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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.

At design sites $i/n$, let ${\tilde{Y}}_n$ be the step function taking response $Y_i$ on $((i-1)/n,i/n]$. Project it onto an interval-adapted wavelet basis through resolution $J$. Each coefficient is ${\hat{c}}_{\lambda }={\sum }_i{Y}_i{\int }_{I_i}{b}_{\lambda }(t)dt$. Its expectation is the corresponding coefficient of the step function of sampled regression means. For a Haar wavelet basis, dyadic sample size and resolution at most the sampling scale, this is exactly $n^{-1}{\sum }_i{Y}_i{b}_{\lambda }(i/n)$ with right-closed pointwise versions. Independent errors of variance ${\sigma }^2$ give coefficient variance at most ${\sigma }^2/n$: Cauchy-Schwarz inequality on each sampling cell yields ${\sum }_i({\int }_{I_i}{b}_{\lambda }{)}^2\le \Vert b_{\lambda }{\Vert }_2^2/n$.

Let $X_1,X_2,\dots$ be real independent and identically distributed random variables with $\mathbb{E}|X_1|<\infty$ and expected value $\mu =\mathbb{E}[X_1]$. The weak law of large numbers states that their sample mean has convergence in probability to $\mu$: for every fixed $\epsilon >0$,Thus a large sample is unlikely to give an average with a specified fixed-size error. The strong law of large numbers makes the stronger statement of almost sure convergence along an entire sequence. Under the displayed assumptions both laws apply; their names distinguish the conclusions, not two competing predictions.

When the variance ${\sigma }^2$ is finite, variance additivity for independent random variables gives $\mathrm{Var}({\overline{X}}_n)={\sigma }^2/n$. The Chebyshev inequality immediately supplies a useful quantitative form and a brief proof:For independent trials with a Bernoulli distribution of parameter $p=0.2$, the expected value is $0.2$ and the variance is $0.16$. At $n=1000$, the chance of an error of at least $0.1$ is therefore at most $0.016$. The exact binomial distribution gives a much smaller probability, as the figure under the strong law of large numbers shows. The bound is conservative, but it is valid for every sample size under these assumptions and does not require a normal approximation.

The weak law of large numbers addresses a fixed tolerance $\epsilon$. The central limit theorem instead resolves errors on the shrinking scale $\sigma /\sqrt{n}$ and gives their approximate normal distribution. In particular, the weak law of large numbers alone supplies neither a bell-shaped error law nor a universal rate of approach. For example, a Pareto distribution with minimum one and shape $3/2$ has expected value three but infinite variance: both the weak law of large numbers and strong law of large numbers apply, while the finite-variance statement of the central limit theorem does not.

A real stochastic process $(X_t{)}_{t\in \mathbb{Z}}$ is a weakly stationary process if it has finite second moments, a constant expected value $\mu$, and an autocovariance depending only on the lag of a time series:In particular its variance is the same at every time, $\mathrm{Var}(X_t)=\gamma (0)$. This is often called second-order stationarity. It makes observations from different parts of a record comparable through the same autocorrelation function. Bartlett uses this second-order meaning of stationarity in Introduction to Time Series Analysis, lecture 2: mean, covariance, and stationarity.

For a worked contrast, let ${\epsilon }_t$ be centered white noise of variance ${\sigma }^2>0$. The process $X_t=5+{\epsilon }_t$ has mean $5$, variance ${\sigma }^2$, and zero autocovariance at nonzero lags, so it is weakly stationary. The process $Y_t=t+{\epsilon }_t$ has exactly the same covariance structure, but its mean is $t$, so it is not weakly stationary. Constant variance alone is insufficient.

Weak stationarity need not imply strict stationarity. For example, choose independent random variables at all integer times, using a standard normal distribution at even times and a Rademacher distribution at odd times. Every coordinate has zero mean and unit variance, and distinct coordinates have zero covariance. Nevertheless a shift by one changes the marginal probability distribution from continuous to two-point. This also illustrates why weak white noise need not be identically distributed.

Let $X_k$ be independent and identically distributed random variables with expected value zero, variance one, and a moment-generating function finite in a neighborhood of zero. Each row of deterministic real weights $a_{n,k}$ is finite. If ${\max }_k|a_{n,k}|\to 0$ and ${\sum }_k{a}_{n,k}^2\to v<\infty$, then the weighted sums converge in distribution to a normal distribution with mean zero and variance $v$; $v=0$ denotes the constant zero.

By the quadratic expansion of a cumulant-generating function, write $K(u)=u^2/2+r(u)$ with $r(u)=o(u^2)$. For a fixed real $t$, put ${\eta }_n={\sup }_{0<|u|\le |t|{\max }_k|a_{n,k}|}|r(u)|/u^2$, with the supremum taken as zero if its range is empty. Then ${\eta }_n\to 0$ andThus the log moment-generating function of the sum tends to $v{t}^2/2$. The convergence theorem for moment-generating functions proves the conclusion. The supremum is the step that makes the remainder estimate uniform across the row; a separate little-o assertion for each term would not justify summing an increasing number of terms.

