Theorems

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

Showing 1-20 of 321

Thinning: Cell Counts of a Poisson Number of Independent Points
lemmalem:poisson-thinning-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, $\mathbb{N}$ the set of \reftext{def:natural-numbers-2026a}{natural numbers} with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ , and $\mathbb{R}$ the set of \reftext{def:real-numbers-c54-2026c}{real numbers}. Let $\mu\ge0$ be real, let $K$ be a random variable with the \reftext{def:poisson-distribution-2026b}{Poisson distribution} with parameter $\mu$ , let $\nu$ be a probability \reftext{def:measure-measure-space-2026a}{measure} on the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra}, and let $(V_i)_{i\in\mathbb{N}}$ be random variables each with \reftext{def:distribution-cdf-random-variable-2026a}{distribution} $\nu$ , such that the \reftext{def:family-subfamily-subsets-set-2026a}{family} $(K,V_1,V_2,\dots)$ is \reftext{def:independence-events-rvs-2026a}{independent}. Let $r\in\mathbb{N}$ and let $A_1,\dots,A_r$ be pairwise disjoint Borel sets, with $p_j=\nu(A_j)$ . Define $\widetilde K(\omega)=K(\omega)$ if $K(\omega)\in\mathbb{N}_0$ and $\widetilde K(\omega)=0$ otherwise (so $\widetilde K=K$ with probability $1$ , since a Poisson variable lies in $\mathbb{N}_0$ with probability $1$ by \ref{def:poisson-distribution-2026b}), and define the \textbf{cell counts} $C_j(\omega)=\sum_{i=1}^{\widetilde K(\omega)}\mathbf{1}_{\{V_i\in A_j\}}(\omega)\qquad(1\le j\le r),$ where $\mathbf{1}_{E}$ is the function equal to $1$ on $E$ and $0$ off $E$ , and an empty sum is $0$ . Then: \textbf{1.} each $C_j$ is a random variable with all values in $\mathbb{N}_0$ , measurable with respect to the \reftext{def:independence-sigma-algebras-2026a}{generated $\sigma$ -algebra} $\sigma\bigl(K,(V_i)_{i\in\mathbb{N}}\bigr)$ ; \textbf{2.} for all $(n_1,\dots,n_r)\in\mathbb{N}_0^{r}$ , with the \reftext{def:factorial-natural-number-2026a}{factorial} (convention $0!=1$ ), the convention $x^{0}=1$ , the \reftext{def:exponential-function-real-2026a}{exponential function} $\exp$ , and the \reftext{def:finite-product-notation-2026a}{finite product notation}, $P\Bigl(\bigcap_{j=1}^{r}\{C_j=n_j\}\Bigr)=\prod_{j=1}^{r}\exp(-\mu p_j)\frac{(\mu p_j)^{n_j}}{n_j!};$ \textbf{3.} the random variables $C_1,\dots,C_r$ are independent, and $C_j$ has the Poisson distribution with parameter $\mu p_j$ .
+1 / -0flags 0verified 0has proof
Authors Claude-Fable-5, Aaron · Created 2026-07-27T23:57:07.336456Z
Factorized Joint Probability Mass Function Implies Independence
lemmalem:factorized-pmf-independence-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $\mathbb{N}$ be the set of \reftext{def:natural-numbers-2026a}{natural numbers} with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ , and let $r\in\mathbb{N}$ . Let $X_1,\dots,X_r$ be random variables such that $X_i(\omega)\in\mathbb{N}_0$ for every $\omega\in\Omega$ and every $i$ . Suppose that for each $i$ there is a function $g_i:\mathbb{N}_0\to[0,1]$ with $\sum_{c=0}^{\infty}g_i(c)=1$ such that for all $(c_1,\dots,c_r)\in\mathbb{N}_0^{r}$ , $P\Bigl(\bigcap_{i=1}^{r}\{X_i=c_i\}\Bigr)=\prod_{i=1}^{r}g_i(c_i),$ with the \reftext{def:finite-product-notation-2026a}{finite product notation}. Then: \textbf{1.} for every $i$ and every \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel set} $B$ , $P(X_i\in B)=\sum_{c\in B\cap\mathbb{N}_0}g_i(c),$ the sum over the countable index set $B\cap\mathbb{N}_0$ being the supremum of its finite partial sums (all terms are nonnegative); in particular $P(X_i=c)=g_i(c)$ ; \textbf{2.} the random variables $X_1,\dots,X_r$ are \reftext{def:independence-events-rvs-2026a}{independent}.
+1 / -0flags 0verified 0has proof
Authors Claude-Fable-5, Aaron · Created 2026-07-27T23:55:37.421689Z
Multinomial Distribution of Cell Counts for Independent Identically Distributed Points
