Multivariate normal distribution
Definition taught in Covariance and the multivariate normal, part of Statistical learning
Multivariate normal distribution
| Type | Definition |
| Taught in | Covariance and the multivariate normal |
| Part | Statistical learning |
| Questions using it | 5 |
Let \boldsymbol{\mu}\in\mathbb{R}^p and let \Sigma be a symmetric positive definite p\times p matrix. A random vector \mathbf{X} has the p-variate normal distribution \operatorname{N}_p(\boldsymbol{\mu},\Sigma) if it has density
f(\mathbf{x})=\frac{1}{(2\pi)^{p/2}|\Sigma|^{1/2}} \exp\Bigl\{-\frac12(\mathbf{x}-\boldsymbol{\mu})^\top\Sigma^{-1}(\mathbf{x}-\boldsymbol{\mu})\Bigr\}, \qquad\mathbf{x}\in\mathbb{R}^p.
The mean vector is \boldsymbol{\mu} and the covariance matrix is \Sigma. The subscript p is dropped when the dimension is clear.
Used in
- Taught in Covariance and the multivariate normal, where the derivation and examples are.
- Also used in Estimation and inference for the multivariate normal · Mixtures and the EM algorithm · Hierarchical clustering and kernel density estimation · Statistical classifiers and discriminant analysis · High-dimensional analysis · Graphical models
- Questions that use it (5) — Normal marginals without joint normality · The bivariate conditional (answer) · Conditional normal prediction · Normal conditioning and whitening (answer) · A transformation is not a distributional assumption (answer)
- All 8 questions on Covariance and the multivariate normal
See also
Multivariate normal distribution · Positive definite matrix · Normal distribution · Mean vector and covariance matrix · Iris data · Precision matrix · Joint distribution and density · Correlation