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The absolute moment of order $p>0$ of a random variable $X$ is $\mathbb{E}[|X{|}^p]$ when this expectation is finite.

A Bayesian network consists of a Directed acyclic graph $G$ whose nodes are random variables, together with conditional distributions satisfying $p(x)={\prod }_{j}p(x_j\mid x_{{\mathrm{pa}}_G(j)})$. Its Directed acyclic graph encodes conditional independence constraints. For unrestricted binary variables it has ${\sum }_j{2}^{|{\mathrm{pa}}_G(j)|}$ free conditional probabilities. A Directed acyclic graph used this way need not have a causal interpretation.

For $r,s\in [0,1]$ and a random variable $U$ with the uniform distribution on $[0,1]$, set $B_r=1_{\{U\le r\}}$ and $B_s=1_{\{U\le s\}}$. Each has the corresponding Bernoulli distribution, and $\mathbb{P}(B_r\ne B_s)=|r-s|$, because disagreement occurs precisely between the two thresholds. This is a maximal coupling of the two laws.

A centered random variable $X$ with $\mathbb{E}[X^2]=\nu$ satisfies the Bernstein moment condition with parameter $b$ whenfor every integer $k\ge 2$. Summing its moment-generating series shows that it is sub-exponential with parameters $(2\nu ,2b)$.

A sequence of random variables $(X_n)$ is bounded in probability when ${\lim }_{K\to \infty }{\sup }_n\mathbb{P}(|X_n|>K)=0$. A sequence converging in probability to a finite random variable has this property, and its product with a sequence tending to zero in probability also tends to zero in probability.

The canonical distance is the L2 norm of an increment of a Gaussian process. It is a pseudometric, since distinct parameters may represent equal random variables almost surely. Its metric covering number appears in the Dudley entropy integral. For standard fractional Brownian motion, $d(s,t)=|s-t{|}^H$; a normalization with twice the covariance function multiplies this distance by $\sqrt{2}$.

For a finite-variance random variable, shift its centred version by $b>0$ and apply Markov inequality to the square. Minimizing $({\sigma }^2+b^2)/(a+b{)}^2$ at $b={\sigma }^2/a$ proves the bound. A two-point distribution attains equality. If the variance is zero, the centred variable vanishes almost surely.

A Cauchy random variable is a random variable having a Cauchy distribution. For a standard Cauchy random variable $V$,

The characteristic function of a real random variable $X$ isIt determines the probability distribution of $X$.

For square-integrable complex random variables, the Hermitian covariance is ${\mathrm{Cov}}_{\mathbb{C}}(Z,W)=\mathbb{E}[(Z-\mathbb{E}Z)\overline{(W-\mathbb{E}W)}]$. Its diagonal is nonnegative and equals $\mathbb{E}|Z-\mathbb{E}Z{|}^2$. The conjugation distinguishes it from the pseudo-covariance.

A concentration inequality bounds the probability that a random variable differs substantially from a typical value such as its expected value or median.

For a real random variable $Y$ and a sigma-algebra $\mathcal{G}$, the conditional characteristic function is $\mathbb{E}[e^{i\theta Y}\mid \mathcal{G}]$. If it equals the nonrandom characteristic function of a probability law for every $\theta$, then $Y$ has that law and is independent of $\mathcal{G}$. To verify the independence assertion, multiply the identity by any bounded $\mathcal{G}$-measurable test variable and use the uniqueness theorem for characteristic functions on the resulting finite measures. Equality for rational $\theta$ extends by continuity, so simultaneous null sets can be arranged when needed.

For an integrable random variable $X$, a sub-sigma-algebra $\mathcal{G}$, and a convex function $\varphi$ with suitable integrability, $\varphi (\mathbb{E}[X\mid \mathcal{G}])\le \mathbb{E}[\varphi (X)\mid \mathcal{G}]$ almost surely. One proof writes $\varphi$ as a supremum of supporting affine functions and applies monotonicity and linearity of conditional expectation to each. This proves the convex-loss form of the Rao-Blackwell theorem.

Let $A,B$ be indicator random variables with conditional independence given a random variable $T$, and suppose $\mathbb{E}[A\mid T]=\mathbb{E}[B\mid T]=g(T)$. Then the law of total expectation gives $\mathbb{E}[AB]=\mathbb{E}[g(T{)}^2]$ and $\mathbb{E}[A]=\mathbb{E}[B]=\mathbb{E}[g(T)]$. Thus their covariance is the variance of $g(T)$ and is nonnegative. The result explains correlated component evolutionary states in a coeval binary population without correlation between initial masses.

The cumulative distribution function of a real random variable $X$ isIt is nondecreasing, right-continuous, tends to zero at $-\infty$, and tends to one at $+\infty$.

A random variable supported on a finite or countable subset of the real line. Its law is specified by a probability mass function. Countable support need not consist of integers: a variable constantly equal to $3/2$ is discrete. Integer-indexed tail-sum formulas therefore require a separate integer-valued hypothesis.

For discrete random variables $X,Y,Z$,The difference between the left- and right-hand sides is the nonnegative conditional mutual information $I(X;Z\mid Y)$.

For a finite-valued random variable, an epsilon-sufficient set $S$ retains probability at least $1-ϵ$. This notion is about covering probability mass and is unrelated to a sufficient statistic. For an IID random variable source, any such set of words with fixed $ϵ<1$ requires exponentially many words at a rate approaching the Shannon entropy.

If $Y$ is a real square-integrable random variable and $X=\mathbb{E}[Y\mid \mathcal{G}]$, then $\mathbb{E}[XY]=\mathbb{E}{X}^2$ and the displayed identity follows by expansion. Thus equality of the second moments forces $X=Y$ as an almost sure equality. This is the equality case of the conditional Jensen inequality for the strictly convex function $z\mapsto z^2$.