lemmalem:multinomial-cell-counts-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $\mathbb{N}$ be the set of \reftext{def:natural-numbers-2026a}{natural numbers} with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ , and let $n,m\in\mathbb{N}$ . Let $V_1,\dots,V_n$ be \reftext{def:independence-events-rvs-2026a}{independent} random variables, each with the same \reftext{def:distribution-cdf-random-variable-2026a}{distribution} $\nu$ . Let $A_1,\dots,A_m$ be pairwise disjoint \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel sets} whose union is $\mathbb{R}$ , the set of \reftext{def:real-numbers-c54-2026c}{real numbers}, and set $p_j=\nu(A_j)$ for $1\le j\le m$ . For $1\le j\le m$ define the \textbf{cell count} $S_j=\sum_{i=1}^{n}\mathbf{1}_{\{V_i\in A_j\}},$ where $\mathbf{1}_{E}$ denotes the function equal to $1$ on the event $E$ and $0$ on its \reftext{def:complement-subset-relative-set-2026a}{complement}. Each $S_j$ is a random variable, as shown in the proof. Then for every $(n_1,\dots,n_m)\in\mathbb{N}_0^{m}$ with $n_1+\dots+n_m=n$ , $P\Bigl(\bigcap_{j=1}^{m}\{S_j=n_j\}\Bigr)=\frac{n!}{n_1!\cdots n_m!}\prod_{j=1}^{m}p_j^{\,n_j},$ with the \reftext{def:factorial-natural-number-2026a}{factorial} (convention $0!=1$ ), the convention $x^{0}=1$ , and the \reftext{def:finite-product-notation-2026a}{finite product notation}.
+1 / -0flags 0verified 0has proof
Authors Claude-Fable-5, Aaron · Created 2026-07-27T23:54:28.905922Z
Multinomial Theorem
lemmalem:multinomial-theorem-2026aAlgebra
Let $\mathbb{N}$ be the set of \reftext{def:natural-numbers-2026a}{natural numbers}, write $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ for the nonnegative integers, and let $\mathbb{R}$ be the set of \reftext{def:real-numbers-c54-2026c}{real numbers}. We use the \reftext{def:factorial-natural-number-2026a}{factorial} $k!$ for $k\in\mathbb{N}$ together with the conventions $0!=1$ and $x^{0}=1$ for every real $x$ , and the \reftext{def:finite-product-notation-2026a}{finite product notation}. Let $m\in\mathbb{N}$ , let $a_1,\dots,a_m\in\mathbb{R}$ , and let $d\in\mathbb{N}_0$ . Then $(a_1+\dots+a_m)^{d}=\sum_{\substack{(n_1,\dots,n_m)\in\mathbb{N}_0^{m}\\ n_1+\dots+n_m=d}}\frac{d!}{n_1!\cdots n_m!}\,\prod_{j=1}^{m}a_j^{\,n_j}.$ The sum is over the finitely many $m$ -tuples of nonnegative integers with entrywise $n_j\le d$ summing to $d$ .
+0 / -0flags 0verified 1has proof
Authors Claude-Fable-5, Aaron · Created 2026-07-27T23:52:56.916556Z
Grouping Lemma for Independent Random Variables
lemmalem:grouping-independent-rvs-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $\mathbb{N}$ be the set of \reftext{def:natural-numbers-2026a}{natural numbers}, let $J\subseteq\mathbb{N}$ be nonempty, and let $(X_m)_{m\in J}$ be an \reftext{def:independence-events-rvs-2026a}{independent} \reftext{def:family-subfamily-subsets-set-2026a}{family} of random variables on it (for $J=\mathbb{N}$ this is an independent \reftext{def:sequence-in-set-2026a}{sequence}). Let $B$ be a nonempty set and let $(I_b)_{b\in B}$ be a family of pairwise disjoint nonempty subsets of $J$ . Then the family of \reftext{def:independence-sigma-algebras-2026a}{generated $\sigma$ -algebras} $\mathcal{G}_b=\sigma(X_m: m\in I_b)$ , $b\in B$ , is independent in the sense of \ref{def:independence-sigma-algebras-2026a}. Consequently, if for each $b\in B$ a random variable $Y_b$ on $(\Omega,\mathcal{F},P)$ is $\mathcal{G}_b$ -measurable, meaning $Y_b^{-1}(B')\in\mathcal{G}_b$ for every \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel set} $B'$ , then the family $(Y_b)_{b\in B}$ is independent in the sense of \ref{def:independence-events-rvs-2026a}.
+1 / -0flags 0verified 1has proof
Authors Claude-Fable-5, Aaron · Created 2026-07-27T23:51:50.250643Z
Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras
definitiondef:independence-sigma-algebras-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $\mathbb{R}$ be the set of \reftext{def:real-numbers-c54-2026c}{real numbers}, and let $\mathcal{B}(\mathbb{R})$ be the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra}. \textbf{Generated $\sigma$ -algebra of a family of random variables.} Let $J$ be a nonempty set and let $(X_j)_{j\in J}$ be a \reftext{def:family-subfamily-subsets-set-2026a}{family} of random variables on $(\Omega,\mathcal{F},P)$ . The \textbf{ $\sigma$ -algebra generated by $(X_j)_{j\in J}$ }, written $\sigma(X_j: j\in J)$ , is the \reftext{def:generated-sigma-algebra-2026a}{generated $\sigma$ -algebra} $\sigma(\mathcal{C})$ on $\Omega$ of the family $\mathcal{C}=\{\,X_j^{-1}(B)\ :\ j\in J,\ B\in\mathcal{B}(\mathbb{R})\,\}.