Since is an even function, is an odd function and its integral over the symmetric interval vanishes. Thus . Its variance is
Consequently
The standard deviation is approximately minute, or seconds.
Treat each observed arrival label as a constraint on an unobserved walking duration. Departures are treated as fixed design information. Writing for the minutes available until work starts, the derived late-label thresholds have frequencies ; the early-label thresholds have frequencies . These are censoring thresholds, not measured journey times. A late label means , and an early label means ; equality has probability zero under the continuous model.
The paper specifies no prior distribution. One explicit proper model is
with independent priors; all times are in minutes. These ranges are a transparent modelling choice, not information deduced from the observations. In particular, this is a prior on the standard deviation , not a uniform prior on the variance . The normal distribution permits negative durations; it is the mathematical approximation requested here, rather than a claim that such durations are physically possible. A truncated model would be a different model.
The following Python implements hard conditioning by rejection sampling. None marks a rejected trace; an accepted Boolean answers whether all five future arrivals are late. The same parameters are shared across past and future days.
import math
import random

LATE = {20: 1, 21: 3, 25: 3, 27: 1, 32: 2}
EARLY = {29: 1, 34: 1, 36: 3, 37: 1, 38: 1, 39: 1, 43: 1, 44: 1}
RECORDS = [(c, True) for c, n in LATE.items() for _ in range(n)]
RECORDS += [(c, False) for c, n in EARLY.items() for _ in range(n)]

def journey_trace(rng):
    mu = rng.uniform(0.0, 60.0)
    sigma = rng.uniform(1.0, 20.0)
    # Twenty latent past durations and five future durations.
    past = [rng.gauss(mu, sigma) for _ in RECORDS]
    future = [rng.gauss(mu, sigma) for _ in range(5)]
    if not all(t > c if late else t < c
               for t, (c, late) in zip(past, RECORDS)):
        return None
    return all(t > 30.0 for t in future)

def accepted_average(trace, n, rng):
    if n <= 0:
        raise ValueError("n must be positive")
    successes = accepted = 0
    while accepted < n:
        result = trace(rng)
        if result is not None:
            accepted += 1
            successes += int(result)
    return successes / n
For example, accepted_average(journey_trace, n, random.Random(2021)) estimates the desired conditional probability from accepted traces. Accepting False is essential: treating a false answer as a rejected trace would condition on the desired conclusion. The model deliberately emphasizes clear semantics; rejection can be inefficient because all twenty past-label constraints must hold together.
Multisensory signals lose synchrony through propagation, receptor and neural delays, and the temporal structure of the event itself. Perceived simultaneity is maintained by tolerance, calibration and inference about common causes, rather than by an infallible clock that makes every channel arrive together. Physical simultaneity, simultaneous receptor stimulation and subjective simultaneity are different quantities.
For an audiovisual event at distance , let auditory and visual propagation speeds be , and processing delays to a specified comparison stage be . A simple internal arrival-time difference, auditory minus visual, is
This follows by subtracting the two sums of travel time and processing delay. At 10 m, using m/s and m/s, the propagation term is about 29.15 ms; at 100 m it is about 291.54 ms. These are calculations under fixed conditions, not perceptual thresholds. Sound speed varies with the medium and temperature. A fixed neural correction could compensate one distance but cannot eliminate propagation differences at every distance.
The neural terms are not fixed constants either. Phototransduction, cochlear transduction, peripheral conduction, stimulus intensity, contrast and the pathway used to define a response can affect latency. A weak or ambiguous stimulus can require more evidence before a decision. Attention, expectations and the selected feature may affect reported timing, without requiring a change in the physical stimulus. A neural onset response, identification response and button press do not mark the same stage. Therefore a reported audiovisual delay should not be casually equated with a single axonal conduction time.
Some apparent desynchronization originates in the source. Lip and jaw movements can precede the acoustic release of speech. A seen hand movement can precede contact and its sound; food handling, oral touch, dissolved tastants and retronasal odor release have different physical time courses. Reverberation can add later auditory energy. Electronic capture, buffering and displays can introduce additional modality-specific delays. The useful goal is consequently to recover whether signals belong to one event and when that event occurred, not to force equal onsets for all its components.
