Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X,Y:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ be real-valued random variables such that $\mathbb E[X^2]<\infty$ and $\mathbb E[Y^2]<\infty$. Define \begin{align*} \operatorname{Cov}(X,Y)&:=\mathbb E\!\left[(X-\mathbb E[X])(Y-\mathbb E[Y])\right],\\ \operatorname{Var}(X)&:=\mathbb E\!\left[(X-\mathbb E[X])^2\right],\qquad \operatorname{Var}(Y):=\mathbb E\!\left[(Y-\mathbb E[Y])^2\right],\\ \sigma_X&:=\sqrt{\operatorname{Var}(X)},\qquad \sigma_Y:=\sqrt{\operatorname{Var}(Y)}. \end{align*} Then \begin{align*} |\operatorname{Cov}(X,Y)|\le \sigma_X\sigma_Y. \end{align*} If $\operatorname{Var}(X)>0$ and $\operatorname{Var}(Y)>0$, and if \begin{align*} \rho(X,Y):=\frac{\operatorname{Cov}(X,Y)}{\sigma_X\sigma_Y}, \end{align*}…

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $X: (\Omega,\mathcal F) \to (\mathbb R,\mathcal B(\mathbb R))$ be a real-valued [random variable](/page/Random%20Variable) such that $\mathbb E[X^2] < \infty$. Then $X \in L^1(\Omega,\mathcal F,\mathbb P)$, and \begin{align*} \mathbb E[|X|] \leq \sqrt{\mathbb E[X^2]}. \end{align*}

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ be a real-valued [random variable](/page/Random%20Variable) such that \begin{align*} \mathbb E[X^2]=\int_\Omega X(\omega)^2\,d\mathbb P(\omega)<\infty. \end{align*} Then $X$ is integrable, $\operatorname{Var}(X):=\mathbb E[(X-\mathbb E[X])^2]$ is finite, and \begin{align*} \operatorname{Var}(X)=\mathbb E[X^2]-(\mathbb E[X])^2. \end{align*}

Let $(X,\tau)$ be a [topological space](/page/Topological%20Space), and let $A \subset X$. Define the derived set of $A$ by \begin{align*} A' := \{x \in X : \text{for every } U \in \tau \text{ with } x \in U,\; U \cap (A \setminus \{x\}) \neq \varnothing\}. \end{align*} Then $A$ is closed in $X$ if and only if $A' \subset A$.

Let $(X,\tau)$ be a $T_1$ [topological space](/page/Topological%20Space), let $A \subset X$, and let $x \in X$ be a [limit point](/page/Limit%20Point) of $A$, meaning that every neighbourhood of $x$ intersects $A \setminus \{x\}$. Then for every neighbourhood $N \subset X$ of $x$, the set $A \cap N$ is infinite.

Let $(X,d)$ be a [metric space](/page/Metric%20Space), let $\tau_d$ be the topology on $X$ induced by $d$, and let $A \subset X$. For $x \in X$, the following are equivalent: 1. $x$ is a [limit point](/page/Limit%20Point) of $A$ with respect to $\tau_d$; that is, every $\tau_d$-[open set](/page/Open%20Set) $U \subset X$ with $x \in U$ satisfies $(U \setminus \{x\}) \cap A \ne \varnothing$. 2. For every $r > 0$, \begin{align*} (B(x,r) \setminus \{x\}) \cap A \ne \varnothing, \end{align*} where $B(x,r) := \{y \in X : d(x,y) < r\}$.

Let $(X,d)$ be a [metric space](/page/Metric%20Space), let $A \subset X$, and let $x \in X$. Then $x$ is a [limit point](/page/Limit%20Point) of $A$ if and only if there exists a sequence $(a_n)_{n=1}^{\infty}$ with $a_n \in A \setminus \{x\}$ for every $n \in \mathbb{N}$ such that $(a_n)$ converges to $x$ in the metric $d$, that is, \begin{align*} \lim_{n \to \infty} d(a_n,x) = 0. \end{align*}

Let $X$ be a set equipped with the discrete topology $\tau = \mathcal{P}(X)$. For every subset $A \subset X$, the derived set $A'$ of $A$ is empty: \begin{align*} A' = \varnothing. \end{align*}

Let $G$ be a finite group, and let $\operatorname{Cl}(G)$ denote the set of conjugacy classes of $G$. Define the complex [vector space](/page/Vector%20Space) of class functions on $G$ by \begin{align*} \operatorname{CF}(G) := \{f: G \to \mathbb{C} \mid f(hgh^{-1}) = f(g) \text{ for all } g,h \in G\}. \end{align*} Then \begin{align*} \dim_{\mathbb{C}} \operatorname{CF}(G) = |\operatorname{Cl}(G)|. \end{align*}

Let $G$ be a group with operation written multiplicatively, and let $A \subset G$. For each $g \in G$, define the [conjugacy class](/page/Conjugacy%20Class) of $g$ in $G$ by \begin{align*} \operatorname{Cl}_G(g) := \{hgh^{-1} : h \in G\}. \end{align*} Then $A$ is conjugation-invariant, meaning that \begin{align*} \forall a \in A,\ \forall h \in G,\quad hah^{-1} \in A, \end{align*} if and only if $A$ is a union of conjugacy classes of $G$, meaning that there exists a subset $S \subset G$ such that \begin{align*} A = \bigcup_{s \in S} \operatorname{Cl}_G(s). \end{align*}