The formula for the probability density function (PDF) of the multivariate normal (Gaussian) distribution for a \(d\)-dimensional random vector \(X = [X_1, X_2,…,X_d]^T\) with mean vector \(\mu = [\mu_1, \mu_2,…,\mu_d]^T\) and covariance \(\Sigma\) (a \(d \times d\) positive definite matrix) is given by:

\(f(x; \mu, \Sigma) = \frac{1}{(2\pi)^{d/2}|\Sigma|^{1/2}} \exp\left( -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu) \right)\)

**Where**:

\(x\) is a realization of the random vector \(X\)

\(|\Sigma|\) denotes the determinant of \(\Sigma\)

\(\Sigma^{-1}\) is the inverse of \(\Sigma\)

\(T\) denotes the transpose of a vector