One solution is a temporal binding window: a range of disparities within which signals can be grouped or judged as simultaneous. The window is measured through a specified task, not a universal duration. In a simultaneity judgment, report probability often has a peak around a point of subjective simultaneity. In a temporal order judgment, performance changes from one reported order to the other across offsets. Those measures can differ because their decisions and criteria differ. A broad simultaneity function can reflect uncertain timing or a liberal response criterion; it does not uniquely measure a long neural integration window.
Consider a Gaussian simultaneity decision model. Let physical auditory-minus-visual onset difference be , stable internal offset , and Gaussian timing noise of standard deviation . If the observer reports simultaneous when , then
The expression is the probability that the noisy difference falls between the two decision bounds. Its center is for symmetric bounds and noise. Increasing makes the observer more tolerant; increasing makes timing less precise. A shift in , a criterion change and extra noise can thus alter different aspects of a measured curve. This is an illustrative model, not a fitted account of the experiment below.
Figure 1.
Original propagation calculation and Gaussian simultaneity-decision examples; model parameters are illustrative, not measured temporal binding windows
. Details.
Temporal recalibration shows that the offset is adaptable. Waka Fujisaki and colleagues repeatedly paired tone pips and ring flashes at a fixed lag, then measured judgments across test lags. Initial adaptation lasted three minutes with brief re-adaptation before tests. Subjective simultaneity shifted toward the experienced lag; changes in an auditory-induced visual illusion gave evidence beyond a simple change in verbal simultaneity responses.[ref] The adjustment was partial, not exact cancellation of the exposure lag. Transfer constrains a purely task-specific response explanation, but does not locate one unique neural clock or exclude all decision contributions.
A complementary strategy is Bayesian causal inference in perception. In a toy Bayesian model, a measured discrepancy has Gaussian variance , while a common-event prior favors discrepancy zero with variance . Completing the square in the product of likelihood and prior proves the Gaussian temporal-discrepancy shrinkage relation
Thus noisy discrepancies can be pulled toward coincidence. This is a conditional model of one-event inference. A system must also allow separate causes: otherwise a loud unrelated sound would be bound to any nearby flash. Spatial agreement, identity, semantic fit, repeated correlation and action–outcome contingencies can affect that causal judgment. A large discrepancy can lead to segregation rather than endless recalibration.
Calibration and tolerance have costs. A wide acceptance range can stabilize a natural event while merging unrelated events; strong adaptation can help with a delayed device but temporarily bias judgments after it is removed. A convincing experiment therefore controls device timing, intensity and spatial cues, distinguishes order from simultaneity reports, and tests transfer and recovery. Perceptual unity is a flexible estimate supported by delayed and noisy evidence; it preserves useful event structure without guaranteeing veridical timing in every circumstance.
The central limit theorem suggests approximating by a normal variable with mean and variance . Thus
Since is integer-valued, the usual continuity correction replaces by :
These are approximation formulas rather than consequences of Chebyshev's bound; the standard deviation is , not .
The long-lived, electrically neutral kaon often traverses the particle detector without decaying into a reconstructable charged final state. Identifying it through a hadronic interaction or missing four-momentum is much less efficient than reconstructing the short-lived kaon through its nearby two-pion decay vertex. The efficiencies must be applied to the observed counts before taking their asymmetry:
Using the leading aligned-amplitude approximation from c,
The exact aligned formula instead gives , an immaterial correction here. Treating the observed counts as independent Poisson random variables and the efficiencies as known, propagation of uncertainty gives
The separate contributions to are and ; the small sample dominates. Linear propagation of uncertainty yields , although the nonlinear extraction and physical boundary make a symmetric confidence interval a rough summary. Unknown efficiency uncertainties, especially that of , would be additional systematic contributions.
The supplied CKM matrix gives , hence and . The asymmetry difference is only about statistical standard deviations, so the measurement is compatible. Without the phase assumption, these data measure to leading order and do not uniquely measure the Cabibbo angle.