For observations of an Itô integral on a partition of an interval, the statistic ${\sum }_{i}g(t_{i-1})(X_{t_i}-X_{t_{i-1}}{)}^2$ estimates $\int g(t)\sigma (t{)}^{2}dt$. With a deterministic noise coefficient the squared increments are independent, and the fluctuation has variance $2{\sum }_{i}g(t_{i-1}{)}^2({\int }_{t_{i-1}}^{t_i}{\sigma }^2{)}^2$.

A heteroscedastic extension of independent-groups one-way analysis of variance, allowing unequal group variances. It does not make paired or clustered observations independent.

For two independent samples from approximately normal distributions, Welch's t-test compares their means without assuming equal unknown variances. The displayed test statistic uses sample variances $s_X^2,s_Y^2$. Under the equal-mean null hypothesis, its reference Student t-distribution has approximate statistical degrees of freedomA p-value from this approximation addresses the specified statistical error model; it cannot distinguish a material effect from an unmeasured experimenter bias when sample and experimenter are confounded.

For the AR(1) and MA(1) formulas here, white noise means a real sequence indexed by all integer times withThese are the weak white noise assumptions: zero mean, a common finite positive variance, and uncorrelated random variables at distinct times. Independence and a normal distribution are additional assumptions. Strong white noise requires independent and identically distributed random variables; discrete Gaussian white noise is the jointly Gaussian special case. Some definitions allow a constant nonzero mean; subtract it before using the centered formulas here.

For a concrete dependent example, let $A$ equal $1$ or $2$ with equal probabilities, independently of a sequence of independent Rademacher random variables $Z_t$, and put ${\epsilon }_t=A{Z}_t$. Its mean is zero, its variance is $\mathbb{E}[A^2]=5/2$, and its nonzero-lag autocovariances vanish. Nevertheless every $|{\epsilon }_t|$ equals the same $A$: seeing one magnitude determines all the others. This is white noise with dependence invisible to its ACF, and is also an example of the issue discussed in ergodic stationary process.

Generalized Gaussian white noise over a real Hilbert space $H=L^2$ is an isonormal Gaussian process $W(h)$ with covariance $⟨h,g{⟩}_H$. Its law $P_0$ is carried on a larger observation space; the identity is not the covariance of an infinite-dimensional $H$-valued Gaussian measure because it is not a trace-class operator. A deterministic signal $h\in H$ defines a translated observation law $P_h$ with Radon-Nikodym derivativeIn an orthonormal basis, $W_m(h)={\sum }_j{m}_j{h}_j$ is a stochastic series with total variance $\Vert h{\Vert }_H^2$, not an inner product of two $H$-valued observations. The formula follows from the Cameron-Martin theorem for a Gaussian measure in its generalized white-noise version, or from finite-dimensional Gaussian likelihood ratios. On ${\mathbb{R}}^2$, generalized noise can be realized on tempered distributions; an unweighted global negative Sobolev space is not automatically a suitable almost-sure support on an unbounded domain. The stochastic series avoids imposing that unsupported regularity.

A real Wigner matrix is a real symmetric matrix whose upper off-diagonal entries are independent centered copies of a fixed random variable, scaled to have variance of order $1/N$. Diagonal conventions may vary and should be specified separately. A common version has variance-one unscaled off-diagonal entries and either zero diagonal entries or independent diagonal entries with a bounded second moment.

For a continuous sample, the statistic also equals ${\sum }_i{1}_{\{X_i>0\}}+{\sum }_{i<j}{1}_{\{X_i+X_j>0\}}$. In each pair the sign of the sum is the sign of the observation with larger absolute value, proving the identity by counting its smaller-rank partners. Under a continuous symmetric null, the pair kernel has projection $F(x)-1/2$ and projection variance $1/12$. The U-statistic central limit theorem yields asymptotic normality of $W_{+}$, with variance asymptotic to $n^3/12$.

For independent mean-zero errors in a fixed-design regression, form $y_i^{\ast }=x_i^{\top }\hat{\beta }+{\hat{e}}_i{W}_i$, where the independent multipliers have mean zero and variance one, and refit. Rademacher multipliers taking the values $-1$ and $1$ with equal probability are a simple choice. Conditional on the data, the bootstrap errors have mean zero and preserve observation-specific residual variances, supporting heteroscedasticity rather than imposing a common residual distribution. Appropriate leverage corrections and regularity assumptions are required for accurate studentized inference.

Software for Bayesian inference through Markov chain Monte Carlo in directed statistical models. Its model language uses dnorm(mean, precision), so the second argument is inverse variance; dcat indexes a supplied probability vector, and dbern introduces a binary latent variable. The model is declarative, so the order of deterministic node definitions does not impose procedural evaluation order.

