10.3 Key definitions and Relationships

Let $𝑿=(X_1,\mathrm{\dots },X_n)$ have a multivariate normal (Gaussian) distribution with a variance matrix of $\mathrm{\Sigma }$ and an expectation vector of $𝝁$. Let $B$ be an $m\times n$ matrix.

1. 1.

   The density is

   $f(𝒙)={\displaystyle \frac{1}{{(2\pi )}^{n/2}{|det(\mathrm{\Sigma })|}^{1/2}}}\exp \left\{-{\displaystyle \frac{1}{2}}{(𝒙-𝝁)}^t{\mathrm{\Sigma }}^{-1}(𝒙-𝝁)\right\}.$

2. 2.

   Decomposition: $X=𝝁+A𝒁$, where $𝒁$ is a vector of $n$ independent standard normal random variables, and $A{A}^t=\mathrm{\Sigma }$.

3. 3.

   Linear transformation: $B𝑿$ has a multivariate normal distribution.

4. 4.

   Marginals: $X_i\sim 𝖭({\mu }_i,{\mathrm{\Sigma }}_{i,i})$, and all pairwise marginals (e.g. of ${(X_1,X_2)}^t$) are bivariate normal.

5. 5.

   Conditionals: the conditionals of a bivariate Gaussian are Gaussian: e.g. $X_1|X_2=x_2$ is Gaussian. This is true more generally for a multivariate Gaussian: $(X_1,X_2)|(X_3=x_3,X_5=x_5,X_7=x_7)$ is bivariate Gaussian.