Probability density function of the multivariate normal distribution

Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Multivariate normal distribution ▷ Probability density function

Theorem: Let $X$ be a random vector following a multivariate normal distribution:

\[\label{eq:mvn} X \sim \mathcal{N}(\mu, \Sigma) \; .\]

Then, the probability density function of $X$ is

\[\label{eq:mvn-pdf} f_X(x) = \frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \cdot \exp \left[ -\frac{1}{2} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) \right] \; .\]

Proof: This follows directly from the definition of the multivariate normal distribution.

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Sources:

original work

Metadata: ID: P34 | shortcut: mvn-pdf | author: JoramSoch | date: 2020-01-27, 15:23.