A psychometric function specifies the probability of a measured response as a physical stimulus variable changes. The response might be “present,” “clockwise,” or correct in a two-alternative forced-choice task. These are different dependent variables. Understanding the function's shape is necessary to distinguish sensory changes from decision, attention and measurement effects. A reported threshold without its task, response definition and performance criterion is incomplete. For example, a visual experiment could use calibrated display visual contrasts, timed two-interval presentations, fixation control and button choices; the observer’s choice supplies the dependent variable.
A smooth curve need not mean that a receptor has a smoothly varying conscious threshold. Consider a noisy internal response , with “yes” reported when . Here is a decision criterion, not necessarily a biological detection limit. If is the noise cumulative distribution function,
With Gaussian noise of standard deviation , symmetry of the normal distribution gives a Gaussian psychometric function
The sigmoid is therefore a distribution tail swept by a changing mean response. At low means, few responses exceed the criterion; near it, many trials switch category; at high means, almost all do. The derivative is , so slope depends on both sensory gain and variability. Moving shifts the curve without necessarily improving sensitivity. If noise vanished and the mean crossed the criterion sharply, this simple model would approach a step rather than a graded transition.
There is no universal sigmoid in every stimulus coordinate. A nonlinear transduction function changes the curve, and a logarithmic intensity axis changes its appearance. As a counterexample, a toy detector responding whenever at least one event occurs in a Poisson distribution with mean has . This is concave on a linear positive-intensity axis. It follows immediately by subtracting the zero-event probability from one. It is a mathematical example, not evidence that human detection universally uses a one-event rule. Variation in attention or decision criterion across trials can broaden a measured curve even if the within-state sensory noise is unchanged.
Task conventions determine the endpoints. A useful psychometric function with guessing and lapses is
Here is the lower asymptote and the upper error asymptote. In balanced two-alternative forced-choice performance, is commonly fixed at . A yes/no response probability does not automatically have this lower bound. In this particular cumulative-Gaussian parameterization, ; thus gives 75% correct only for and . Other function families use different scale parameters. Nor is automatically the fraction of inattentive trials: if a fraction of trials involves random two-choice guessing, those trials contribute an upper error probability .
Figure 1.
Original theoretical noisy-decision and psychometric-curve examples; the right panel is a phenomenological family, not observed participant data
. Details.
The relation to signal detection theory is equally important. Under independent equal-variance Gaussian signal and noise samples, define sensitivity index . In an ideal two-interval comparison, the response difference has mean and variance , giving
This Gaussian forced-choice sensitivity relation follows by standardizing the Gaussian difference. It is a specified decision model, not an identity for every forced-choice experiment. In a yes/no task, hits and false alarms are both needed: under the corresponding equal-variance model, . A criterion change can increase both rates while leaving unchanged. The phenomenological fit above need not have parameters identical to these internal-response parameters.
These distinctions affect experiments. Felix A. Wichmann and N. Jeremy Hill used simulation and statistical modeling to show that unmodeled stimulus-independent errors can bias threshold and slope estimates, even when lapses are uncommon.[ref] Ignoring high-intensity errors can make a function appear shallower and falsely suggest worse sensory precision. A broad group-average curve can likewise reflect a mixture of individually sharp functions with different locations.
At predetermined stimulus levels, independent trials give counts . The parameter-dependent log-likelihood is . Fitting this binomial likelihood, checking goodness of fit and quantifying uncertainty are preferable to drawing a curve through noisy proportions. Easy trials help constrain lapses; levels near the transition constrain location and slope. Randomization, stable calibration and checks for learning or fatigue help distinguish a stationary sensory relation from drift. Adaptive procedures require inference appropriate to their sampling design.
Thus the function is both a description and a testable model. Its shape determines what a threshold means, how an experiment should sample, and which claims about sensory mechanisms are warranted.
Assuming independent Gaussian measurement errors with the quoted standard deviations, use the inverse-variance weighted mean:
Here and , giving
The consistency statistic is for four degrees of freedom, so there is no evident excess scatter requiring error rescaling. A shared systematic error would require a covariance matrix and would not be reduced by this averaging rule.
