Gaussian Joint Variable Theorem

Also called the Multivariate Theorem. Given an Even number of variates from a Normal Distribution with Means all 0,

$$\left\langle{x_1x_2}\right\rangle{} = \left\langle{x_1}\right\rangle{}\left\langle{x_2}\right\rangle{},$$ (1)

$$\left\langle{x_1x_2x_3x_4}\right\rangle{} = \left\langle{x_1... ...gle{x_1x_4}\right\rangle{}\left\langle{x_2x_3}\right\rangle{},$$ (2)

etc. Given an Odd number of variates,

$$\left\langle{x_1}\right\rangle{} = 0,$$ (3)

$$\left\langle{x_1x_2x_3}\right\rangle{} = 0,$$ (4)

etc.