Multivariate Gaussian distribution

A multivariate Gaussian distribution is the law of a Random vector $X\in {\mathbb{R}}^d$ such that every linear combination $a^{T}X$ has a Normal distribution, allowing a constant as a zero-variance Normal distribution. Its Mean vector $\mu$ and positive-semidefinite covariance matrix $\Sigma$ determine its Characteristic function:If $\Sigma$ is positive definite, the law has the normalized Probability density functionA singular covariance matrix is also permitted, but this density formula is not: the law is concentrated on the affine subspace $\mu +\mathrm{range}\Sigma$. For example, $(Z,Z)$ with $Z$ standard normal has covariance $\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$ and lives on the diagonal, so it has no two-dimensional Lebesgue density.

A construction valid in both cases is $X=\mu +AZ$, where $Z$ has independent standard normal coordinates and $A{A}^T=\Sigma$. Its Characteristic function follows by multiplying the one-dimensional characteristic functions. For $\mathrm{rank}\Sigma =r$, one can use a matrix $A$ with $r$ columns.

Normal marginals alone do not imply a multivariate Gaussian law. Let $Z$ be standard normal and let $S$ be an independent Rademacher random variable. Both $Z$ and $SZ$ are standard normal, but their sum is zero with probability $1/2$ and otherwise equals $2Z$, so that nonconstant sum is not normal. Joint Gaussianity is the stronger condition needed for Gaussian moment and linear-transformation rules.

For independent standard normal random variables $X,Y$, put $X=R\cos \Theta$, $Y=R\sin \Theta$, with $R\ge 0$ and $0\le \Theta <2\pi$. The polar coordinates Jacobian determinant is $r$, so the joint probability density is $r{e}^{-r^2/2}/(2\pi )$ on $r>0$. It factorizes into a Rayleigh distribution density for $R$ and a continuous uniform distribution density for $\Theta$. Consequently the radius and angle are independent. The change of variables for probability densities $S=R^2$ gives $f_S(s)=e^{-s/2}/2$ for $s>0$, the exponential distribution of rate $1/2$, also the chi-squared distribution with two degrees of freedom. Measurable transformation preserves independence, giving $\Theta \perp S$. The angle at $R=0$ is arbitrary on an event of probability zero.

Thus a sector with angular length $\ell$ has probability $\ell /(2\pi )$, independently of any radial restriction. For instance $Y>|X|$ occupies a sector of angle $\pi /2$ and has probability $1/4$.

For centered jointly Gaussian random variables, an odd product has zero expectation, and an even product's expectation is the sum over all pairings of the products of the pair covariances. In particular,For a proof, the moment-generating function is $\exp (t^T\Sigma t/2)$. Differentiating once in each variable at zero leaves precisely products of covariance entries indexed by complete pairings. No independence assumption between the coordinates is needed.

For a centered scalar normal random variable of variance $v$, the Isserlis theorem sums complete pairings of $2n$ factors. The first factor has $2n-1$ partners, then $2n-3$ choices remain for the first unpaired factor, giving $(2n-1)!!=(2n)!/(2^{n}n!)$. Odd moments vanish. Summing the moment-generating function gives $\exp (v{s}^2/2)$. This is a scalar formula, not a formula for the length of a vector.

Every linear combination $a^{T}X$ of the coordinates of a jointly normal random vector is a normal random variable. Its mean is $a^T\mu$ and its variance is $a^T\Sigma a$.