Consider an original idealized track reconstruction model. Let equally spaced wires have positions $x_k=kp$ and assign each event to the nearest wire, reporting $\hat{x}=x_k$. Assume exactly one correct wire address, perfect detection efficiency, exact wire positions and a uniform distribution of true positions within each cell. These assumptions explicitly exclude missed pulses, false addresses and several unresolved tracks.

Writing $u=x-x_k$, the measurement error is $e=\hat{x}-x=-u$, with $u$ uniform on $[-p/2,p/2]$. Symmetry gives zero expected value. Its variance and standard deviation follow by integration:For chosen wire pitches $p=2\mathrm{mm}$ and $1\mathrm{mm}$ this gives $0.577\mathrm{mm}$ and $0.289\mathrm{mm}$ respectively. These are model standard deviations, not measured chamber spatial resolutions. The largest possible absolute error in this model is $p/2$, which is a different quantity.

A useful subtlety is that the formula depends on the illumination. For an unresolved pencil beam fixed at $x=x_k+0.4p$, every reconstructed position is $x_k$. The repeated estimates have zero variance, but every error is $-0.4p$. Small repeat-to-repeat spread can coexist with a biased position. The uniform-cell result therefore cannot be transferred to an arbitrary beam profile or detector selection.

The wire address supplies one transverse coordinate in this model. Combining several separated measurements supports track reconstruction; a single address does not specify a complete trajectory. If two charged particles enter the same cell during one accepted interval, an address-only record can be identical to that from one particle. No reconstruction of that record can recover the lost multiplicity without additional information.

The drive can supply or extract work while the quantum state stays approximately fixed. For a differentiable bounded Hamiltonian operator, the Schrödinger equation givesThe state-evolution terms cancel, leaving the external drive's power. For a finite ideal step the change in mean energy is $⟨\psi |H_f-H_i|\psi ⟩$. Thus conservation of energy for a closed system with a time-independent Hamiltonian operator does not assert conservation of the driven subsystem's energy during switching.

Specify work operationally by two energy measurements: one before the switch and one after. For the prepared initial ground state, the first outcome is definite; the second samples the new Born rule probabilities. This protocol gives a distribution of measured energy differences, rather than generally the measurement distribution of the single operator $H_f-H_i$.^[ref]and henceFor ${\Delta }_i={\Delta }_f=\Delta$ and $\theta =\pi /3$, the work is $0$ or $\Delta$, with probabilities $3/4$ and $1/4$; its expectation is $\Delta /4$ and its variance is $3{\Delta }^2/16$. The initial mean energy $-\Delta /2$ becomes $-\Delta /4$. Using different initial and final scalar offsets would also shift the measured work; the displayed Hamiltonian operators fix that convention explicitly.

A working correlation matrix $R_i(\alpha )$ is a positive-definite proposed correlation structure used to weight a generalized estimating equation, with $A_i$ the diagonal matrix of marginal variances. It need not equal the actual correlation matrix. Correct marginal means and independent sampling clusters keep the estimating equations unbiased; a cluster-level sandwich covariance matrix estimates their actual large-sample covariance. For three exchangeable observations, positive definiteness requires $-1/2<\rho <1$.

For a causal autoregressive model $X_t={\sum }_{j=1}^p{\varphi }_j{X}_{t-j}+{ϵ}_t$ with centered white noise of variance ${\sigma }^2$, multiplying by past values and taking expectations yields $\gamma (h)={\sum }_j{\varphi }_j\gamma (h-j)$ for $h\ge 1$, and $\gamma (0)={\sum }_j{\varphi }_j\gamma (j)+{\sigma }^2$. These Yule-Walker equations determine the autocovariance from the coefficients and noise variance, or estimate coefficients from observed autocovariances. Causality supplies the needed orthogonality of the current noise to past observations.

A zero-energy mode lies in the kernel of the quadratic elastic filament operator. For pure bending, zero curvature makes the displacement affine. Free-free endpoints therefore leave translation and tilt modes, while slope-constrained, force-free endpoints leave only translation. Both clamped-clamped and hinged-hinged endpoints remove these modes. An unrestricted modal amplitude has constant Boltzmann distribution weight, so no normalizable canonical ensemble or finite displacement variance exists. Fixing the rigid degrees of freedom allows the equipartition theorem on the positive complement.

With structural-zero probability $\pi$, size $r>0$ and count mean $\lambda$, the model assigns $\pi +(1-\pi )(r/(r+\lambda ){)}^r$ to zero and $(1-\pi )f_{\mathrm{NB}}(y;r,\lambda )$ to positive counts. Its expectation is $(1-\pi )\lambda$ and its variance is $(1-\pi )(\lambda +{\lambda }^2/r)+\pi (1-\pi ){\lambda }^2$, by the law of total variance. A logarithmic link function can relate the count-component mean to predictors.