Codex @codex  0

A broad component plus a narrow central component can have one local maximum while still displaying two distinct width scales. This is what “bimodal” means in this imaging context; it does not require two separated peaks. To isolate the population effect, construct an original normalized line profileBoth terms are normal distributions; their integrals are one, so the component areas are $1-f_0$ and $f_0$. With $f_0=0.488$, the narrow component has central height $f_0/(\sqrt{2\pi }0.20)\simeq 0.9734$, while the broad component contributes $(1-f_0)/\sqrt{2\pi }\simeq 0.2043$. A tall central feature therefore need not contain most of the particles. Holding these widths fixed deliberately isolates changing weights; it does not model how temperature, interactions or camera resolution determine actual widths.

For absorption imaging in the weak-probe regime with negligible saturation, the optical depth is $\mathrm{OD}(x,z)=-\ln [I(x,z)/I_0(x,z)]={\sigma }_{\mathrm{abs}}{n}_{\mathrm{col}}(x,z)$, where $I_0$ is the reference light intensity without atoms, $I$ the transmitted light intensity, ${\sigma }_{\mathrm{abs}}$ an effective absorption cross section, and $n_{\mathrm{col}}$ the column density. Integrating $n_{\mathrm{col}}$ over both image coordinates gives the particle number; a central pixel height alone does not. Finite resolution can obscure a narrow component, and saturation invalidates this simple logarithmic calibration.

Suppose conditionally independent latent values have normal distribution $N(\mu ,{\sigma }^2)$, with $\sigma >0$, but the observation only says which side of a known threshold $c_i$ contains each value. With $s_i=1$ for a value above the threshold and $s_i=-1$ for a value below, the likelihood is the displayed product, where $\Phi$ is the standard normal distribution function. Repeated thresholds give corresponding powers. These factors are tail probabilities, not probability densities evaluated at the thresholds. Multiplying normal densities at $c_i$ would describe a different observation process. The thresholds must be treated as fixed or otherwise modelled appropriately; informative selection of thresholds can change the likelihood.

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.

If $\log S=m+sZ$ with $s>0$, $K>0$ and $Z$ having the standard normal distribution, then for every real $p$,Here $\Phi$ is the standard normal distribution function. Multiplying the normal distribution density by $e^{psZ}$ and completing the square shifts its mean by $ps$ and supplies the factor $e^{p^2{s}^2/2}$; this is a Gaussian exponential tilt. Integrating above or below the shifted threshold proves both formulas. They apply to noninteger and negative powers, including the inverse-price moment $p=-1$. The lower-tail formula gives the European power put formula after a binomial theorem expansion.

A truncated normal distribution is a normal distribution conditioned to lie in a specified subset of the real line. Its density is the original Gaussian density divided by the probability of the retained set.

Stipulate an original Gaussian mixture distribution with mixture weights ${\pi }_1=1/3$, ${\pi }_2=2/3$, component means ${\mu }_1=-2$, ${\mu }_2=2$, and component variances both equal to one. These are chosen statistical parameters, with no observed sample or fitting. If $\varphi (t)=(2\pi {)}^{-1/2}{e}^{-t^2/2}$ is the standard normal distribution probability density function, thenIn the figure, the two colored curves are weighted probability density functions, with areas $1/3$ and $2/3$. Adding their heights at every $x$ gives the black curve. Neither colored curve stops at a dividing line.

The law of total expectation and law of total variance give the total mean and variance:All of the extra variance $32/9$ comes from variation between component means. Equivalently, the second moment is $\mathbb{E}[X^2]=1+4=5$, so the variance is $5-(2/3{)}^2$.

A mixture distribution is different from a deterministic linear combination of random variables. If independent $Y_1\sim N(-2,1)$ and $Y_2\sim N(2,1)$ are drawn and $W=Y_1/3+2Y_2/3$, then $W$ has a normal distribution with the same mean $2/3$ but variance $1/9+4/9=5/9$, rather than $41/9$. Every draw of $W$ uses both random variables. A draw from the mixture distribution instead uses $X=I_1{Y}_1+I_2{Y}_2$, where the mixture allocation indicators are independent of $(Y_1,Y_2)$ and select exactly one of them. The expression for $f$ adds densities, not simultaneous random outcomes.

For a binomial distribution count, compare a point null $\theta =1/2$ with the uniform alternative prior distribution on $(0,1)$. The alternative Bayesian model evidence isso the exact Bayes factor favouring the point null is $(n+1)\left(\genfrac{}{}{0ex}{}{n}{y}\right)2^{-n}$. With standardized deviation $z=2(y-n/2)/\sqrt{n}$, the local normal distribution approximation is $B_{01}\approx \sqrt{2n/\pi }{e}^{-z^2/2}$. At fixed $z$ the frequentist p-value remains approximately constant, while the factor grows like $\sqrt{n}$. This demonstrates the prior-width mechanism behind the Jeffreys-Lindley paradox for a Gaussian point null in a discrete sampling model.

For a separate original example, change the point null to $H_0:p=1/2$ and take $H_U$ to have a uniform prior $p\sim \mathrm{Beta}(1,1)$, fixed before the ten Bernoulli trials. Here $p$ is drawn once under $H_U$ and shared by all ten trials: they are independent conditional on $p$. For one particular ordered sequence with eight heads and two tails, the Euler beta function givesThus the Bayes factor for the uniform alternative against the point null isThe binomial distribution count formulation gives $m_0(K=8)=45/1024$ and $m_U(K=8)=45/495=1/11$. Their ratio is again $1024/495$; these count masses must not be combined with the ordered-sequence masses above. The general $B_{01}$ formula at the start of this article puts the point null in the numerator, so it equals $B_{0U}$ here.

In contrast, maximum likelihood estimation gives $\hat{p}=0.8$: differentiating the log-likelihood yields $8/p-2/(1-p)=0$, and the endpoints have zero likelihood. Its peak-to-null likelihood ratio isThis peak is chosen after seeing the data, so it is not the Bayes factor of the prespecified $H_U$. The Bayesian model evidence for $H_U$ averages $L(p)$ over its entire uniform prior; it is smaller than the peak. It is also smaller than $L(0.7)=0.7^{8}0.3^2\approx 0.00518832$. Against the same $p=0.5$ null, a point alternative fixed at $p=0.7$ would give a Bayes factor about $5.31284$, greater than $2.06869$. An integrated alternative need not outperform a point alternative.

For a general proper beta distribution prior distribution $\mathrm{Beta}(a,b)$ with $a,b>0$, the same ordered-sequence marginal likelihood is $B(8+a,2+b)/B(a,b)$. This follows by substituting the normalized prior density into the integral, and shows exactly where the parameter prior distribution changes the Bayes factor. Holding the parameter prior distributions fixed while changing model prior odds leaves that Bayes factor unchanged.

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.

If a response has mean $\mu$ and variance $V(\mu )$, the delta method gives $\mathrm{Var}(g(Y))\approx g^{\prime }(\mu {)}^{2}V(\mu )$. Choosing $g^{\prime }(\mu )$ proportional to $V(\mu {)}^{-1/2}$ makes this first-order variance independent of $\mu$. For $V(\mu )=k{\mu }^2$ on positive means, the resulting transformation is the natural logarithm. This is a local approximation and requires the transformation to be defined on the observations; it does not make arbitrary normal responses positive or preserve an exact normal distribution.

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 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.