A small constant input can repeatedly produce the same analogue-to-digital converter code, so averaging those codes cannot reveal its position within a conversion interval. dither makes adjacent codes toggle; their relative frequencies then encode changes smaller than one conversion step. With adequate oversampling, low-pass filtering and averaging reduce the added, sufficiently uncorrelated noise while retaining the slow signal.
For independent sample errors, the averaged standard deviation scales as ; ideally four times as many samples gain about one effective bit. For example a constant input rounded to levels separated by would always return zero. Adding uniform dither on gives output with probability and zero otherwise: the expected average is , with error . The benefit concerns the filtered estimate, not a more accurate individual conversion. Excessive dither wastes range, and averaging does not correct calibration bias, correlated drift or arbitrary converter nonlinearity.[ref]
For independent Gaussian measurements with correctly specified standard deviations, model residuals give the chi-squared test statistic
For a linear fit of identifiable parameters under the null model, it has a chi-squared distribution with degrees of freedom. The upper-tail p-value tests unusually large residuals; a nonlinear fit generally uses this distribution only approximately. Correlated Gaussian errors instead require .
For counts, Pearson's chi-squared test uses , with calculated under the null hypothesis. For multinomial categories and fitted parameters the usual asymptotic degrees of freedom are ; the subtraction of one reflects the fixed total count. Sparse expected counts can invalidate this approximation, requiring a suitable exact or simulation-based test. Contingency-table independence gives degrees of freedom.
A small p-value indicates tension with the specified model and error assumptions; it is not the probability that the null is true. Excessive may reflect a wrong model, underestimated errors, correlations or outliers. An implausibly small value may suggest overestimated errors or hidden constraints. For an adequate fitted model is typically near one, with fluctuations, rather than required to equal one.
The central estimate is . propagation of uncertainty must include the occurrence of in both numerator and denominator. With angular derivatives measured per radian,
Interpreting the two quoted errors as independent standard deviations, convert them to radians before calculating
If the quoted errors are instead absolute bounds, the first-order worst-case estimate is . Correlated errors add to the variance. Treating and as independent would lose their shared measurement uncertainty.
In the high-resolution quantization approximation, a smooth input distribution varies little across one conversion interval, so rounding to the nearest level gives an approximately uniform quantization error . Therefore
The input must stay within the analogue-to-digital converter's range. A large standard deviation alone does not guarantee uniform quantization error: an input concentrated exactly on widely separated conversion levels has zero error. Truncation without a half-step correction also has a different mean and root mean square.
Bayes' theorem updates a parameter distribution using measured data:
The likelihood describes the measurement process, the prior distribution encodes information before the present data, and the denominator normalizes the posterior distribution. Include instrumental noise and known selection effects in the likelihood. nuisance parameters, such as an uncertain background, should be jointly inferred and marginalized, rather than silently fixed.
For independent readings with a Gaussian distribution with known common variance and prior distribution , completing the square gives
The posterior distribution is : prior distribution and data contribute in proportion to precision. A broad prior distribution recovers the usual estimate with standard deviation . Report a posterior distribution estimate and credible interval, specify the prior distribution, and examine sensitivity when the data are weak. A credible interval is a posterior distribution probability statement about the parameter conditional on the model; it is not defined by the repeated-sampling coverage of a confidence interval.
A chi-squared test compares discrepancies with their expected random size. For independent Gaussian observations with known standard deviations, a fitted model gives
Under a correct regular model with fitted parameters, its reference statistical degrees of freedom are usually . Correlated errors instead require with the correct covariance matrix. The reduced statistic has mean about one and standard deviation , so it should not be forced to equal exactly one.
For binned counts, Pearson's statistic is . With a fixed total and populated categories, fitting distribution parameters generally gives . The approximation needs adequately populated expected bins and the appropriate dependence assumptions. Sparse data may require a Poisson likelihood, exact calculation or simulated reference distribution rather than a casual chi-squared approximation.
The upper-tail probability measures how unusual the discrepancy would be under the null model; it is not the probability that the model is true. A large statistic can indicate a wrong model, underestimated noise or unmodelled systematic effects; an unusually small one can indicate overestimated errors or correlations. Residual plots and checks of the noise model are needed alongside the numerical test.
