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For a random walk with centered independent identically distributed increments of variance one, the maximum of its polygonal interpolation is the maximum at its grid vertices. The maximum functional is Lipschitz for the supremum norm, so the continuous mapping theorem applied to the Donsker invariance principle proves the displayed convergence in distribution. The Brownian reflection principle then gives upper-tail limits $2(1-\Phi (a))$ at all $a>0$, since the limiting maximum has no atom there.

The exact range is a separation beyond which the semivariogram reaches its sill. A Gaussian semivariogram with nonzero structured variance approaches its sill asymptotically and therefore has infinite exact range. Software range parameters may instead specify a scale.

For $n\ge 3$ independent labels with three type probabilities, conditioning the indicator of an ordered $(0,1,2)$ initial triple on the multinomial distribution type counts gives the displayed unbiased estimator. The Rao-Blackwell theorem reduces its variance strictly when all three probabilities are positive. At a boundary parameter with a missing type, both variances vanish.

If a set partition refines a rectangle $U\times V$ of a graph into rectangles $U_a\times V_b$, write $d=d(U,V)$ for its edge density of a bipartite graph. The weighted variance identity isIt follows by expanding the square and using ${\sum }_{a,b}|U_a||V_b|d(U_a,V_b)=|U||V|d$. In particular, equitable regularity energy cannot decrease under refinement. If a witness rectangle has relative sizes at least $\epsilon$ and edge density of a bipartite graph discrepancy greater than $\epsilon$, its contribution, after division by the squared total vertex count, exceeds ${\epsilon }^4|U||V|/N^2$.

Regression diagnostics check assumptions and influential observations in a fitted regression function. Residual-versus-fitted plots can reveal omitted mean structure or nonconstant variance; a quantile-quantile plot checks a specified error distribution. Regression leverage measures unusual predictor configurations, and Cook's distance combines leverage and residual size to measure coefficient sensitivity. Independence may require checking collection order and the study design in addition to residual plots.

A regression outlier has a response unusually far from its fitted conditional mean. A standardized regression residual adjusts for regression leverage; an externally studentized residual also estimates noise variance with the observation deleted. Selecting the largest residual among many observations changes its reference distribution. An outlier need not be an influential observation, and unusual valid data should not be discarded solely for being unusual.

For a correctly specified full-rank fixed-design Gaussian linear regression with $p<n$, dividing the residual sum of squares by the residual degrees of freedom $n-p$ gives an unbiased estimator of the noise variance. The projection removes the whole deterministic mean, so the numerator's expectation is $(n-p){\sigma }^2$. A misspecified subset model need not have this property.

Applying a measurable per-claim retention $g$ to independent claim sizes preserves the compound Poisson distribution form. The transformed severity is the pushforward measure of the original severity law. With count parameter $\lambda$, the retained aggregate has expected value $\lambda \mathbb{E}g(X)$ and variance $\lambda \mathbb{E}[g(X{)}^2]$. Zero retained marks may be removed by Poisson thinning.

If $S$ has exponential distribution with expected value $\mu$, the tail integral formula for moments gives $\mathbb{E}\min (S,M)=\mu (1-e^{-m})$ and $\mathrm{Var}(\min (S,M))={\mu }^2(1-2m{e}^{-m}-e^{-2m})$, where $m=M/\mu$. At matching retained expected value, the excess variance under quota share reinsurance is $2{\mu }^2{e}^{-m}(m-1+e^{-m})\ge 0$.

For $M>1$ completed-data estimates $Q_m$ with variances $U_m$, pool using $\overline{Q}=M^{-1}{\sum }_m{Q}_m$, $\overline{U}=M^{-1}{\sum }_m{U}_m$, $B=(M-1{)}^{-1}{\sum }_m(Q_m-\overline{Q}{)}^2$, and $T=\overline{U}+(1+M^{-1})B$. The total variance includes within-imputation uncertainty, between-imputation uncertainty and finite-$M$ simulation uncertainty.

For equal independent binomial samples, the displayed normal-approximation size per group targets a two-sided significance level $\alpha$ and power $1-\beta$ at specified alternative rates, with $\bar{p}=(p_1+p_0)/2$. Its first variance term sets the null critical value and its second the alternative distribution. A numerical sample size is not determined by an effect ratio alone; power, sidedness, allocation and baseline rates must also be specified. Rare-event discreteness and clustered designs can materially change the achieved power.

The second moment of a real random variable is $\mathbb{E}[X^2]$. Its variance is $\mathbb{E}[X^2]-(\mathbb{E}X{)}^2$ when these quantities are finite.

Let $B=0$ with probability $\pi$, and otherwise let $B$ have a Gamma distribution with mean one and variance $\tau$. Given $B$, counts $Y_j$ are conditionally independent Poisson random variables with means $B{\mu }_j$. For $q=1-\pi$, the law of total variance and law of total covariance giveEach component has a zero-inflated negative binomial distribution, but the components are not independent after marginalizing $B$. The zero component is shared by the whole profile. At $\tau =0$, interpret the positive Gamma component as a point mass at one.

For fixed $0<\lambda <1$ in $G(n,\lambda /n)$, the largest component of a graph has the displayed order with high probability. A dominating Galton-Watson process gives the tail bound $\mathbb{P}(T\ge j)\le e^{-(\lambda -1-\log \lambda )(j-1)}$ by the exponential Markov inequality. For the lower bound, the tree-component expectation in the Erdős-Rényi model diverges at $j=\lfloor (1-\eta )\log n/(\lambda -1-\log \lambda )\rfloor$. The disjoint-set covariance factor and the exclusion of overlaps give relative variance tending to zero, so the second moment method supplies a tree component of that order.

For a long potential approximately constant over the short-scale region, local-type primordial non-Gaussianity gives ${\phi }_s=(1+2f_{\mathrm{NL}}{\phi }_{G,L}){\phi }_{G,s}$, to first order in the long mode and ignoring short-mode squares. A linear cosmological Poisson equation therefore givesto the same order. The variance is conditional on the long mode.

The sill is the large-distance level of a semivariogram when that limit exists. For covariance tending to zero, the sill is the total marginal variance; a nugget plus a structured component has total sill ${\tau }^2+{\sigma }^2$. A nonzero constant covariance component is invisible to the semivariogram.

For known zero mean, invertible observation covariance $\Sigma$, target covariance vector $c$ and target variance $v_0$, simple kriging predicts $c^T{\Sigma }^{-1}z$ with error variance $v_0-c^T{\Sigma }^{-1}c$. Completing the covariance quadratic form proves optimality. A multivariate normal distribution makes this the exact conditional mean and variance.