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When a person's latent slope has normal distribution $N({\beta }_1,{\tau }_1^2)$, its $u$th quantile is ${\beta }_1+{\tau }_1{\Phi }^{-1}(u)$. Multiply by an elapsed time to obtain the corresponding latent change quantile. This describes population variation between people; it is distinct from a confidence interval for the mean slope and from a prediction interval that includes measurement error.

Independent uniform distribution draws $U_1,U_2$ on $(0,1)$ produce independent standard normal distribution draws $X_1=\sqrt{-2\log U_1}\cos (2\pi U_2)$ and $X_2=\sqrt{-2\log U_1}\sin (2\pi U_2)$. The proof transforms an independent uniform angle and radial density $r{e}^{-r^2/2}$ through the polar-coordinate Jacobian.

At integer times, $B_n/\sqrt{n}$ has the same standard normal distribution for every $n$. The event that its limit superior exceeds a fixed finite level is a tail event of the independent unit-time Brownian increments, since changing any finite initial segment contributes only a term tending to zero. The Kolmogorov zero-one law applies. Its probability is positive because each fixed-time exceedance has the same positive probability, giving a positive probability of infinitely many exceedances by the decreasing union-of-tail-events argument. Thus its probability is one. Intersecting over integer levels proves the stated divergence, without the lower iterated-logarithm bound.

An increment of a Brownian motion between deterministic times $s<t$ is $W_t-W_s$. It has normal distribution $N(0,t-s)$ and is independent of the filtration up to time $s$. Increments on disjoint time intervals are independent. Their covariance is the length of overlap of the corresponding intervals; this also follows from the Brownian covariance kernel.

For every fixed mesh $1/k$, independent increments with the normal distribution imply increments of absolute value greater than one occur infinitely often almost surely, by the second Borel-Cantelli lemma. A countable intersection over $k$ contradicts uniform continuity on $[0,\infty )$.

For $\mu =\mathcal{N}(0,\Sigma )$ with injective covariance operator of a Gaussian measure, its translated law ${\mu }_h=\mathcal{N}(h,\Sigma )$ is an equivalent probability measure to $\mu$ if and only if $h\in E_{\mu }$, the Cameron-Martin space of a Gaussian measure. For such $h$,The series has mean-square convergence and converges almost surely under $\mu$, with normal distribution $\mathcal{N}(0,\Vert h{\Vert }_{E_{\mu }}^2)$. Its exponential density has expected value one by the moment-generating function of a normal distribution. Finite-dimensional projections give the formula by ratios of multivariate normal densities; convergence of these likelihood ratios gives the infinite-dimensional result. The expression $⟨h,x{⟩}_{E_{\mu }}$ is only formal when the sample is outside $E_{\mu }$. For $h\notin E_{\mu }$, translation gives mutually singular measures.

The delta of the call price in an arithmetic stock model with interest is the normal distribution function at its standardized discounted moneyness. It lies strictly between zero and one before maturity. Its maturity limit is the call payoff derivative away from the strike, an exceptional event of probability zero under the Gaussian pricing law.

A deviance residual is the signed square root of an observation's contribution $d_i$ to fitted-model deviance. For a Poisson regression, $d_i=2[y_i\log (y_i/{\hat{\mu }}_i)-(y_i-{\hat{\mu }}_i)]$, with $0\log 0=0$. Its squared sum is the Poisson deviance. Leverage and estimated dispersion can further standardize these residuals; a normal quantile-quantile plot is a diagnostic approximation, not a requirement that count errors have a normal distribution.

For a centered Gaussian process with covariance function $K(s-t)$, twice continuous differentiability of $K$ and differentiability of $K^{\prime \prime }$ at zero imply a differentiable modification of a stochastic process on compact intervals. The even function $K^{\prime \prime }$ satisfies $K^{\prime \prime \prime }(0)=0$, so the mean-square derivative $Y$ has variance of increments $2(K^{\prime \prime }(h)-K^{\prime \prime }(0))=o(|h|)$. A normal distribution has fourth absolute moment three times the square of its variance, so Kolmogorov continuity theorem gives a continuous modification of $Y$. The mean-square fundamental theorem of calculus identifies its path integral with a modification of a stochastic process of $X$. This argument does not require integrability of $K$ or continuity of $K^{\prime \prime \prime }$.

An Edgeworth expansion approximates a probability density function by a normal distribution multiplied by Hermite polynomials encoding higher cumulants. For a centered variable of variance ${\sigma }^2$ and third cumulant ${\kappa }_3$,The correction follows from inverse-transforming the cubic term of the cumulant-generating function. It is an asymptotic approximation near the bulk of the distribution; truncation can give a negative result in far tails.

