Gaussian Distribution

Antithetic sampling requires the perturbation distribution to be symmetric about the origin (such as a Gaussian distribution); otherwise the mean of paired perturbations is not zero and the centroid anchoring property no longer holds.

Adding two independent Gaussian distributions yields a Gaussian distribution, with the means added and the variances added.

Under short-range dependence, the paper proves that the iterates are asymptotically Gaussian at the √γ scale, with the asymptotic variance jointly determined by the population curvature and the long-run variance of the expectile score.

EL-F gives a tail bound with super-polynomial decay for the regret beyond the forced exploration budget, and on the two-armed Gaussian instance precisely characterizes the exponent of the linear regret deviation, while also establishing a matching lower bound for the class of strategies under an equivalent per-instance expected regret budget constraint.

Under the multivariate Gaussian distribution, mutual information is fully determined by the covariance matrix and can be expressed as the ratio of the determinant of the joint covariance matrix to the determinant of the marginal covariance matrices.

Prior research has shown that when each pairwise source–target variable joint distribution follows a Gaussian distribution, the BROJA optimization problem can be reduced to a finite-dimensional numerical optimization problem.

When ρ is a Gaussian distribution with covariance matrix Σ, the prior Gibbs distribution corresponding to the moment measure has covariance Σ⁻¹, causing the prior to be highly concentrated at zero when ρ is dispersed, making the Lipschitz constant of the transport map explode.

When the prediction error is assumed to follow a Gaussian distribution with zero mean and fixed variance, taking the negative logarithm of the likelihood function and expanding it yields a leading term that is exactly in the form of mean squared error.

Under the assumption of a Gaussian output distribution, minimizing negative log-likelihood (NLL) and minimizing mean squared error (MSE) are mathematically completely equivalent.

The Gaussian distribution PDF contains the term exp(-x²/2σ²), which underflows to zero when x is far from the mean or σ is extremely small; therefore, in practice the log-PDF is commonly computed in log space to ensure numerical stability.

The central limit theorem states that the standardized mean of independent and identically distributed random variables (with finite variance) approaches a standard normal distribution as the number of variables grows large enough, regardless of the type of the original distribution.

The tail of a Gaussian distribution decays at a rate of exp(-x²/2), and the probability of an event exceeding 3σ is approximately 0.27%.

The tail decay of a Gaussian distribution is exponential, so extreme values are almost entirely unexpected.

The ubiquity of the Gaussian distribution stems from the central limit theorem, which states that the sum of a large number of independent random variables tends toward a normal distribution.

The tails of the Gaussian distribution decay exponentially fast, causing models based on the Gaussian assumption to be sensitive to outliers.

In a univariate Gaussian distribution, the mean μ controls the location and the variance σ² controls the degree of dispersion.