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Let a constant input lie a fraction $f$ of the way between two consecutive quantizer levels separated by $\Delta$. With independent uniform additive dither spanning one step and nearest-level rounding, the two output codes occur with probabilities $1-f$ and $f$. Their mean equals the input and their sample-average variance is the displayed expression. This simple model shows both the sub-step information and the remaining noise cost.

Official documentation for stats::summary.lm. It defines the coefficient columns, residual variance estimate, degrees of freedom and coefficient of determination.

A symmetric stable distribution has this characteristic function, with $c>0$ and location $\mu$. At location zero it is a strictly stable distribution: weighted sums of independent copies have scale factor $({\sum }_j|a_j{|}^{\alpha }{)}^{1/\alpha }$. The index $\alpha$ need not permit a finite mean or variance. The scale in this convention is $c^{1/\alpha }$, so the coefficient of $|t{|}^{\alpha }$ and the scale parameter should not be confused.

For the random adjacent transposition shuffle on $S_n$ with $n\ge 2$, the position $f(\sigma )={\sigma }^{-1}(i)$ of one label is uniform on $\{1,\dots ,n\}$ in equilibrium, so ${\mathrm{Var}}_{\pi }(f)=(n^2-1)/12$. Each adjacent transposition moves the label with probability $2/n$, giving $\mathcal{E}(f,f)=(n-1)/n^2$. The Poincare inequality for a reversible Markov chain variational formula therefore gives $\gamma \le 12/[n^2(n+1)]$. In particular the continuous uniform distribution on $(0,1)$ has variance $1/12$, not $1/6$.

A bead in a liquid keeps undergoing Brownian motion even when an optical trap holds its average position fixed. For a stationary harmonic optical trap in thermal equilibrium at absolute temperature $T$, the equipartition theorem assigns mean potential energy $k_{B}T/2$ to this one quadratic coordinate. With $k_B$ the Boltzmann constant and $x=0$ at equilibrium,At the same temperature, quadrupling the spring constant divides the position variance by four and the standard deviation by two. The Boltzmann factor gives the complete equilibrium probability density function, a Gaussian distribution:The prefactor makes ${\int }_{-\infty }^{\infty }p(x)dx=1$; its second moment is ${\int }_{-\infty }^{\infty }{x}^{2}p(x)dx=k_{B}T/k$. The probability density function has units of inverse length, and its area gives probability. This density is exact for the ideal quadratic model. In a finite optical trap it approximates the occupied well when thermal excursions remain within the harmonic region and escape is negligible during observation. Driven motion, appreciable nonconservative forces or laser heating need not give this equilibrium density.

Here $\ell =\sqrt{k_{B}T/k_0}$, $q=x/\ell$ and $\kappa =k/k_0$. The plotted density per unit $q$ is $p_q(q)=\ell p(\ell q)=\sqrt{\kappa /(2\pi )}{e}^{-\kappa q^2/2}$. Each curve has total area one; their dimensionless standard deviations are $1$, $1/\sqrt{2}$ and $1/2$. The curves are exact model densities, not noisy experimental data.

Brownian motion becomes a measuring tool: a calibrated position measurement supplies ${\sigma }^2=⟨(x-⟨x⟩{)}^2⟩$, and the known temperature supplies $k=k_{B}T/{\sigma }^2$. Once $k$ is known, a slowly applied constant external force $f$ shifts the mean by $⟨x⟩=f/k$, so $f=k⟨x⟩$ in the same harmonic range. Fluctuations therefore calibrate the conversion from displacement to force. Svoboda and Block used this equipartition theorem method when calibrating their bead traps. ^[ref].

For the example stiffness $0.01pN/nm$ and $T=300\mathrm{K}$, $k_{B}T\approx 4.142\mathrm{pN}\mathrm{nm}$ and $\sigma \approx 20.35\mathrm{nm}$. Using a variance measured in uncalibrated detector units would not determine that physical stiffness. Measurement noise, finite exposure time, drift and insufficient sampling of the correlated Brownian motion can bias the estimate; record positions over many optical trap relaxation times and account for the detector response. The deterministic relaxation curve above describes relaxation without thermal forcing; an individual thermal trajectory continues to fluctuate.

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 $k_{B}T{L}^3/(192A)$.

The thermal diffusion length $\ell (t)=\sqrt{\alpha t}$ is the characteristic length over which thermal conduction changes a temperature distribution during time $t$ in the constant-thermal diffusivity model. It is a scale, not a sharp front or distance travelled at a constant speed: the heat equation has infinite propagation speed mathematically. Different definitions of penetration depth introduce numerical factors. In particular, the normalized one-dimensional heat kernel has spatial variance $2\alpha t$, so its root-mean-square width is $\sqrt{2\alpha t}$.

At $t=10\mathrm{s}$, the model materials have thermal diffusion lengths $4.47$ and $2.24\mathrm{mm}$. For $L=10\mathrm{mm}$ their thermal diffusion times are $50$ and $200\mathrm{s}$. These values describe the same heat equation as the thermal diffusion rate of a Fourier mode; the different numerical factors come from the definition of the scale.