With unknown independent covariate and centered error distributions, take the regular mean-preserving error tangent space, let $h={\partial }_{\theta }{g}_{\theta }$, $\rho =-f^{\prime }/f$ and ${\tau }^2=E_f{\epsilon }^2>0$. Under $E_f\rho =0$, $E_f(\epsilon \rho )=1$ and finite second moments, orthogonal projection removes $(E_{v}h)(\rho -\epsilon /{\tau }^2)$ from $h\rho$. Thus the efficient score and efficient information are $\tilde{\ell }=(h-E_{v}h)\rho +(E_{v}h)\epsilon /{\tau }^2$ and $\tilde{I}={\mathrm{Var}}_v(h)E_f{\rho }^2+(E_{v}h{)}^2/{\tau }^2$. The reduction to $h\epsilon /{\tau }^2$ is valid for a normal distribution of errors or for constant $h$, but need not hold otherwise. Centered covariates and logistic distribution errors give the counterexample $\tilde{\ell }=X\tanh (\epsilon /2)$.

In the expectation-maximization algorithm for a stationary autoregressive process of order one, a single missing interior value has a normal distribution conditional on its neighbors. The E-step uses both its conditional mean and its conditional second moment, not just mean imputation. The M-step maximizes the expected stationary complete-data log-likelihood, including the initial-observation density and the conditional-variance contribution to the two adjacent innovation squares.

For every $\theta \in \mathbb{R}$ and standard Brownian motion $B$,is a positive martingale. Applying Itô formula cancels the drift, and its expectation is one by the moment-generating function of the normal distribution.

The fixed-design nonparametric regression model observes $Y_i=m(x_i)+\sigma {ϵ}_i$ at deterministic design points $x_i$. For independent and identically distributed random variables ${ϵ}_i$ with the standard normal distribution, comparing two mean functions has Kullback-Leibler divergence ${\sum }_i(m_1(x_i)-m_0(x_i){)}^2/(2{\sigma }^2)$.

For a stationary unit-variance normal distribution autoregression with coefficient $a\in (-1,1)$, its vector of length $T\ge 2$ has precision matrix $J$ that is tridiagonal: the two endpoint diagonal entries are $1/(1-a^2)$, interior entries are $(1+a^2)/(1-a^2)$, and adjacent off-diagonal entries are $-a/(1-a^2)$. An independent observation $r=\lambda Z+\eta$, with covariance $vI$, updates precision and information vector toThe conditional law is multivariate normal distribution $N(Q^{-1}h,Q^{-1})$.

For a normal distribution $N(\lambda ,{\sigma }^2)$, the logarithm of its moment-generating function is $s\lambda +s^2{\sigma }^2/2$. Dividing by $s>0$ gives its effective bandwidth. Independent demands add these bandwidths. For total mean $m$ and positive total variance $v$, optimizing a Chernoff bound with target $e^{-\gamma }$, $\gamma >0$, yields the sufficient margin $C\ge m+\sqrt{2\gamma v}$.

A Gaussian likelihood is a likelihood function obtained by modeling the observations conditionally with a normal distribution.

A Gaussian linear mixed model combines fixed effects, random effects with a normal distribution, and independent Gaussian noise. For $b\sim N(0,D)$ and $\epsilon \sim N(0,R)$ independently, its marginal distribution is $Y\sim N(X\beta ,V)$ with $V=ZD{Z}^T+R$. The random effects induce dependence among observations sharing their design columns. A random-intercept linear mixed model and a random-slope linear mixed model are special cases.

For $n$ independent observations with a normal distribution $N(\mu ,{\sigma }^2)$, $\sigma >0$ fixed, every estimator has ${\sup }_{\mu }{\mathbb{E}}_{\mu }|T-\mu |\ge \sigma /(4\sqrt{2n})$. Compare ${\mu }_0=0$ and ${\mu }_1=\sigma /\sqrt{2n}$: the joint Kullback-Leibler divergence is $1/4$, so the total variation–Hellinger–relative entropy inequality bounds total variation distance by $1/2$. Apply the Le Cam lower bound under absolute-error loss. Degenerate zero-noise laws do not satisfy a positive lower bound.

A mean-zero normal distribution prior of variance $s^2$ on the log baseline multiplies the multinomial kernel by the displayed correction, up to a constant independent of coefficients. The factor is not constant in $S$, so the posterior equivalence is approximate. Dominated convergence makes it tend to one as $s^2$ tends to infinity.