Interpret the quoted r.m.s. as the root-mean-square fluctuation about the mean, namely the standard deviation . Standardization gives . The supplied cumulative Gaussian curve gives , hence
The exact Gaussian value is . A raw r.m.s. temperature of could not coexist with a mean of ; the spread interpretation is required.
A Gaussian distribution with mean and standard deviation has probability density
Its symmetry, finite variance and simple behavior under linear combinations make it useful for repeated measurements. Roughly , and of observations lie within one, two and three true standard deviations of the mean. These are coverage probabilities for a specified distribution, not guarantees that every experimental error is normal.
Why it appears. The central limit theorem explains why many sums of small independent finite-variance contributions approach a Gaussian law after centering and scaling, provided no single contribution dominates. Detector noise, small mechanical disturbances and multiple electronic fluctuations can combine approximately this way. The theorem does not remove a common calibration bias, and it does not justify assuming normal tails for rare large disturbances or arbitrarily correlated errors. A large-mean Poisson distribution also approaches a Gaussian with variance equal to its mean, making normal approximations useful for high-count photon or particle data; low counts require the discrete model.
A classical particle at thermal equilibrium in a one-dimensional harmonic potential has a Gaussian position distribution directly from the Boltzmann distribution:
Thus a Gaussian law can arise from the energy dependence of equilibrium probability, rather than only from an informal appeal to many independent errors.
Precision of repeated measurements. For independent observations with the same mean and variance, the sample mean has variance . For Gaussian observations this mean is exactly Gaussian:
Consequently quadrupling the number of independent readings halves the random standard error. When the true variance is unknown, a small-sample confidence interval for a normal mean uses a Student t-distribution with statistical degrees of freedom; replacing that by a normal quantile can overstate precision. Shared systematic errors and correlations do not decrease by the same averaging rule.
Combining results and fitting models. Independent Gaussian estimates of one quantity, with known variances , combine through the inverse-variance weighted mean
For a model with independent zero-mean Gaussian noise, maximizing the likelihood is equivalent to minimizing the weighted least-squares statistic
For a correct linear model with known independent Gaussian errors and fitted parameters of full rank, the minimized statistic has a chi-squared distribution with degrees of freedom. Nonlinear fits only inherit that familiar approximation under additional regularity and adequate-data assumptions. Known correlated Gaussian errors require the full inverse covariance matrix in the quadratic form.
These properties support confidence intervals, consistency checks, calibration and error propagation. They also give practical diagnostics: inspect residuals for trends, changing spread, correlations or outliers before trusting a Gaussian fit. Gaussian models connect physical fluctuations to quantitative uncertainty, but their assumptions must be checked rather than inferred from the shape of a convenient bell curve.
A chi-squared test compares residuals with the random scatter predicted by a model. For independent measurements with known Gaussian standard deviations and predictions , form
If the mean model and error assumptions are correct, a full-rank linear fit of parameters gives a chi-squared distribution with statistical degrees of freedom. With a fully specified prediction and no fitted parameters, . Nonlinear fits usually make this an approximation. Independent external calibration is not the same as fitting a parameter to these test data.
Under the reference distribution, the mean statistic is and its variance is . The upper-tail p-value quantifies how unusual a discrepancy at least this large would be under the assumed model. A small value suggests model failure or incorrect uncertainties; it is not the probability that the model is true. An unusually small statistic can also reveal inflated uncertainties or correlations. A rejection threshold should be specified in advance.
For independent Poisson distribution counts, the model variance is its expected count . The Pearson chi-squared statistic is . The estimated-error alternative is common for large counts but can differ substantially when counts are small. In that regime the Poisson deviance or a direct likelihood calculation is preferable, with simulation if an accurately calibrated goodness-of-fit probability is required.
Write and use the Heisenberg uncertainty relation separately in the three Cartesian directions. Since a position coordinate is confined to an interval of length , Popoviciu's inequality on variances gives . Consequently , and the expectation value of the kinetic energy obeys
Thus an uncertainty-based lower estimate is . This confinement energy bound from position support is not sharp: the exact ground-state energy for a cubical infinite square well is . The Heisenberg uncertainty principle establishes the confinement scale ; identifying a box size with an approximate position standard deviation changes its numerical coefficient.
