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For a positive quadratic bending operator $K$ with a real orthonormal basis of eigenfunctions, the equipartition theorem gives $C(x,y)=k_{B}T{\sum }_n{W}_n(x)W_n(y)/{\mu }_n$, where $K{W}_n={\mu }_n{W}_n$. This is the inverse-operator Green function multiplied by Boltzmann constant and temperature. Unconstrained zero-energy filament modes prevent a normalizable canonical ensemble, and negative modes signal an unstable quadratic model.

For both ends clamped and $K=A{\partial }_x^4$, the diagonal is $C(x,x)=k_{B}T{x}^3(L-x{)}^3/(3A{L}^3)$. This differs from a clamped-free tip variance. The center variance is $k_{B}T{L}^3/(192A)$.

For positive claim sizes with finite second moment in a compound Poisson distribution aggregate of parameter $\lambda$, the two excess of loss reinsurance payouts are $\min (X,M)$ and $(X-M{)}_{+}$. The derivative of their total aggregate variance is $2\lambda (M\overline{F}(M)-\mathbb{E}[(X-M{)}_{+}])$. Thus the displayed condition characterizes stationarity. When the tail probability is positive it says that the retention equals the mean residual life. Global minimality requires an additional sign or comparison argument.

A treatment contrast is a linear contrast of cell means comparing treatment responses, with coefficients adding to zero. Pairwise differences and marginal factorial differences are examples. Its variance depends on the true allocation units and their covariance matrix.

For two centered predictors with squared norms $n$ and correlation coefficient $\rho$, the ordinary least squares estimate of either partial slope has variance ${\sigma }^2/[n(1-{\rho }^2)]$. Compared with the same predictor alone and the same error variance, the inflation factor is $1/(1-{\rho }^2)$. The eigenvalues of the slope Gram matrix are $n(1+\rho )$ and $n(1-\rho )$.

In a centered MA(1) process, $X_{T+2}$ is uncorrelated with every observation through time $T$. Its best linear predictor from that information is therefore zero, with error variance ${\sigma }^2(1+{\theta }^2)$. One-step innovation formulas using $X_{T+1}$ cannot be substituted when that observation is unavailable.

An unbiased likelihood estimator has expectation equal to the likelihood function at each parameter. A pseudo-marginal algorithm requires this estimator to be nonnegative. Importance sampling provides $\hat{L}=M^{-1}{\sum }_j\ell (F_j)p_{\theta }(F_j)/g_{\theta }(F_j)$ for independent random variables $F_j\sim g_{\theta }$, provided the proposal covers the target support. Relative variance affects the mixing time even though unbiasedness guarantees the correct marginal target.

A UMVCUE has minimum conditional variance among all estimators unbiased in the selected experiment for every parameter. For independent Gaussian stage estimates $A,V$ with information $I_1,J$, continuation $A\ge c$, total information $I_2=I_1+J$ and pooled estimate $T$, Rao-Blackwellization of the fresh estimate gives $T-(I_{1}s/J)\varphi ((T-c)/s)/\Phi ((T-c)/s)$, where $s^2=1/I_1-1/I_2$. The conditional family has complete sufficient statistic $T$, so the Lehmann–Scheffé theorem establishes the optimum. A conditional-likelihood bias correction does not automatically have this unbiasedness property.

For unknown drift $X\beta$, a full-column-rank design matrix $X$, a known positive-definite matrix observation covariance matrix $\Sigma$, and target drift row $x_0^T$, universal kriging minimizes prediction variance subject to $X^{T}w=x_0$. Equivalently, with $\hat{\beta }=(X^T{\Sigma }^{-1}X{)}^{-1}{X}^T{\Sigma }^{-1}z$, predict $x_0^T\hat{\beta }+c^T{\Sigma }^{-1}(z-X\hat{\beta })$. The first term is the estimated trend alone; the second is a correlated residual prediction.

For a normal distribution $N_n(\mu ,{\sigma }^{2}I)$ with unrestricted $\mu \in {\mathbb{R}}^n$ and unknown ${\sigma }^2$, an integrable data-only unbiased estimator of the mean-vector prediction risk of a fixed rank-$k$ orthogonal projection matrix exists exactly when $2k=n$. To prove necessity, randomize the mean by independent Gaussian variance $v$: conditioning adds $(n-k)v$ to the squared bias term, whereas the marginal variance identity would add $kv$. When $2k=n$, the residual sum of squares itself is unbiased.

The raw same-observable time correlation $C(t,\tau )=⟨X(t)X(t+\tau )⟩$. For a stationary process, it equals $⟨X{⟩}^2+\mathrm{Cov}(X(t),X(t+\tau ))$. Its connected part is the autocovariance; unlike the normalized statistical autocorrelation, no division by the variance is imposed. The distinction matters for fluctuations around a nonzero equilibrium point.

For Gaussian initial conditions and the Einstein-de Sitter second-order standard perturbation theory density kernel, the unsmoothed density contrast obeysThere are three choices of the second-order field and two connected Wick contractions; the angular average of $F_2$ is $5/7+(2/7)/3=17/21$. Thus the reduced skewness is $6(17/21)=34/7$. Smoothing changes this result, and the unsmoothed calculation requires a finite variance.

A Poisson distribution of parameter $a$ conditioned to lie in $\{0,\dots ,C\}$. It is the occupancy law of a unit-resource loss network. Its mean is the carried load of an Erlang loss resource, $a[1-E(C,a)]$. Differentiating the mean with respect to $\log a$ gives its variance, which is positive for $a>0$ and $C\ge 1$; this proves strict monotonicity of the carried load.

For exponential distribution claims of mean $\mu$ and compound Poisson distribution count parameter $\lambda$, the total party variance under excess of loss reinsurance is $g(M)=2\lambda {\mu }^2(1-(M/\mu )e^{-M/\mu })$. Its derivative is $2\lambda e^{-M/\mu }(M-\mu )$, so $M=\mu$ is the unique global minimum, with value $2\lambda {\mu }^2(1-e^{-1})$.

For any aggregate with positive finite variance $v$, quota share reinsurance with retained fraction $\alpha$ has total party variance $v({\alpha }^2+(1-\alpha {)}^2)=v/2+2v(\alpha -1/2{)}^2$. Equal sharing uniquely minimizes this objective. The two party totals remain dependent; adding their individual variances is a different objective from the fixed variance of their sum.

The variance of an estimator is the variance of its sampling distribution under the parameter value $\theta$.

The variance of hypothetical means in the Bühlmann model is the between-risk variance of the conditional claim expected value. It creates the shared covariance $a$ between different years of the same risk and determines how much individual experience should influence the premium.