Multivariate normal distribution

If $X,Y$ are independent and identically distributed random variables with a centered multivariate normal distribution and covariance matrix $\Sigma$, then $(U_{\theta },V_{\theta })$ has the same probability law as $(X,Y)$. Its two marginal covariance matrices are $\Sigma$ and its cross-covariance is zero. The pair has a multivariate normal distribution, so the two components are independent. This remains valid when $\Sigma$ is singular.

A Gaussian random vector has a multivariate normal distribution; equivalently, every real linear combination of its coordinates is a normal random variable, allowing zero variance. Its expected value vector and covariance matrix determine its law. Subvectors are independent when their cross-covariances vanish, a feature particular to joint Gaussianity.

For centered, unit-variance jointly Gaussian random variables with correlation coefficient $\rho$, the displayed identity holds. Realize the pair as projections onto unit vectors of angle $\theta =\arccos \rho$. Rotational symmetry of the two-dimensional Gaussian random vector direction makes sign disagreement have probability $\theta /\pi$. The expectation is therefore $1-2\theta /\pi =(2/\pi )\arcsin \rho$. Perfectly correlated and anticorrelated endpoints follow directly; the sign at zero does not affect the result.

A Gaussian random vector with independent normal coordinates of expectation zero and variance one. Its distribution is invariant under orthogonal transformations. In particular, its direction in a fixed two-dimensional plane is uniform, which underlies the Gaussian sign-correlation identity.

For a positive-definite matrix $\Sigma$, the distribution $N_p(\mu ,\Sigma )$ has density

If $X\sim N(\mu ,\Sigma )$ and $A$ and $c$ are deterministic, thenIn particular, $a^{T}X$ is normal with mean $a^T\mu$ and variance $a^T\Sigma a$.

Two subvectors of a jointly multivariate normal distribution are independent exactly when their cross-covariance matrix is zero.

For a jointly Gaussian vector, conditional independence is equivalent to vanishing conditional cross-covariance. Equivalently, two coordinates are conditionally independent given all the others exactly when their entry in the precision matrix is zero.

For means ${\mu }_i$, standard deviations ${\sigma }_i$, and correlation $\rho$, the standardized quadratic form in the density is

A bivariate standard normal distribution is a bivariate normal distribution whose two marginal means are zero and marginal variances are one; its remaining parameter is their correlation.

For independent $Z_i\sim N_p(0,\Sigma )$ with known zero mean,

Writing $a={\Sigma }_{11}$, $c={\Sigma }_{12}$, and $b={\Sigma }_{22}$, the vector $({\hat{\Sigma }}_{11},{\hat{\Sigma }}_{12},{\hat{\Sigma }}_{22})$ has asymptotic covariance

Conditionally on $X_2=x_2$,

Any jointly normal random variables with zero covariance are independent.

Orthogonal projections of an isotropic Gaussian vector onto orthogonal subspaces are independent.