Joint distribution and density
Definition taught in Covariance and the multivariate normal, part of Statistical learning
Joint distribution and density
| Type | Definition |
| Taught in | Covariance and the multivariate normal |
| Part | Statistical learning |
The joint distribution function of \mathbf{X} is
F(\mathbf{x})=\mathbb{P}(\mathbf{X}\le\mathbf{x})=\mathbb{P}(X_1\le x_1,\ldots,X_p\le x_p), \qquad \mathbf{x}\in\mathbb{R}^p.
If F is absolutely continuous, \mathbf{X} has a joint density
f(\mathbf{x})=\frac{\partial^p F(\mathbf{x})}{\partial x_1\cdots\partial x_p}
almost everywhere, and
F(\mathbf{x})=\int_{-\infty}^{x_p}\cdots\int_{-\infty}^{x_1} f(u_1,\ldots,u_p)\,\,\mathrm{d} u_1\cdots\,\mathrm{d} u_p.
Used in
- Taught in Covariance and the multivariate normal, where the derivation and examples are.
- All 8 questions on Covariance and the multivariate normal
See also
Joint probability density · Cumulative distribution function · Mean vector and covariance matrix · Correlation · Multivariate normal distribution · Squared Mahalanobis distance · Linear images of a normal vector · Conditional distribution of a multivariate normal