}{c}\int_{\mathbb{R}}\mathbf{1}_{B}\,g\,d\lambda,$ with $\mathbf{1}_B$ the indicator of \ref{def:simple-function-integral-2026a}. By claims 1 and 2 of \ref{thm:gaussian-integral-2026a}, $N$ is a probability \reftext{def:measure-measure-space-2026a}{measure} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ . Its \reftext{def:distribution-cdf-random-variable-2026a}{cumulative distribution function} is denoted $\Phi(t)=N\bigl((-\infty,t]\bigr),$ and $\Phi$ is continuous at every point (claim 3 of \ref{thm:gaussian-integral-2026a}). A random variable is called \textbf{standard normal} (or \textbf{standard Gaussian}) if its distribution is $N$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T05:51:58.337064Z
Strong Law of Large Numbers under a Fourth Moment Bound
theoremthm:strong-law-large-numbers-fourth-moment-2026aProbability
Let $(X_m)_{m\in\mathbb{N}}$ be a \reftext{def:independence-events-rvs-2026a}{sequence of independent and identically distributed random variables} on a probability space $(\Omega,\mathcal{F},P)$ such that $X_1^{4}$ has finite \reftext{def:expectation-variance-2026a}{expectation} (hence so do $X_1$ , $X_1^{2}$ , and $X_1^{3}$ , since $|x|^{k}\le 1+x^{4}$ for $k\in\{1,2,3\}$ and every real $x$ , and monotonicity applies by \ref{thm:linearity-monotonicity-integral-2026a}). Write $\mu=\mathbb{E}[X_1]$ and $S_n=X_1+\cdots+X_n$ for $n\in\mathbb{N}$ . Then $\frac{S_n}{n}\longrightarrow\mu\quad\text{almost surely}\qquad(n\to\infty),$ in the sense of \ref{def:convergence-modes-2026a}. The proof rests on the fourth-moment bound $\mathbb{E}\bigl[(S_n-n\mu)^{4}\bigr]\le C n^{2}$ for a constant $C$ depending only on the distribution of $X_1$ , together with \ref{lem:markov-chebyshev-2026a} applied to $(S_n-n\mu)^{4}$ and the first \reftext{lem:borel-cantelli-2026a}{Borel–Cantelli lemma}.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T05:11:18.499961Z
Weak Law of Large Numbers
theoremthm:weak-law-large-numbers-2026aProbability
Let $(X_m)_{m\in\mathbb{N}}$ be a \reftext{def:independence-events-rvs-2026a}{sequence of independent and identically distributed random variables} on a probability space $(\Omega,\mathcal{F},P)$ such that $X_1$ and $X_1^{2}$ have finite \reftext{def:expectation-variance-2026a}{expectation}, and write $\mu=\mathbb{E}[X_1]$ . For $n\in\mathbb{N}$ let $S_n=X_1+\cdots+X_n.$ Then $\frac{S_n}{n}\longrightarrow\mu\quad\text{in probability}\qquad(n\to\infty),$ in the sense of \ref{def:convergence-modes-2026a}. Explicitly, for every $\varepsilon>0$ and every $n$ , $P\Bigl(\Bigl|\frac{S_n}{n}-\mu\Bigr|\ge\varepsilon\Bigr)\ \le\ \frac{\operatorname{Var}(X_1)}{n\,\varepsilon^{2}},$ which tends to $0$ as $n\to\infty$ ; the bound combines \ref{lem:markov-chebyshev-2026a} with the additivity of the variance over independent summands from \ref{lem:expectation-product-independent-2026a}.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T05:10:58.131306Z
Expectation of a Product of Independent Random Variables
lemmalem:expectation-product-independent-2026aProbability
Let $X$ and $Y$ be \reftext{def:independence-events-rvs-2026a}{independent} \reftext{def:probability-space-random-variable-2026a}{random variables} on a probability space $(\Omega,\mathcal{F},P)$ , each with finite \reftext{def:expectation-variance-2026a}{expectation}. Then the product $XY$ (a random variable, since $XY=\tfrac{1}{4}\bigl((X+Y)^2-(X-Y)^2\bigr)$ and sums, differences, and squares of random variables are random variables by Step 0(a) of the proof of \ref{thm:linearity-monotonicity-integral-2026a} and the power argument of \ref{def:expectation-variance-2026a}) has finite expectation, and $\mathbb{E}[XY]=\mathbb{E}[X]\,\mathbb{E}[Y].$ Consequently, if $X_1,\dots,X_r$ are independent random variables each having finite expectation and finite second moment, then for $i\ne j$ , $\mathbb{E}\bigl[(X_i-\mathbb{E}[X_i])(X_j-\mathbb{E}[X_j])\bigr]=0,$ and the \reftext{def:expectation-variance-2026a}{variance} is additive over independent summands: $\operatorname{Var}(X_1+\cdots+X_r)=\operatorname{Var}(X_1)+\cdots+\operatorname{Var}(X_r).$
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T05:10:53.017879Z

Showing 1-20 of 315

Pythagorean Theorem in Euclidean Space

Difference, Dot Product, and Orthogonality in Rn\mathbb{R}^nRn

Time Change of the Homogeneous Poisson Process

Kolmogorov Forward Equations for the Inhomogeneous Poisson Process

Existence of the Inhomogeneous Poisson Process

Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process

Poisson Distribution

Agreement of the Riemann and Lebesgue Integrals for Continuous Functions on a Closed Interval

Moments and Stability of the Standard Normal Distribution

Existence of Independent Sequences with Prescribed Distributions

Joint Distribution, Expectations, and Block Independence for Independent Random Variables

Smooth Test Function Criterion for Convergence in Distribution

Taylor Expansion with Third-Order Remainder Bound

Change of Variables for Expectations

Central Limit Theorem

The Gaussian Weight Defines a Probability Distribution

Standard Normal Distribution

Strong Law of Large Numbers under a Fourth Moment Bound

Weak Law of Large Numbers

Expectation of a Product of Independent Random Variables

Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$