Ohm's law gives the mean electric current
For an estimate, assume the shot noise counting model: electron transfers are independent, so the number recorded in an interval has a Poisson distribution with expectation value and variance . One measurement has relative standard deviation . Averaging independent intervals reduces this to , so
about minutes in total.
This is a one-standard deviation, shot noise estimate. A resistance and mean voltage alone do not specify the actual current fluctuations: a real resistor also has Johnson-Nyquist noise, and electron-transfer correlations can change the shot noise. Its temperature, measurement bandwidth and noise model would be needed for an unconditional acquisition time.
For independent small uncertainties interpreted as standard deviations, propagation of uncertainty gives
The angular uncertainty is in radians. Since , the allowed angular contribution to the fractional uncertainty of the numerical aperture is only . Therefore
The uncertainty in the refractive index already contributes . If the quoted errors instead mean strict worst-case bounds, first-order absolute errors must be added rather than added in quadrature; that convention would require . The result uses the usual independent statistical-error convention.
A localized quantum particle is described by a wave packet: a coherent quantum superposition of momentum components, rather than a single delocalized plane wave. The Born rule assigns the position probability density . A useful normalized initial Gaussian wave packet is
Its mean position and momentum are and , and its standard deviations satisfy , . This unchirped initial state saturates the Heisenberg uncertainty relation.
For a free particle, the momentum-space Schrödinger equation multiplies each amplitude by . The inverse Fourier transform is a Gaussian integral, giving the free Gaussian wave packet evolution
One way to obtain this expression explicitly is to write the initial momentum amplitude as proportional to , multiply by the free evolution phase and complete the square in . Taking the modulus squared removes the position-dependent phase and leaves
The expectation value follows uniform classical motion while the density spreads. A narrower initial packet needs a wider momentum distribution and therefore spreads more rapidly. At long times its width grows linearly with ; its variance grows quadratically, unlike the linear variance growth of ordinary diffusion. The probability continuity equation preserves total probability despite the decreasing peak height. The mean momentum and the mean energy remain constant, since the momentum populations do not change.
The same result has a wave interpretation. For a free nonrelativistic particle , so the group velocity is , equal to at the packet center. The curvature makes different momentum components travel with different group velocities, producing dispersion. The phase velocity is for this energy convention and does not describe the motion of the envelope. The evolving quadratic spatial phase is essential to the spreading; a density snapshot alone does not specify the full quantum state.
A freely spreading Gaussian wave packet remains a pure Gaussian state but develops position-momentum correlations. Its symmetrized covariance is . Consequently
Equality in the Robertson-Schrödinger uncertainty relation survives at every time, whereas the simpler product equals only initially. A chirped Gaussian wave packet can already have . More generally, gives . A sufficiently negative initial covariance permits initial contraction before subsequent spreading.
Figure 1.
Free Gaussian spreading, constant coherent-state width and the breathing width of a squeezed state
. Details.
A potential energy changes this behavior. A quadratic potential preserves the Gaussian form because the evolution of position and momentum is linear. For a quantum harmonic oscillator, the Gaussian covariance in a harmonic potential is
A coherent state has , and zero centered covariance. Its width is constant, and its center follows . In oscillator notation its coherent state label rotates as , apart from the common ground-energy phase of the state vector. A squeezed coherent state generally has a width that breathes at . Thus being Gaussian does not imply either free spreading or constant width; the Hamiltonian operator and initial covariance determine the result.
For a general nonquadratic potential, different parts of the packet experience different nonlinear forces, and an initially Gaussian profile need not remain Gaussian. Ehrenfest theorem gives the exact relations and . Replacing the last mean by is exact for a quadratic potential and approximate for a packet narrow compared with the scale on which the force varies. This connects packet motion to the correspondence principle without treating a quantum state as a definite classical path. Scattering can split a packet into reflected and transmitted components, while confinement can produce oscillations and interference between discrete energy eigenstates. Their relative phases evolve even when their populations remain constant. Localization, spreading and coherent motion all follow from the same linear unitary time evolution.