For a single mode in thermal equilibrium, its quantum harmonic oscillator energy ladder gives a geometric distributionIts probability generating function is $F(z)=(1-q)/(1-qz)$. Differentiating twice gives $F^{\prime \prime }(1)=2q^2/(1-q{)}^2=2{\mu }^2$, hence $g^{(2)}(0)=2$. At the chosen mean $\mu =2$, $q=2/3$, $p_0=1/3$, $p_1=2/9$, $p_2=4/27$, and $\mathrm{Var}N=6$. Rare high occupations increase the pair count even though the mean matches the other preparations. The geometric weights also arise from a Gaussian mixture of coherent states.^[ref]

Single-mode selection is essential to this factor of two. If $M$ independent equally populated thermal modes are collected together, let $N={\sum }_{j=1}^M{N}_j$ with $⟨N_j⟩=\mu /M$. Adding their variances gives $\mathrm{Var}N=\mu +{\mu }^2/M$, hence $g^{(2)}(0)=1+1/M$. Averaging over many unresolved modes can make thermal pair counts approach the independent-count reference.

Model an ideal resistor of constant electrical resistance $R>0$ at uniform absolute temperature $T$ by its Thévenin equivalent: a zero-mean random voltage source in series with a noiseless $R$. In the classical range its one-sided, open-circuit voltage power spectral density, per hertz, iswhere $k_{\mathrm{B}}$ is the Boltzmann constant. Thus a positive-frequency interval of width $df$ contributes $4k_{\mathrm{B}}TRdf$ to the mean square voltage. This is approximately white noise only where the electrical resistance is effectively constant and $hf\ll k_{\mathrm{B}}T$, with $h$ the Planck constant. The ideal constant spectrum cannot describe arbitrarily high frequencies.^[ref]

For a real voltage, an equivalent two-sided power spectral density is $S_v^{(2)}(f)=2k_{\mathrm{B}}TR$ over signed frequencies. With a real linear time-invariant system of frequency response $H(f)$, the same output mean square isAngle brackets denote an expected value; for this zero-mean noise the mean square is its variance. Do not integrate $4k_{\mathrm{B}}TR$ over both signs of frequency: that counts the same fluctuations twice. All integrals here use frequency $f$ in hertz, not angular frequency $\omega =2\pi f$. The amplitude spectral density is $\sqrt{4k_{\mathrm{B}}TR}$ in $\mathrm{V}/\sqrt{\mathrm{Hz}}$, whereas the root mean square voltage requires a specified measurement signal bandwidth.

Suppose a probability density function and its third derivative are bounded. A square-integrable unit-integral kernel for density estimation with first two moments zero and ${\mu }_3=\int |u{|}^3|K(u)|du<\infty$ has bias of a kernel density estimator at most $\Vert f^{(3)}{\Vert }_{\infty }{\mu }_3{h}^3/6$, by Taylor theorem with Lagrange remainder and moment cancellation. Independence gives variance at most $\Vert f{\Vert }_{\infty }\Vert K{\Vert }_2^2/(nh)$. The Cauchy-Schwarz inequality bounds the expected centred absolute deviation by the square root of this variance. Adding the bias proves the inequality uniformly in $x$. The choice $h=n^{-1/7}$ gives expected absolute error $O(n^{-3/7})$.

Independent stochastic, approximately fixed-noise and fractional-response contributions add in variance. The relative resolution is $A/\sqrt{E}\oplus B/E\oplus C$. Signal-counting and sampling fluctuations enter $A$; fixed equivalent energy noise enters $B$; residual response nonuniformity enters $C$. The powers approximate a fitted energy range rather than describing every leakage or Pile-up effect exactly.

A time series is a sequence of observations indexed by time: hourly temperatures, daily sales, or successive measurements of a fluctuating signal. A probabilistic model treats these observations as one sample path of a stochastic process $(X_t)$. For equally spaced observations, take $t\in \mathbb{Z}$ and measure a lag of a time series in observation intervals. The order matters: shuffling a record preserves its observed values but destroys information about how one value follows another.

Two basic ways to model temporal dependence are an autoregressive model, which feeds past observations into the next value, and a moving-average model, which combines present and past white noise. An AR(1) carries a disturbance forward with diminishing weights; an MA(1) carries it into just two observations. Their autocorrelation functions make this difference visible. A weakly stationary process supplies a common mean and a lag-dependent autocovariance with which to describe that dependence.

These models help forecast future observations and quantify the uncertainty of averages. Successive measurements need not provide as much information as the same number of independent random variables: the long-run variance of a stationary process accounts for their accumulated covariances. Trend and seasonality require attention before fitting a stationary model; differencing and a seasonal difference operator can help, but do not automatically make every process stationary. See Peter Bartlett, Introduction to Time Series Analysis, lecture 1: objectives and models.

The three illustrated sample paths have zero mean and unit variance, so the visual comparison concerns dependence rather than scale. The AR(1) paths start from their stationary normal distribution, independently of future discrete Gaussian white noise; the MA(1) path includes the noise preceding its first observation. These are finite random examples, not exact pictures of an ACF. Compare their theoretical autocorrelation functions in autocorrelation.

