The Multivariate Gaussian Distribution

A vector-valued random variable X = X 1 · · · X n T is said to have a multivariate normal (or Gaussian) distribution with mean µ ∈ R n and covariance matrix Σ ∈ S n ++ 1 if its probability density function 2 is given by p(x; µ, Σ) = 1 (2π) n/2 |Σ| 1/2 exp − 1 2 (x − µ) T Σ −1 (x − µ). We write this as X ∼ N (µ, Σ). In these notes, we describe multivariate Gaussians and some of their basic properties. 1 Relationship to univariate Gaussians Recall that the density function of a univariate normal (or Gaussian) distribution is given by p(x; µ, σ 2) = 1 √ 2πσ exp − 1 2σ 2 (x − µ) 2. Here, the argument of the exponential function, − 1 2σ 2 (x − µ) 2 , is a quadratic function of the variable x. Furthermore, the parabola points downwards, as the coefficient of the quadratic term is negative. The coefficient in front, 1 √ 2πσ , is a constant that does not depend on x; hence, we can think of it as simply a " normalization factor " used to ensure that 1 √ 2πσ ∞ −∞ exp − 1 2σ 2 (x − µ) 2 = 1.