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For independent and identically distributed exponential random variables of rate $\lambda >0$, the sample mean has large-deviation speed $n$ and the displayed good rate function, infinite for $x\le 0$. The cumulant-generating function is $\log [\lambda /(\lambda -\theta )]$ for $\theta <\lambda$. Its Legendre-Fenchel transform has maximizing parameter $\theta =\lambda -1/x$. For $x\le 0$, sending $\theta \to -\infty$ gives infinite cost. The cost has its unique zero at $1/\lambda$ and diverges at both ends of its positive domain.

For sample means of Poisson random variables of mean $\lambda >0$, the cumulant-generating function is $\lambda (e^{\theta }-1)$. Its Legendre-Fenchel transform is the displayed good rate function, with $0\log 0=0$, and is infinite for $x<0$. Near its zero, $I_P(\lambda +h)=h^2/(2\lambda )+O(h^3)$. This local quadratic explains the Poisson moderate deviation principle.

If the Cramér–Lundberg ruin asymptotic has a finite positive constant, compare it with the leading nonzero term of a finite exponential representation. An adjustment coefficient smaller than that term's decay rate would give limit zero; a larger coefficient would give infinity. Thus the two rates agree and the prefactor is the leading coefficient. Terms with zero coefficient are absent and do not determine a decay rate.

Subtracting the uniform-flow pressure $-\mu Ux/k$ leaves $(\mu Ua/k)\alpha e^{-\xi }\cos \eta /(1+\alpha )$. The coordinate map has $r\sim a{e}^{\xi }/2$, so $e^{-\xi }\sim a/(2r)$, giving the displayed dipole coefficient. The one-half is essential. In the perfect-conductor limit the independent expression $p=-(\mu U/k)\mathrm{Re}\sqrt{z^2-a^2}$ has the same far-field expansion.

For $d{C}_t=a(b-C_t)dt+\sigma \sqrt{C_t}d{W}_t$, the Itô formula applied to $Z=\sqrt{C}$ gives the displayed dynamics before hitting zero. In particular $2ab={\sigma }^2$ does not eliminate the reciprocal drift. The cancellation condition is $4ab={\sigma }^2$. Under that condition $C$ can be represented as the square of a signed Ornstein-Uhlenbeck process with drift $-aY/2$ and volatility $\sigma /2$; its principal square root is the reflected process $|Y|$, not an unrestricted signed OU process. The mean of $C$ is $b+(C_0-b)e^{-at}$ for all admissible positive parameters.

The Pearson correlation coefficient is the covariance divided by the product of the standard deviations. For real random variables with finite, strictly positive variances,It is the expected value of the product of the two centered variables after dividing each by its standard deviation. The Cauchy-Schwarz inequality gives $-1\le {\rho }_{X,Y}\le 1$. This removes measurement units, while retaining the direction and strength of the linear relationship. Constant random variables have zero standard deviation, so their Pearson correlation coefficient is undefined.

For the covariance model $Y=2X+Z$, with $X,Z$ independently uniform on $\{-1,0,1\}$, the calculation givesThe positive correlation is less than one because the independent $Z$ term introduces extra variation. For $Y=-2X+Z$ the correlation is $-2/\sqrt{5}$. Both examples appear in the covariance diagram.

For nonzero real $a,c$, the correlation of $aX+b$ and $cY+d$ is $\mathrm{sgn}(ac){\rho }_{X,Y}$. Positive rescaling and added offsets therefore leave the correlation unchanged; reversing one scale reverses its sign. Equality $|{\rho }_{X,Y}|=1$ means that one random variable is an affine function of the other almost surely, with a nonzero slope of the corresponding sign. This is the equality condition in the Cauchy-Schwarz inequality, and includes perfect positive correlation.

Zero correlation says that the centered variables are orthogonal in their square-integrable inner product. It permits nonlinear statistical dependence, as the example $Y=X^2$ in uncorrelated random variables demonstrates. More generally, a correlation is a statement about a joint probability distribution; identifying effects of interventions is the separate task of causal inference. This distinction keeps an observed association from being mistaken for a demonstrated mechanism.

An early source for standardized association is Francis Galton's 1888 paper “Co-relations and their Measurement, chiefly from Anthropometric Data”. He compared stature and forearm measurements and changed each measurement to units of its own variability so the two regression directions could be compared. His scale used probable error, so it should not be identified word for word with the modern standard deviation formula above. ^[ref].

For a positive-definite covariance matrix, this is Euclidean distance after whitening. It accounts for both scales and correlations and is invariant under consistent nonsingular linear changes of coordinates. Estimating the covariance can be unstable with small samples, and a singular covariance requires restricting to a nonsingular subspace rather than claiming an ordinary metric on the full space.

This unit-variance covariance matrix has eigenvalue $1+(p-1)r$ on the constant-vector direction and eigenvalue $1-r$ on its orthogonal complement. It is positive semidefinite exactly when $-1/(p-1)\le r\le 1$. For positive $r$, the first population principal component is the normalized sum of the centered variables.

For a random vector with covariance $\Sigma$ and positive coordinate variances, let $D=\mathrm{diag}(\Sigma )$. Its correlation matrix is the covariance of $D^{-1/2}(X-\mathbb{E}X)$, so it is positive semidefinite with unit diagonal. Each entry is the correlation of the corresponding pair of variables. In particular $\mathrm{tr}(C)=p$, explaining why an eigenvalue in correlation-based principal component analysis represents that eigenvalue divided by $p$ of the total variance. The empirical version is the sample correlation matrix.