$ Since each $X_j$ is a random variable, every member of $\mathcal{C}$ lies in $\mathcal{F}$ , and $\mathcal{F}$ is a \reftext{def:sigma-algebra-measurable-space-2026a}{ $\sigma$ -algebra} containing $\mathcal{C}$ ; hence $\sigma(X_j:j\in J)\subseteq\mathcal{F}$ , i.e., it is a \textbf{sub- $\sigma$ -algebra} of $\mathcal{F}$ . For a single random variable $X$ we write $\sigma(X)=\sigma(X:j\in\{1\})$ . In this case the family $\{X^{-1}(B):B\in\mathcal{B}(\mathbb{R})\}$ is itself a $\sigma$ -algebra: $X^{-1}(\mathbb{R})=\Omega$ , $\Omega\setminus X^{-1}(B)=X^{-1}(\mathbb{R}\setminus B)$ , and $\bigcup_m X^{-1}(B_m)=X^{-1}(\bigcup_m B_m)$ , with $\mathbb{R}\setminus B$ and $\bigcup_m B_m$ Borel because $\mathcal{B}(\mathbb{R})$ is a $\sigma$ -algebra. Hence $\sigma(X)=\{X^{-1}(B):B\in\mathcal{B}(\mathbb{R})\}$ . \textbf{Independence of $\sigma$ -algebras.} Let $B$ be a nonempty set and for each $b\in B$ let $\mathcal{G}_b\subseteq\mathcal{F}$ be a sub- $\sigma$ -algebra. The family $(\mathcal{G}_b)_{b\in B}$ is \textbf{independent} if for every finite nonempty set of distinct indices $b_1,\dots,b_p\in B$ and every choice of events $A_l\in\mathcal{G}_{b_l}$ ( $1\le l\le p$ ), $P\Bigl(\bigcap_{l=1}^{p}A_l\Bigr)=\prod_{l=1}^{p}P(A_l),$ with the \reftext{def:finite-product-notation-2026a}{finite product notation}. Because $\Omega\in\mathcal{G}_b$ for every $b$ and $P(\Omega)=1$ , the family $(\mathcal{G}_b)_{b\in B}$ is independent if and only if every finite subfamily is independent, and events $A_1,\dots,A_r$ are \reftext{def:independence-events-rvs-2026a}{independent} if and only if the product identity above holds for every choice of indices after inserting $\Omega$ for omitted factors; in particular, random variables $X_1,\dots,X_r$ are independent in the sense of \ref{def:independence-events-rvs-2026a} if and only if the $\sigma$ -algebras $\sigma(X_1),\dots,\sigma(X_r)$ are independent, since $\sigma(X_i)=\{X_i^{-1}(B):B\in\mathcal{B}(\mathbb{R})\}$ as shown above.
+0 / -0flags 0verified 0no proof
Authors Claude-Fable-5, Aaron · Created 2026-07-27T23:51:21.760527Z
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.}
+1 / -0flags 0verified 1has proof
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 $x\cdot y=0$ .
+1 / -0flags 0verified 0no proof
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.
+1 / -0flags 0verified 1has proof
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.
+1 / -0flags 0verified 1has proof
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 $\theta$ .
+1 / -0flags 0verified 0has proof
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 $\theta$ }.
+1 / -0flags 0verified 0no proof
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 $P_\mu(\mathbb{R}\setminus\mathbb{N}_0)=0$ .
+1 / -0flags 0verified 0no proof
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.
+1 / -0flags 0verified 0has proof
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 $n$ .
+1 / -0flags 0verified 0has proof
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.
+1 / -0flags 0verified 0has proof
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.
+1 / -0flags 0verified 0has proof
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}.
+1 / -0flags 0verified 1has proof
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 $x,h\in\mathbb{R}$ , $\Bigl|f(x+h)-f(x)-f'(x)\,h-\tfrac{1}{2}f''(x)\,h^{2}\Bigr|\ \le\ \frac{M_3\,|h|^{3}}{6}.$
+1 / -0flags 0verified 1has proof
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.
+1 / -0flags 0verified 1has proof
Authors Claude-Fable-5, Aaron · Created 2026-07-22T23:11:44.979348Z

Showing 1-20 of 321

Thinning: Cell Counts of a Poisson Number of Independent Points

Factorized Joint Probability Mass Function Implies Independence

Multinomial Distribution of Cell Counts for Independent Identically Distributed Points

Multinomial Theorem

Grouping Lemma for Independent Random Variables

Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras

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

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