Historically, Yule's 1927 study investigated disturbed periodicities in sunspot observations; it should not be read as the discovery of the particular AR(1) examples here. Slutsky studied how summing random inputs could produce wave-like records, a warning against inferring a periodic cause from appearance alone. The editorial note to his 1937 English paper identifies the 1927 Russian original. See G. Udny Yule, On a method of investigating periodicities in disturbed series, with special reference to Wolfer's sunspot numbers (1927) and Eugen Slutsky, The Summation of Random Causes as the Source of Cyclic Processes (1937), including the editorial note on the 1927 original.

For diffusion constant $D>0$, Gaussian increments have variance $2D\Delta t$. Their convolution gives a path weight proportional to $\exp [-\sum (\Delta x{)}^2/(4D\Delta t)]$ with measure factor $(4\pi D\Delta t{)}^{-M/2}$ for $M$ slices. The continuum kinetic action is notation for this limiting measure; typical Brownian paths have no ordinary derivative.

The total differential is the linear map that combines the first-order responses to simultaneous changes in all coordinates. For a differentiable real-valued function at $a$, it can be writtenHere $h$ is an input vector; $d{x}_j$ denotes the coordinate linear functional $h\mapsto h_j$. The coordinate formula and the Fréchet derivative describe the same linear map. The defining approximation isThe little o notation says the remaining approximation error becomes small relative to the input norm, across every direction of approach.^[ref]

For $f=x^2+xy+2y^2$ at $a=(1,1)$, $d{f}_a(h,k)=3h+5k$. A simultaneous change $(h,k)=(0.02,-0.01)$ gives the linear approximationThe exact algebra isConsequently the exact change is $0.0104$, and the approximation error is $0.0004$. If both input changes are halved, the linear prediction becomes $0.005$ and the error becomes $0.0001$: the first-order response halves, while the quadratic error falls by a factor of four. The right panel under directional derivative displays this comparison as the displacement is scaled.

For a quick check of differentiability, $|hk|\le (h^2+k^2)/2$ givesDividing by $\sqrt{h^2+k^2}$ makes this bound tend to zero. The same calculation proves that the tangent plane to the graph of a function at $(1,1,4)$ is $z=4+3(x-1)+5(y-1)$. It approximates the entire local surface, rather than only two coordinate slices.

The total differential is useful for estimating the combined effect of small input changes without reevaluating a complicated function. For example, if $|h|\le 0.02$ and $|k|\le 0.01$, then $|d{f}_a(h,k)|\le 0.11$. The exact remainder bound from the displayed polynomial is at most $0.0008$, so this example has $|\Delta f|\le 0.1108$ throughout that input rectangle. These are deterministic bounds; a variance calculation additionally needs a joint probability distribution and its covariances. A first-order estimate is also local: large changes require control of the remainder, not merely knowledge of the partial derivatives at one point.

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.

An optical pulse with constant group delay over its spectrum. At fixed spectral amplitude, its power-weighted temporal variance is minimal.

For independent Poisson process arrivals of geometric translational bursts with mean $b$, arrival rate $\lambda$, and unit protein removal rate $\alpha N$, the stationary mean is $\bar{N}=\lambda b/\alpha$. Starting with zero molecules, the expected value is $m(t)=\bar{N}(1-e^{-\alpha t})$. The variance satisfies $v^{\prime }=\lambda b(1+2b)+\alpha m-2\alpha v$. An integrating factor gives the displayed expression, and its stationary limit is $\bar{N}(1+b)$. Concentration fluctuations require a specified cell volume to convert concentrations into molecule counts.

For a dilute classical ideal gas whose atoms carry independent nondegenerate internal levels $0,ϵ$, the canonical partition function factorizes into translation and $(1+e^{-\beta ϵ}{)}^N$. The upper-level probability is $p=(1+e^{\beta ϵ}{)}^{-1}$. Taking two derivatives of its logarithm gives the title sum of independent variances. The internal contribution freezes exponentially at low temperature and tends to $N{ϵ}^2/4$ at high temperature. The translational classical result cannot be continued through quantum degeneracy without changing the model.

The expected number of proteins produced by one messenger RNA before its removal. With translation rate $r$ and exponential removal rate ${\alpha }_R$, $b=r/{\alpha }_R$. Independent Poisson translation during an exponential lifetime yields a geometric burst count on nonnegative integers, with variance $b(1+b)$. In the short-message-lifetime limit with independent unit protein removal, the protein Fano factor tends to $1+b$.

A nonconstant smooth periodic roll obeys $\mu ⟨T_0^{\prime 2}⟩-⟨T_0^{\prime \prime 2}⟩-⟨T_0^{\prime 4}⟩=0$. The orthogonal-roll perturbation raises the increasing-gradient functional for poorly conducting convection by the displayed amount. The variance is strictly positive: a nonconstant periodic function has a derivative that vanishes somewhere and is nonzero somewhere. The Hessian of $V$ therefore has a positive direction. Its self-adjoint linearized evolution operator has a positive Rayleigh quotient and hence a positive eigenvalue, proving transverse linear instability wherever a nonconstant roll exists.