For a two-dimensional positive-definite covariance matrix $\Sigma$, the centered covariance ellipse is the image of the unit circle under its positive definite square root of a matrix. Equivalently, it satisfies $x^T{\Sigma }^{-1}x=1$. Its semi-major and semi-minor axes have lengths given by the square roots of the larger and smaller eigenvalues, along the corresponding eigenvectors. The ellipse encodes a second-moment geometry; a probability-coverage interpretation needs additional information about the probability distribution and the chosen scale.

In independent trials with two outcomes of probabilities $p$ and $q=1-p$, let $N$ include the first trial and the trial that first completes the pair. Conditional on the first outcome, $N-1$ has a geometric distribution of parameter $q$ or $p$, with mixture weights $p,q$. The law of total expectation and law of total variance giveThe final term in the unsimplified variance is the variance of the two conditional means; omitting it is incorrect unless $p=q$. Both outcomes must have positive probability for these finite formulas.

For independent coupon labels arriving at rate $\lambda$, Poisson thinning makes the type-specific first arrival times independent exponentials of rates $\lambda p_j$. Completion time is their maximum. Its distribution is $\mathbb{P}(T<t)={\prod }_j(1-e^{-\lambda p_{j}t})$. The completion count $L$ depends only on the labels and is independent of the exponential interarrival sequence. Therefore $T={\sum }_{k\ge 1}{E}_k{1}_{\{L\ge k\}}$ and nonnegative integration gives the displayed identity. Completion count and its expectation do not depend on arrival rate.

For a Poisson random measure on ${\mathbb{R}}^d$ with Lebesgue measure intensity and a nonnegative compactly supported continuous function $f$, put $m_r=\mu (B(0,r))$ and $a_r={\int }_{B(0,r)}{e}^{-f}d\mu$. Continuity gives $a_r/m_r\to e^{-f(0)}$. With $L=\mathbb{E}{e}^{-M(f)}$, the count-weighted Laplace functional of a Poisson random measure and independent inside/outside counts give respectivelyBoth converge to the displayed value. The conditioning is meaningful for every $r>0$; it is not conditioning on a positive-probability point at the origin.

If a Banach space $Y$ has cotype 2 constant $C$, every bounded linear operator $T:{\ell }_{\infty }^N\to Y$ satisfies ${\pi }_2(T)\le 3^{1/4}C{\pi }_4(T)$ and ${\pi }_2(T)\le 4\sqrt{3}{C}^2\Vert T\Vert$, uniformly in $N$. The first bound follows from Pietsch factorization theorem and the fourth moment bound for Rademacher sums. For the second, 2-summing domination gives a weighted coordinate $L^2$ norm; clipping coordinates at a variable threshold yields ${\pi }_4(T)\le 2\sqrt{\Vert T\Vert {\pi }_2(T)}$, which absorbs the first bound.

Dualize the Euler sequence on complex projective space and tensor by the hyperplane line bundle to obtain $0\to {\Omega }^1(1)\to {\mathcal{O}}^{n+1}\to \mathcal{O}(1)\to 0$. On global sections the last map sends $(c_i)$ to ${\sum }_i{c}_i{X}_i$, an isomorphism onto the linear forms. Left exactness of sheaf cohomology therefore gives the asserted vanishing. No higher-cohomology assumption is needed for this final kernel calculation.

On a Riemann surface, a meromorphic n-differential is a meromorphic section of the $n$th power of the canonical bundle. In a coordinate it is $F(z)(dz{)}^n$, with coefficients transforming by the $n$th power of the derivative. At a ramification coordinate $w=t^e$, its pullback has order $e{\mathrm{ord}}_w\omega +n(e-1)$. At a modular cusp, the exponential coordinate gives $dz=(h/(2\pi i))dq/q$, so a weight-$2n$ form of cusp order $v$ produces a differential of order $v-n$.

In conformal Newtonian gauge, a free scalar has no linear anisotropic stress. Eliminating its perturbation with the momentum constraint gives ${\Phi }_k^{\prime \prime }+3{\Phi }_k^{\prime }/\tau +k^2{\Phi }_k=0$. Substitution of ${\Phi }_k=(-\tau {)}^{-1}y(-k\tau )$ gives the order-one Bessel equation. The displayed Hankel function has positive incoming frequency $e^{-ik\tau }$ at early times.

A coordinate of a vector in a chosen basis is one entry of its coordinate vector. The corresponding coordinate functional in the dual basis extracts it. For an orthonormal basis, it can instead be read as an inner product with the basis vector, as in orthonormal eigenbasis. For a nonorthogonal basis, ordinary dot products need not give the coordinates.

A value of $g\ne 1$ is a convex lattice subgroup maximal among those omitting $g$. Zorn's lemma supplies a value: the union of a chain of convex lattice subgroups omitting $g$ still omits $g$. Such subgroups are prime. Their unique cover is the smallest convex lattice subgroup properly containing them.

A proper convex lattice subgroup $P$ is prime when $a,b\ge 1$ and $a\wedge b=1$ imply $a\in P$ or $b\in P$. Equivalently the ordered right cosets of $P$ form a totally ordered set. Prime refers to this order property, and does not require a normal subgroup or prime index.