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Gaussian noise is an additive random error with a normal distribution, often assumed independent across observations and centered at zero.

A Gaussian practical-null mixture compares a narrow centered normal distribution prior with a wider centered normal distribution prior. Neither component is a point mass. With observed mean $y\mid \beta \sim N(\beta ,1/n)$ and prior precisions $q_0>q_1>0$, its model predictive variances are $V_i=1/n+1/q_i$, and the Bayes factor isThe overall posterior density is a Bayesian model averaging mixture with component means $ny/(n+q_i)$. A wide-prior penalty can favor the practical null near zero; its dominance depends quantitatively on both prior scales and the observation, not merely on the observation being of order $n^{-1/2}$.

Gaussian process classification links latent variables from a Gaussian process to class probabilities. A binary probit regression likelihood is ${\prod }_i\Phi ((2y_i-1)f_i)$ under conditional independence. The posterior distribution of the latent values is not a normal distribution.

A random element $G$ of a Hilbert space is Gaussian when every continuous linear functional of $G$ has a normal distribution. Its mean and covariance operator determine its distribution.

A Gaussian random variable is a random variable whose probability distribution is a normal distribution.

A random variable $X$ in a real separable Banach space $B$ is Gaussian when $\ell (X)$ has a normal distribution for every continuous linear functional $\ell \in B^{\ast }$. It is centered when all these expected values vanish. Its probability law is a Gaussian measure. Degenerate normal distributions are allowed.

For a regression function $g_{\theta }$, let $h_{\theta }={\partial }_{\theta }{g}_{\theta }$. With independent centered normal distribution error of known variance ${\sigma }^2$, the parametric score function is $h_{\theta }(X)\epsilon /{\sigma }^2$. It has zero conditional expectation given the covariate, so unknown covariate-density nuisance directions do not remove any of its Fisher information.

With the convention $d_{\mathrm{Hell}}^2=\frac{1}{2}\int (\sqrt{p}-\sqrt{q}{)}^2$, the Hellinger distance between two normal distributions with ${\sigma }_1,{\sigma }_2>0$ isTo prove it, integrate the geometric mean of their probability density functions. Completing the square makes that geometric mean the density of a normal distribution times the displayed prefactor and exponential. For example, writing $S={\sigma }_1^2+{\sigma }_2^2$, that normal distribution has mean $({\theta }_1{\sigma }_2^2+{\theta }_2{\sigma }_1^2)/S$ and variance $2{\sigma }_1^2{\sigma }_2^2/S$. The identity $d_{\mathrm{Hell}}^2=1-\int \sqrt{pq}$ then proves the formula.

For equal unit variances, this becomes $1-e^{-({\theta }_1-{\theta }_2{)}^2/8}$. More generally, put $B=\sqrt{2{\sigma }_1{\sigma }_2/S}\le 1$ and $x=({\theta }_1-{\theta }_2{)}^2/(4S)$. Then $1-B{e}^{-x}=(1-B)+B(1-e^{-x})$, $1-B=({\sigma }_1-{\sigma }_2{)}^2/[S(1+B)]$, and $1-e^{-x}\le x$, proving

The ordinary time integral of standard Brownian motion is a centered Gaussian process with continuously differentiable paths and derivative $B_t$. For $0\le s\le t$, its covariance is $\mathbb{E}(I_s{I}_t)=s^2(3t-s)/6$, obtained by integrating the Brownian covariance $\min (u,v)$. Consequently $I_t$ has normal distribution $N(0,t^3/3)$. This time integral is distinct from the Itô integral with respect to Brownian motion.

For finite fourth moment and positive variance, kurtosis is $\mathbb{E}[(X-\mathbb{E}X{)}^4]/\mathrm{Var}(X{)}^2$. The normal distribution has kurtosis $3$. A random scale can increase kurtosis even when the conditional distribution is normal.

Near a stable deterministic equilibrium, replace the drift by its linearization and the diffusion coefficient by its equilibrium value. The resulting Ornstein-Uhlenbeck approximation has a stationary normal distribution.

The times of local maxima of a Brownian motion path are dense in the half-line almost surely, and every such local maximum is a strict local maximum. Density follows from nowhere monotonicity of Brownian motion and the extreme value theorem. Strictness follows by checking that maxima over separated rational compact time intervals are unequal; the increment across their gap supplies an independent nondegenerate normal distribution.

A logistic-normal count model takes $Y_i\mid {\theta }_i\sim \mathrm{Bin}(n_i,{\mathrm{logit}}^{-1}{\theta }_i)$ and ${\theta }_i=x_i^T\beta +\lambda Z_i+{\eta }_i$, where $Z$ is a stationary unit-variance autoregressive process of order one with coefficient $a$ and the independent ${\eta }_i$ have normal distribution with variance $v$. The marginal latent variance is ${\lambda }^2+v$, and its off-diagonal covariance is ${\lambda }^2{a}^{|i-j|}$. Random success probabilities produce overdispersion relative to the binomial distribution.

A positive random variable is log-normal when its logarithm has a normal distribution. If $Z\sim N(0,1)$ and $X=e^Z$, then $\mathbb{E}[X^n]=e^{n^2/2}$.

Here $Z$ has the standard normal distribution. Applied to the generating function $e^{sx-s^2/2}$, the Gaussian moment-generating function replaces $s$ by $e^{-t}s$, proving the Hermite eigenvalue identity. Positivity and preservation of Gaussian measure then give the $L^2$ contraction and identify the full Ornstein-Uhlenbeck semigroup by polynomial density.

A normal approximation replaces a sampling distribution by a normal distribution, often through a central limit theorem. In a Wald test, approximate statistical power is determined by the standardized difference under the alternative. This yields Neyman allocation for fixed treatment-specific variances.